High-dimensional chaos in dissipative and driven dynamical systems
Z.E. Musielak and D.E. Musielak
Department of Physics, The University of Texas at Arlington, Arlington, TX 76019, USA∗
Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior
of these systems is significantly different as compared with the behavior of systems with less than
two degrees of freedom. These findings motivated us to carry out a survey of research focusing
on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos, and the
persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews
recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and
Van der Pol oscillators, modified canonical Chua’s circuits, and other dynamical systems and maps,
and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and
synchronization developed for high-dimensional dynamical systems are also reviewed.
I.
INTRODUCTION
Dynamical systems are self-contained entities that
evolve in time independently of the real world that surrounds them. Typically, these systems are classified as
conservative if their state-space volume is conserved, and
as non-conservative otherwise. In Newton’s mechanics,
dynamical systems are called deterministic because their
present and future states are fully determined by their
initial conditions. The full predictability of the system’s
behavior in time is the basis for the classical mechanical description of dynamical systems. Poincaré (1890,
1892) was the first to demonstrate that the sensitivity of
the 3-body problem to initial conditions was so strong
that no reasonable prediction in time could be possible.
The sensitivity of a dynamical system to arbitrarily small
changes in its initial conditions and the resulting lack of
predictability of its future states have become the definition of deterministic chaos.
A century of analyzing conservative and nonconservative dynamical systems has led to the striking
discovery that a simple, but nonlinear, deterministic system with only a few degrees of freedom can generate
chaotic behavior. This chaotic behavior is the fundamental property of the system and it cannot be removed by
gathering more information about the system. Clearly,
this is a paradox because the chaotic behavior is generated by the fixed deterministic rules that do not themselves involve any element of chance. Since the system is
deterministic, its future is (in principle) completely determined by the past. In practice, however, very small
uncertainties in the system’s initial conditions are quickly
amplified and, as a result, the system’s behavior becomes
unpredictable. The history of dynamical systems and
chaos is described by Aubin & Dalmedico (2002).
In this review, we concentrate on dissipative and driven
nonlinear dynamical systems that exhibit chaos, and we
classify them into two groups, namely, low and high di-
∗ Electronic
address: [email protected];[email protected]
mensional systems. The low-dimensional (LD) systems
are those that have one-and-a half degrees of freedom,
which means that their behavior can be described in
three-dimensional (3D) phase space. On the other hand,
all dynamical systems that have two or more physical
degrees of freedom and require four-dimensional (4D)
or higher-dimensional phase space are classified as highdimensional (HD) systems. Our definition of HD systems
also includes systems in which the state-space dimension
approaches infinity; however, the emphasis of this paper
is on HD systems that can be derived from LD systems
by adding a finite number of degrees of freedom. The dimension of a system always means its dimension in phase
space, unless stated otherwise. Hence, we refer to dynamical systems as 3D systems, 4D systems, and so on.
Examples of Low Dimensional or LD systems include
the following dynamical systems that are described in
3D phase space: the Lorenz model (Lorenz 1963), the
Duffing oscillator (Duffing 1918; Ueda 1979, 1980), the
Van der Pol oscillator (Van der Pol 1922, 1926; Smale
1967), the dissipative, driven and nonlinear pendulum
(D’Humiéres et al. 1982; Blackburn et al. 1987, 1989;
Baker & Gollub 1990), the Rössler model (Rössler 1976),
and the Belousov-Zhabotinsky reaction (Zhabotinsky
1991). The 1D logistic map (May 1976), and the 2D
Hénon map (Hénon 1969) are also classified as LD systems (e.g., Lauwerier 1986a, b). Description of these
systems and discussion of their chaotic behavior can be
found in books on nonlinear dynamics and chaos (e.g.,
Thompson & Stewart 1986; Sagdeev et al. 1988; Jackson 1990; Ott 1993; Hilborn 1994; Strogatz 1994; Tél &
Gruiz 2006; Ivancevic & Ivancevic 2007).
In the real world, most dynamical systems are highdimensional (HD) and many examples of such systems
are described in this review. We concentrate on HD dissipative and driven nonlinear systems and give a special
emphasis to those HD systems that can be constructed by
adding degrees of freedom to already known LD systems.
The main advantage of this approach is that it allows us
to study the effects caused by increasing the number of
degrees of freedom on the system’s chaotic behavior, and
it helps us to make direct comparison between the LD
and HD systems. The latter is important because it may
2
guide us in finding basic rules that govern the chaotic
behavior of the HD systems.
In general, there are two methods to construct HD
systems from LD systems. One method is used to extend
LD systems to higher dimensions by adding higher-order
modes or new states. Examples include extension of the
3D Lorenz model to higher dimensions by taking into
account higher-order modes (e.g., Curry 1978; Howard &
Krishnamurti 1986; Thieffault & Horton 1996; Reiterer et
al. 1998; Tong & Gluhovsky 2002; Humi 2004; Musielak
et al. 2005b; Chen & Price 2006; Roy & Musielak 2007a,
b, c), or adding new states to the 3D Lorenz system (e.g.,
Li et al. 2005), or adding modes to other LD systems
(e.g., Rössler 1979, 1983; Harrison & Lai 2000).
In the other method, the physical degrees of freedom
are added by coupling together LD systems. Typical
examples include coupled Lorenz systems (e.g., Pazó et
al. 2001; Pazó & Matias 2005), coupled logistic maps or
other maps coupled together (e.g., Harrison & Lai 2000),
3D maps (e.g., Stefanski 1998), coupled Rössler systems
(e.g., Harrison & Lai 1999), coupled Duffing oscillators
(e.g., Ueda 1980, 1991; Ueda et al. 1993; Savi & Pacheco
2002; Musielak et al. 2005a), coupled Van der Pol oscillators (e.g., Kapitaniak & Steeb 1991), and other LD
systems coupled together (e.g., Kapitaniak & Chua 1994;
Harrison & Lai 2000; Kapitaniak et al. 2000; Ivancevic
& Ivancevic 2007).
The HD systems produced by these two methods are
important because they allow us to naturally extend the
LD systems to higher dimensions and investigate changes
in their behavior caused by additional degrees of freedom.
Since each HD system has the corresponding LD system
as its subset, in principle the former can be reduced to the
latter and the behavior of both systems can be compared.
Here, we describe many examples of such systems and
discuss the obtained results. There are also numerous HD
dynamical systems for which the LD component cannot
be clearly identified; several such systems are discussed
by Ivancevic & Ivancevic (2007).
The field of research of HD nonlinear dynamical systems that exhibit chaos is often called high-dimensional
chaos. We define High Dimensional Chaos (HDC) as the
phenomena when a HD system exhibits chaos and its
chaotic attractor has non-integer dimension greater than
three (see Sec. II). On the other hand, our definition of
low-dimensional chaos (LDC) is restricted to 3D dynamical systems exhibiting chaotic behavior (e.g., Hilborn
1994).
Since chaotic HD systems have complex time signals,
they are not as vulnerable to unmasking procedures as
are chaotic LD systems. This suggests that HDC may
improve security and high quality synchronization. As
a result, there is great interest in HDC and its practical applications to improve secure communication (e.g.,
Peng et al. 1996; Lakshmanan & Murali 1996; Blakely
and Gauthier 2000; Ivancevic & Ivancevic 2007, and references therein). Other interesting applications of HDC
include studies of neural networks and their dynamics
(e.g., Kaneko 1998; Albers & Sprott 2006). In some
HDC applications, the considered systems are not dissipative; therefore, they are not included in this paper.
For discussion of those systems, see Ozorio de Almeida
(1988), Kaneko & Konishi (1994) and Ivancevic & Ivancevic (2007).
New results obtained in the field of HDC clearly show
that the HD systems behave differently than the LD systems; more specifically, the HD systems are more resistant to the onset of chaos than the LD systems, and they
transition to fully developed chaos via routes that may be
different than those observed in the LD systems. Hence,
the general rules governing the LD and HD chaotic systems are not the same, and one of the most challenging
problems in the field of nonlinear dynamics and chaos is
to discover the rules that govern HDC.
The main purpose of this review is to summarize the
results already obtained in the field of HDC, make a comprehensive comparison between the LD and HD systems,
and formulate general rules that govern HDC. A brief
summary of the most fundamental results obtained for
the LD systems is given in Sec. II. High-dimensional
Lorenz and Duffing systems are described in Sec. III and
IV, respectively. Other HD systems exhibiting chaos are
presented in Sec. V. General rules of HDC are formulated in Sec. VI. Control and synchronization of HDC is
discussed in Sec. VII, and conclusions are given in Sec.
VIII.
II.
OVERVIEW OF LOW-DIMENSIONAL
DYNAMICAL SYSTEMS
A.
Selected systems
For the last four decades, extensive studies of numerous nonlinear systems have been performed. The main
emphasis of these studies has been on LD systems that
are dissipative and driven. The fact that the behavior
of these 3D systems can easily be displayed graphically
has helped to establish the range of physical parameters
for which the onset of chaos and fully developed chaos
are observed, and to determine several basic routes via
which the systems transition to chaos. The systems investigated range from fluid motions and all different types
of oscillators to chemical reactions and biological systems
describing the rhythms of life.
Among many known LD dynamical systems, the
Lorenz model (Lorenz 1963; Sparrow 1982), the Duffing oscillator (Duffing 1918; Ueda 1979, 1980), the Van
der Pol oscillator (Van der Pol 1922, 1926, 1927; Smale
1967), the Rössler model (Rössler 1976), the dissipative,
driven and nonlinear pendulum (D’Humiéres et al. 1982;
Blackburn et al. 1987, 1989; Baker & Gollub 1990), and
the Belousov-Zhabotinsky reaction (Zhabotinsky 1991)
are considered to be prototypes of all LD systems that
exhibit chaos. These systems are described by ordinary
differential equations, which means that their evolution
3
in time is continuous. There are also systems that are described by difference equations, or maps, and their evolution is discrete in time. The 1D logistic map (May
1976, 1980) and the 2D Hénon map (Hénon 1969, 1976)
have played an important role in establishing the field of
nonlinear dynamics and chaos.
The Lorenz system was developed by the MIT meteorologist Edward Lorenz in 1963. It models 2D RayleighBénard convection in a fluid that is treated in the Boussinesq approximation (Saltzman 1962), and it was originally designed to make weather predictions. The physical relevance of this model to describe either the fluid
convection or the weather behavior is questionable because no strong chaos, so prominent in the Lorenz system, has ever been observed in the full Boussinesq system with sufficiently high resolution (e.g., Chen & Price
2006). Despite this limitation, the so-called generalized
Lorenz equations were used to develop a comprehensive
theory of turbulent flows (Ivancevic & Ivancevic 2007).
The Lorenz model has been extensively studied in the
literature (e.g., Sparrow 1982; Schmutz & Rueff 1984;
Thompson & Stewart 1986; Jackson 1990; Argyris et al.
1993); its importance is that it shows a variety of phenomena typical for 3D dissipative and driven dynamical systems, and that it was the first system in which a
strange (chaotic) attractor was discovered (Lorenz 1963).
Lorenz-like chaotic dynamics has been observed in
many systems. For example in electrical and optical systems (e.g., Cuomo et al. 1993; Pazo et al. 2001; Li et
al. 2005; see also Jackson 1990 and Hilborn 1994, and
references therein), in lasers described by the LorenzMaxwell-Haken equations (Haken 1983, 1993; Arecchi
1988), and in some communications systems (e.g., Van
Wiggeren & Roy 1998; Wang et al. 2001; Yang 2004).
The Lorenz model has been investigated as a chaotic
cryptosystem (e.g., Jiang 2002; Solak 2004; Orue et al.
2006) and also used as an example to control chaos
(e.g., Mirus & Sprott 1999; Lü et al. 2002; GonzálezMiranda 2004). Moreover, the so-called generalized
Lorenz (Vanecek & Celikovsky 1996) and Lorenz-like (Lü
et al. 2004; Liu et al. 2004) systems have been introduced
and their studies showed the existence of new chaotic attractors (Mello et al. 2008). Since these generalized systems are 3D systems (see also Chechin & Ryabov 2004),
we classify them here as LD dynamical systems.
The Duffing oscillator is named after the German electrical engineer Georg Duffing who first introduced it in
1918 (see Duffing 1918). Since then, the system has become the classical model for mechanical and electrical oscillators, and many other physical phenomena. Specific
examples include electric circuits with nonlinear inductance, damped and driven nonlinear mechanical oscillators, impact oscillators, mechanical systems that contain
gears, backlash or deadband regions, Josephson junctions, plasma oscillations, optical bistability, ship dynamics, and many others (e.g., Nayfeh & Mook 1979; Novak
& Frehlich 1982; Hayashi 1985; Thompson & Stewart
1987; Andronov et al. 1987; Moon 1987, 1992; Jordan
& Smith 1990; Kahn 1990; Hilborn 1994; Trueba et al.
2003; Ivancevic & Ivancevic 2007). Some applications
of this system required significant modifications. An
important example is the so-called Holmes-Duffing system, which approximates oscillations of a damped beam
driven by a periodic force applied at its tip (Holmes 1979;
Holmes & Holmes 1981; Holmes & Whitley 1983).
The original Duffing system constitutes a paradigm in
the study of nonlinear oscillators. The most comprehensive investigation of the chaotic behavior of this system
was performed by Ueda (1979, 1980, 1991) and Ueda et
al. (1993). They established the range of physical parameters responsible for the onset of chaos and also determined the system’s route to chaos. Other important
studies of the Duffing model were done by Kozlowski et
al. (1995), Benner (1997), Kenfack (2003), Murakami et
al. (2003), Musielak et al. (2005a), Wang et al. (2006)
and Xu et al. (2008).
The Van der Pol oscillator, originally introduced in
1922 (see Van der Pol 1922), was the first relaxation
oscillator used to model the human heartbeat in 1928.
The system was extensively studied by Smale (1967) and
played an important role in introducing Smale’s horseshoe map; more recent studies of this system include
Kapitaniak & Steeb (1991), Poliashenko et al. (1991,
1992), Chen & Chen (2008), and Elabbasy & El-Dessoky
(2008). Note that the Van der Pol and Duffing oscillators can also be combined together so that the Van der
Pol-Duffing system is obtained (e.g., Woafo et al. 1996).
In 1976, Otto E. Rössler discovered a particularly simple system of equations that has only one nonlinear term,
known as the most elementary geometric construction of
chaos in a 3D dynamical system (Rössler, 1976). The set
of equations that Rössler developed was found useful in
modeling equilibrium in chemical reactions. Rössler also
extended his system to 4D and demonstrated the existence of high-dimensional chaos (Rössler 1979). Other
studies addressed 3D Rössler systems coupled together
(e.g., Harrison & Lai 2000; Yanchuk & Kapitaniak 2001),
and 3D Rössler systems coupled to other LD systems
(e.g., González-Miranda 2004; Belykh et al. 2008).
The Belousov-Zhabotinski (BZ) reaction describes the
model of chemical oscillations in a reactor (Zhabotinsky,
1991). In 1951, B. Belousov discovered the oscillation
of a solution color during the cerium-catalyzed oxidation
of citric acid by bromate. He found that the oscillation
period decreases with increasing temperature. However,
Belousov did not discover the mechanism of the oscillations. In 1961, A.M. Zhabotinsky began study of the
Belousov reaction in a continuous-flow stirred-tank reactor, replacing the original citric acid with malonic acid to
improve the optical contrast of the color oscillations. After Zhabotinsky provided the experimental evidence for
the existence of homogeneous, oscillating chemical reaction described by Belousov, it was termed the BelousovZhabotinsky (BZ) reaction. The BZ system is studied
in nonlinear dynamics because it represents an oscillator
in the sense that the composition of the reactants oscil-
4
lates in time, sometimes chaotically (e.g., Richetti et al.
1987; Petrov et al. 1993; Flesselles et al. 1998; Hamik
et al. 2001). A simplified version of this reaction is the
so-called Brusselator model, which is described by a set
of three nonlinear, first-order, ordinary differential equations (e.g., Nicolis & Prigogine 1989, Hilborn 1994, and
references therein).
The fact that a pendulum can also behave chaotically,
clearly shows the fundamental role played by nonlinearity, damping and a driving force in generating chaotic
behavior (e.g., Baker & Gollub 1990). There are many
studies of different dynamical pendulum systems, including a damped and parametrically forced pendulum (e.g.,
Kim & Lee 1996), a driven pendulum (e.g., Heng & Martienssen 1992), a generalized perturbed pendulum (e.g.,
Trueba et al. 2003), a forced spherical pendulum (e.g.,
Tritton 1986; Tritton & Groves 1999), a double pendulum (e.g., Skeldon & Mullin 1992), coupled pendulums
(e.g., Smith et al. 2003), and other variety of pendulums
(e.g., Mawhin 1997, and references therein).
The logistic map is a simple 1D model for generic population growth, which takes into account birth, death
and migration of members of a population (e.g., Hilborn
1994). In more recent studies two logistic maps coupled
together and also coupled to other maps were considered
(e.g., Harrison & Lai 2000; Ivancevic & Ivancevic 2007).
Finally, the 2D Hénon map is a Poincaré section (see
Sec. II.B) of the 3D Lorenz system (e.g., Benedicks &
Carleson 1991). Other studies of this map include its 3D
generalization (e.g., Baier & Klein 1990; Stefanski 1998),
and construction of the time-delayed Hénon map (e.g.,
Sprott 2006).
The behavior of the above LD dynamical systems is
consistent with the Poincaré-Bendixon theorem (e.g.,
Hirsch & Smale 1974), which precludes the existence of
other than steady, periodic, or quasiperiodic attractors in
autonomous systems defined on one- or two-dimensional
manifolds. Thus, the minimal dimension for the existence
of chaos is three, which means that the system must be
described in 3D phase space (see Sec. I). However, chaos
can also occur in dynamical systems with the fractional
order < 3 and examples of such systems include nonautonomous Duffing systems with the order 2.1 (Arena
et al. 1997), Chua’s circuits with the order 2.7 (Hartley
et al. 1995; Hartley & Lorenzo 2002), and the so-called
Lorenz-Chen-Lü (LCL) system with the order 2.76 (Wu
et al. 2008). We decided to exclude these systems from
our paper because their high-dimensional extensions have
not yet been developed. Hence, all LD and HD systems
considered here are of an integer order.
B.
Basic techniques
Basic techniques to investigate the chaotic behavior of
a system and the onset of chaos include phase portraits,
Poincaré sections, power spectra, and plots of Lyapunov
exponents (e.g., Jackson 1990; Hilborn 1994). The main
advantage of phase portraits is that they show the evolution of a system in phase space and the formation of its
attractor. To study nonlinear systems that evolve continuously in time, Poincaré introduced the concept of a
section, which cuts across the orbits in phase space and
allows one to observe the system’s evolution in discrete
time steps. The method is now called the Poincaré section and its main advantage is that it replaces the original
continuous system by a discrete, iterative map whose dimension is reduced by one. Power spectra can be used
to determine the system’s type of attractor and its transition to chaos.
The fundamental property of any chaotic system is its
extreme sensitivity to arbitrarily small changes in initial
conditions. This property can be quantified by showing that the distance between two initially close trajectories diverges exponentially as |δx(t)| ≈ |δx(0)| exp(λt)
when λ is positive. The result was first obtained by Lyapunov (1907, 1949), who demonstrated that the trajectories cannot diverge faster than exponentially; λ is now
called the Lyapunov exponent. In reality, the divergence
or convergence changes locally, so it must be monitored
along each trajectory. As a result of this divergence or
convergence, a sphere of initial conditions evolves into
an ellipsoid, whose principal axes are in the directions of
expansion and contraction. Hence, the Lyapunov exponents are the average rates of expansion or contraction
along the principal axes of the ellipsoid (e.g., Wolf et al.
1985; Eckmann et al. 1985; Wolf 1986; Baker & Gollub
1990; Bryant et al. 1990; Geist et al. 1990).
The number of Lyapunov exponents for a dynamical
system is the same as the number of dimensions of phase
space required to analyze the system; this means that 3D
dynamical systems are characterized by 3 different Lyapunov exponents. Denoting the largest Lyapunov exponent as λmax , one may calculate the so-called Lyapunov
time TL ≈ − ln |δx/Ds |/λmax , where Ds is the characteristic size of the system. The meaning of TL is that the
system’s time evolution can only be predicted up to the
Lyapunov time. As discussed by Ivancevic & Ivancevic
(2007), this is a rather crude estimate of the system’s
predictability because of several limitations of this approach.
For a 3D dissipative system, the sum of all Lyapunov
exponents must be negative. However, if one Lyapunov
exponent is positive, then the system becomes sensitive
to small changes in initial conditions. In other words,
calculating the Lyapunov exponents one may determine
whether the system is chaotic or not. The exponents
are also used to classify attractors. There are four basic
types of attractors: a fixed point, which is represented by
three negative Lyapunov exponents; a limit cycle, which
is characterized by two negative and one zero Lyapunov
exponents; a torus, with one negative and two zero Lyapunov exponents; and a strange attractor, with one negative, one zero and one positive Lyapunov exponents.
Detailed description of system’s attractors, especially its
strange attractor, is given by Jackson (1990), Ott (1981,
5
1993), and Ruelle (1989); see also Feigenbaum (1987),
Grebogi et al. (1987) and Argyris et al. (1993).
The above classification of the system’s attractors
shows that a positive Lyapunov exponent is an indicator of chaos. However, for a system to be chaotic, its
trajectories must not only diverge but eventually they
have to come close to each other and mix; an important
result first obtained by Smale (1967) was that stretching
and folding of the trajectories was required to develop
chaotic behavior and form a strange (or chaotic) attractor. Mechanisms of generating strange attractors are discussed by Gilmore (1998) and Tsankov & Gilmore (2003),
who showed that the ’stretch-fold-squeeze’ mechanism
generates the Rössler strange attractor, and the ’stretchroll’ mechanism is responsible for the origin of the Duffing strange attractor. Examples of strange attractors
for different dynamical systems are given by Holden &
Muhamad (1986), Thompson & Stewart (1987), Jackson (1990), Hilborn (1994), Strogatz (1994), Tél & Gruiz
(2006), and other books devoted to the field of nonlinear
dynamics and chaos.
An important property of any strange attractor is that
it has a non-integer dimension, which is often called fractal dimension (Mandelbrot 1983). In general, the dimension of a strange attractor is calculated using either the
Hausdorff dimension, the capacity dimension, or the correlation dimension (Russell et al. 1980; Peitgen et al.
1992). There is also an important relationship between
Lyapunov exponents and dimension (Kaplan & Yorke
1979). This so-called Kaplan-Yorke relation defines the
Lyapunov dimension, DL , as DL = 1 − λ1 /λ2 , where λ1
and λ2 are positive and negative Lyapunov exponents,
respectively. Note that this relation can easily be generalized to higher dimensions (e.g., Baker & Gollub 1990).
The Lorenz strange attractor has a Hausdorff dimension of 2.06 (Lorenz 1984). As already noted above, the
2D Hénon map is a Poincaré section of the 3D Lorenz
system. According to Hénon (1969, 1976), the strange
attractor for this map is smooth in one direction and a
Cantor set (Cantor 1883) in another. Numerical calculations give a Hausdorff dimension of 1.26 (Russell et al.
1980) and a correlation dimension of 1.42 (Grassberger
1983, 1986) for the Hénon strange attractor.
C.
Routes to chaos
In 3D dynamical systems, transition to fully developed
chaos takes place via either local or global bifurcations
(e.g., Hilborn 1994). In the case of local bifurcations, the
three basic routes to chaos are: the period-doubling cascade (Feigenbaum 1978, 1979, 1980), the quasi-periodic
route (Ruelle & Takens 1971), and the intermittency
route (Pommeau & Manneville 1980). However, for the
global bifurcations, the following two routes to chaos have
been identified: the crisis route (Grebogi et al. 1983), and
the route that involves chaotic transients (e.g., Jackson
1990). These routes are important because they repre-
sent universal features of a system and they are independent of the system’s physical properties.
The period-doubling route to chaos begins with a limit
cycle that has a fixed period; when this limit cycle becomes unstable, another limit cycle is formed with the
double period - in the process of forming the next stable cycles, the period always doubles until it reaches infinity and the system’s behavior becomes chaotic. The
famous bifurcation diagram for the 1D logistic map led
to the discovery of the period-doubling cascade and the
fundamental number that is associated with the perioddoubling sequences; this number is now called the Feigenbaum constant (Feigenbaum 1979, 1980). The Duffing
oscillator and the dissipative, driven and nonlinear pendulum transition to chaos via this route (e.g., Ueda 1979,
1980; Novak & Frehlich 1982; D’Humiéres et al. 1982;
Blackburn et al. 1987, 1989).
In the quasi-periodic scenario, an unstable limit cycle introduces another frequency that allows forming a
torus, which may decay and lead to chaos; this route to
chaos is often called the Ruelle -Takens-Newhouse scenario after the authors who first proposed it and proved
that only a 3-frequency torus is allowed before chaos sets
in (Ruelle & Takens 1971; Newhouse et al. 1978; Ruelle 1989; also Grebogi et al. 1985). An example of
such system that transitions to chaos via this route is
the Belousov-Zhabotinsky chemical reaction (Argoul et
al. 1987; Richetti et al. 1987). The route was also identified as one of many possible routes leading to turbulent
convection (e.g., Gollub & Benson 1980; Baker & Gollub
1990).
The intermittency route is characterized by irregularly
occuring bursts of chaos interspersed with intervals of periodic behavior. These chaotic bursts become longer and
longer when the system’s control parameter is increased,
and eventually they totally replace the periodic motion
(Manneville & Pommeau 1979; Pommeau & Manneville
1980). This behavior is caused by collisions between unstable and stable fixed points (or limit cycles) in phase
space and the resulting disappearance of some of these
points (or limit cycles). The route does not have clearcut precursors because the unstable points may not be
visible, and yet it was observed in experiments involving Rayleigh-Bénard convection (e.g., Berge et al. 1980,
1984; Gollub & Benson 1980; Dubois et al. 1983), nonlinear oscillators (e.g., Jeffries & Perez 1982; Huang & Kim
1987), and lasers (e.g., Harrison & Biswas 1987; Arecchi
1988; Sacher et al. 1989; Tang et al. 1991).
There are three different types of intermittency,
namely, Type I or tangent bifurcation intermittency,
Type II or Hopf bifurcation intermittency, and Type III
or period-doubling intermittency; detailed description of
these types of intermittency, their experimental verification, and a list of relevant references can be found in
Hilborn (1994).
In the crisis route, a chaotic attractor and its basin of
attraction can suddenly disappear or reappear, or change
in size as a result of changing the system’s control param-
6
eter (Grebogi et al. 1982, 1987). Typically, two different
types of crisis are identified: a boundary crisis that is
related to the disappearance (or decrease in size) of the
chaotic attractor and its basin of attraction, and an interior crisis that describes the reappearance (or increase
in size) of the attractor and its basin of attraction. The
main reason for this rather strange behavior is the attractor’s interaction with either an unstable fixed point
or an unstable limit cycle (Grebogi et al. 1983, 1985;
also Hilborn 1994). The route was observed in experiments involving nonlinear oscillators (e.g., Hilborn 1985),
a gravitationally buckled and parametricly driven magnetoelastic ribbon (e.g., Ditto et al. 1989), and lasers
(e.g., Dangoisse et al. 1986).
Finally, the chaotic transient route is observed when
the system’s trajectories interact with various unstable
fixed points and limit cycles; the result is that homoclinic and heteroclinic orbits may suddenly appear and
they may strongly influence the trajectories passing near
them. Studies of the 3D Lorenz model have shown that
its transition to fully developed chaos occurs via chaotic
transients with both homoclinic and heteroclinic bifurcations and with the homoclinic orbits being present (for
details, see Sparrow 1982, 1986; Thompson & Stewart
1986; Jackson 1990; Kennamer 1995).
Detailed discussion of the above routes to chaos can be
found in many books on nonlinear dynamics and chaos
(e.g., Thompson & Stewart 1987; Jackson 1990; Hilborn
1994; Strogatz 1994; Tél & Gruiz 2006) as well as in some
reviews (e.g., Eckmann 1981; Argyris et al. 1993).
D.
From low to high-dimensional systems
The results described above were obtained for 3D nonlinear dynamical systems, which are classified here as LD
systems. The main dynamical property of these systems
is that they are characterized by only one positive Lyapunov exponent, and that their strange attractors have
non-integer dimensions less than three. We refer to the
field of research of the LD systems that exhibit chaos as
low-dimensional chaos (LDC).
An important next step is to extend these LD systems to higher dimensions in phase space, and to study
the effects caused by increasing the number of degrees
of freedom. As discussed in Sec. I, there are two general methods to construct HD systems from the already
known LD systems. One method involves adding higherorder modes, or new states, to the original three-modes
systems. In the other method, the physical degrees of
freedom are added by coupling together the LD systems.
Since each HD system constructed by these methods has
the corresponding LD system as its subset, detailed comparison between the behavior of the LD and HD systems
can be made. Many examples of such systems are presented and discussed below.
We refer to high-dimensional chaos (HDC) when a HD
system, whose phase space dimension exceeds 3D, ex-
hibits either chaos (one positive Lyapunov exponent) or
hyperchaos (more than one positive Lyapunov exponent),
and when the non-integer dimension of its strange attractor is greater than three, including cases with multiple
zero Lyapunov exponents. The main advantage of this
general definition of HDC is that it clearly separates the
LD systems from the HD systems constructed by adding
degrees of freedom. This allows us to include most of the
results obtained so far for chaotic HD systems.
III.
HIGH-DIMENSIONAL LORENZ MODELS
A.
Saltzman’s equations and the Lorenz model
Let us consider a 2D model of Rayleigh-Bénard convection in a fluid that is treated in the Boussinesq approximation (Saltzman 1962) and described by the following
set of hydrodynamic equations:
∂Vz
∂Vx
+
=0,
∂x
∂z
(1)
∂Vx
∂Vx
∂Vx
1 ∂p
+ Vx
+ Vz
+
∂t
∂x
∂z
ρ ∂x
−ν
∂ 2 Vx
∂ 2 Vx
+
∂x2
∂z 2
=0,
(2)
∂Vz
∂Vz
∂Vz
1 ∂p
+ Vx
+ Vz
+
− gαT
∂t
∂x
∂z
ρ ∂z
−ν
∂ 2 Vz
∂ 2 Vz
+
2
∂x
∂z 2
∂T
∂T
∂T
+ Vx
+ Vz
−κ
∂t
∂x
∂z
=0,
∂2T
∂2T
+
2
∂x
∂z 2
(3)
=0,
(4)
where Vx and Vz are the horizontal and vertical components of the fluid velocity, and the fluid physical parameters are the density ρ, pressure p and temperature T .
In addition, ~g = −g ẑ is gravity, α is the coefficient of
thermal expansion, ν is the kinematic viscosity, and κ is
the coefficient of thermal diffusivity.
We assume that the fluid is confined between two
horizontal surfaces located at z = 0 and z = h with
T (z = 0) = T0 + ∆T0 and T (z = h) = T0 , and
that the temperature varies between the surfaces as
T (z) = T0 + ∆T0 (1 − z/h). We denote the temperature changes due to convection by θ(x, z, t) and write
T (x, z, t) = T0 +∆T0 (1−z/h)+θ(x, z, t). Using the continuity equation (Eq. 1), we introduce the stream function
ψ, which is defined by Vx = −∂ψ/∂z and Vz = ∂ψ/∂x.
To express the hydrodynamic equations in terms of θ and
ψ, we differentiate Eqs (2) and (3) with respect to z and
7
x, respectively, and then subtract the former equation
from the latter. This gives
∂ψ ∂
∂ψ ∂
∂
∇2 ψ −
∇2 ψ +
∇2 ψ
∂t
∂z ∂x
∂x ∂z
−gα
∂θ
− ν∇4 ψ = 0 ,
∂x
(5)
where ∇4 = ∂ 4 /∂x4 + ∂ 4 /∂z 4 . The energy equation (Eq.
4) can also be expressed in terms of ψ and θ, and we
obtain
∂ψ ∂θ
∂θ ∆T0 ∂ψ ∂ψ ∂θ
−
−
+
− κ∇2 θ = 0 .
∂t
h ∂x
∂z ∂x
∂x ∂z
(6)
We introduce the following dimensionless quantities:
x∗ = x/h, z ∗ = z/h, t∗ = tκ/h2 , ψ ∗ = ψ/κ, θ∗ =
θαgh3 /κν, ∇∗ = ∇/h. Then, Eqs (5) and (6) become
∂ψ ∗ ∂
∂
∇∗2 ψ ∗ −
∇∗2 ψ ∗
∗
∗
∗
∂t
∂z ∂x
+
∂ψ ∗ ∂
∂θ∗
∗2 ∗
∇
ψ
−
σ
− σ∇∗4 ψ ∗ = 0 ,
∂x∗ ∂z ∗
∂x∗
(7)
and
∂θ∗
∂ψ ∗ ∂ψ ∗ ∂θ∗ ∂ψ ∗ ∂θ∗
−
R
− ∗
+
− κ∇∗2 θ∗ = 0 , (8)
∂t∗
∂x∗
∂z ∂x∗ ∂x∗ ∂z ∗
where σ = ν/κ is the Prandtl number and R =
αgh3 ∆T0 /κν is the Rayleigh number. We refer to this
set of equations as Saltzman’s equations.
According to Saltzman (1962), one may impose the
following boundary conditions: ψ ∗ = 0, ∇∗2 ψ ∗ = 0 and
θ∗ = 0 at both surfaces located at z = 0 and z = h, and
write the solutions as double Fourier expansions of ψ ∗
and θ∗
ψ(x∗ , z ∗ , t∗ ) =
∞
X
∞
X
Ψ(m, n, t∗ )
m=−∞ n=−∞
h
m
n ∗ i
,
x∗ +
z
·exp 2πhi
L
2h
(9)
and
θ(x∗ , z ∗ , t∗ ) =
∞
X
∞
X
Θ(m, n, t∗ )
m=−∞ n=−∞
h
m
n ∗ i
·exp 2πhi
,
x∗ +
z
L
2h
(10)
where L is the characteristic scale representing periodicity 2L in the horizontal direction, and the modes of the
Fourier expansions are labelled by two integers m and n
that correspond to the horizontal and vertical directions,
respectively.
Saltzman (1962) expressed Ψ and Θ in terms of
their real and imaginary parts, Ψ(m, n) = Ψ1 (m, n) −
iΨ2 (m, n) and Θ(m, n) = Θ1 (m, n) − iΘ2 (m, n), which
do not show explicitly time-dependence, and substituted
the above solutions into Eqs (7) and (8). The general
result was a set of first-order differential equations for
the Fourier coefficients Ψ1 , Ψ2 , Θ1 and Θ2 . However,
when the theory was applied to describe cellular convective motions originating from small perturbations, Saltzman fixed the vertical nodal surfaces of the convective
cells by excluding all Ψ2 (m, n) and Θ1 (m, n) modes. In
addition, he did not consider shear flows and assumed
that the Ψ1 modes with m = 0 were zero.
To derive the simplest model that approximates the
2D Rayleigh-Bénard convection, Lorenz (1963) used the
minimal truncation of Saltzman’s equations and selected
the following three Fourier modes (see Table 1): Ψ1 (1, 1),
which describes the circulation of the convective roll,
and Θ2 (1, 1) and Θ2 (0, 2), which approximate the horizontal and vertical temperature differences in the convective roll, respectively. He introduced the variables
X(τ ) = Ψ1 (1, 1), Y (τ ) = Θ2 (1, 1) and Z(τ ) = Θ2 (0, 2),
and derived the following set of equations:
dX
= −σX + σY ,
dτ
(11)
dY
= −XZ + rX − Y ,
dτ
(12)
dZ
= XY − bZ ,
dτ
(13)
where τ = π 2 (1 + a2 )t∗ is a dimensionless time, a = h/L
is the aspect ratio, b = 4/(1 + a2 ) and r = R/Rc with
Rc = π 4 (1 + a2 )3 /a2 .
Extensive studies of the Lorenz model (e.g., Lorenz
1963, 1984; Sparrow 1982, 1986; Schmutz & Rueff 1984;
Thompson & Stewart 1986; Argyris et al. 1993) showed
that at r = 24.75 the system exhibits fully developed chaos and the Lorenz chaotic (strange) attractor
is formed (see Fig. 1). An important result was obtained
by Tucker (1999) who presented a formal mathematical
proof for the existence of the Lorenz strange attractor;
before this analytical result was obtained, the existence of
the Lorenz attractor was only verified numerically (Stewart 2000).
Numerous attempts have been made to extend the 3D
Lorenz system to higher dimensions by selecting higherorder Fourier modes (e.g., Curry 1978; Howard & Krishnamurti 1986; Kennamer 1995; Thieffault & Horton
1996; Reiterer et al. 1998; Tong & Gluhovsky 2002; Humi
2004; Musielak et al. 2005b; Chen & Price 2006; Roy
& Musielak 2007a,b,c). In the following subsection, we
give a brief description of the methods used to develop
these models and the required validity criteria. Then, we
present and discuss the models in Sec. III.C and D.
8
tems (e.g., Howard & Krishnamurti 1986; Thieffault &
Horton 1996; Humi 2004; Chen & Price 2006).
According to Saltzman (1962), the dimensionless kinetic, K ∗ , and potential, U ∗ , energy can be expanded
into spectral components as
K∗ =
∞
∞
1 X X 2
δ (m, n)|Ψ(m, n)|2
2 m=−∞ n=−∞
(14)
∞
∞
1σ X X
|Θ(m, n)|2
2 R m=−∞ n=−∞
(15)
and
U∗ = −
FIG. 1: Lorenz chaotic attractor.
B.
where δ 2 (m, n) = (2πa)2 m2 + π 2 n2 .
The total energy of the system, E ∗ = K ∗ + U ∗ , is
conserved in the dissipationless limit, σ → 0, when the
following condition is satisfied
Methods and validity criteria
lim
σ→0
In general, there are three basic methods of adding the
higher-order modes to those originally selected by Lorenz
(1963). In the first method, one keeps m (horizontal
modes) fixed and changes n (vertical modes), which is
essentially the vertical mode truncation. In the second
method, one changes m and keeps n fixed, which means
that this is the horizontal mode truncation. Finally, in
the third method, both m and n are changed and, therefore, we call this method the horizontal and vertical mode
truncation (Roy & Musielak 2007a,b,c). The models constructed by using one of these methods are called here
HD Lorenz models, and the modes selected for different
models are summarized in Table 1. Note that the models
are constructed in such a way that the original 3D Lorenz
model is always their subset (see Sec. III.C and D).
There are two validity criteria that the constructed
models must satisfy in order to be considered physically
meaningful systems. The first criterion requires that energy is conserved in the dissipationless limit. Some higher
dimensional models of Rayleigh-Bénard convection based
on finite Fourier mode truncations do not conserve energy
in the dissipationless limit, as already recognized in the
literature (Treve & Manley 1982; Thieffault 1995; Thieffault & Horton 1996; Tong & Gluhovsky 2002).
There are several important reasons to have energy
conserving HD Lorenz systems. One is the fact that the
effects of non-conservation energy can be large and that
they can significantly affect the energy flow in the dissipative regime. In addition, the thermal flux in steady-state
is described correctly only by energy conserving systems.
Moreover, the energy conserving truncations represent
the whole system more accurately and they reduce nonphysical numerical instabilities. The second requirement
is that the HD Lorenz models must have bounded solutions; systems with unbounded solutions are considered
to be non-physical (Curry 1979). This criterion has been
used by many authors to validate their HD Lorenz sys-
dE ∗
=0
dτ
(16)
To derive the condition for bounded solutions, one introduces the quantity Q∗ (Thieffault & Horton 1996) defined as
2
X
X
2
∗
∗
2
Q =K +2
Θ(0, n) −
|Θ(m, n)| +
n
m,n>0
n>0
(17)
note that all selected modes are included in Q∗ , so if one
of them diverges, then Q∗ also diverges. We take the
time derivative of Q∗ and write
dQ∗
≤ [−min(2ν, κ) Q∗ + 4κn0 ]
dτ
(18)
where n0 is the number of the Θ(0, n) modes (see Sec.
III.C). To obtain dQ∗ /dτ < 0, we must have Q∗ >
4κn0 /min(2ν, κ), which is the condition for bounded solutions.
Applying the above criteria to the Lorenz model
(Lorenz 1963) described in Sec. III.A, it can be shown
that the modes Ψ1 (1, 1), Θ2 (1, 1) and Θ2 (0, 2) selected by
Lorenz conserve energy in the dissipationless limit (see
Eq. 16), and that solutions of Eqs (11), (12) and (13)
are bounded (see Eq. 18).
C.
From 3D to 9D models
The method of horizontal mode truncation can be used
to construct HD Lorenz models. In this method, one
fixes the value of n = 1 and adds the Fourier modes
with different m. Roy & Musielak (2007a) added Ψ1 (2, 1)
and Θ2 (2, 1) to the original three Lorenz modes and constructed a 5D Lorenz model (see Table 1). Studies of
this model show that its solutions are bounded and that
it does conserves energy in the dissipationless limit. The
system transitions to fully developed chaos at r = 22.5,
9
which is a lower value than the value of r = 24.75 found
for the 3D Lorenz system. Moreover, the system’s route
to chaos is the same as that observed in the original
Lorenz model (see Sec. II.C).
In the vertical mode truncation, one fixes the value of
m by taking m = 1 and adds the vertical modes in both
the stream function ψ and the temperature variations θ
(see Eqs. 9 and 10). Following this procedure, one may
select Ψ1 (1, 2) and Θ2 (1, 2), and add these two modes
to the original Lorenz modes. However, the resulting 5D
model cannot be treated as a new HD system because the
first-order differential equations describing the evolution
of Ψ1 (1, 2) and Θ2 (1, 2) are decoupled from the Lorenz
equations (see Roy & Musielak 2007b). The same is true
when only one mode, either Ψ1 (1, 2) or Θ2 (1, 2), is selected and a 4D model is constructed.
Since neither 5D nor 4D models constructed by the
method of vertical mode truncation are new HD models,
one must add more higher-order modes. The next mode
that must be added is Θ2 (0, 4) and the rules of the vertical mode truncation with m = 1 do not apply to the selection of this mode; actually, its selection is required by the
energy conservation principle (see Thiffeault and Horton
1996). Hence, to construct a 6D Lorenz model, one takes
the three Lorenz modes and adds Ψ1 (1, 2), Θ2 (1, 2) and
Θ2 (0, 4). It is easy to show that in the resulting 6D system, there is no coupling between the Lorenz modes and
the higher-order modes (Roy & Musielak 2007b). One
may continue adding modes and construct 7D and 8D
systems, and find out that they are not new HD systems because of the missing coupling with the 3D Lorenz
model.
An interesting result was obtained by Kennamer
(1995) who used the vertical mode truncation and selected the modes: V (τ ) = Ψ1 (1, 3), W (τ ) = Θ1 (1, 3) and
S(τ ) = Θ2 (0, 4) (see Table 1). As a result, he was able
to extend the 3D Lorenz model to six dimensions (6D).
This model is described by the following set of first-order
differential equations:
dX
= −σX + σY ,
dτ
(19)
dY
= −XZ + rX − Y + ZS − 2W S ,
dτ
(20)
dZ
= XY − bZ − XV − Y S ,
dτ
(21)
dV
= XZ − 2XW + rS − cV ,
dτ
(22)
dW
= 2XV + 2Y S − 4bW ,
dτ
(23)
dS
σ
= V − cσS ,
dτ
c
(24)
FIG. 2: Projections of the chaotic attractor of the 6D Lorenz
model on the XY Z and V W S phase space are plotted for the
control parameter r = 41.0 (after Kennamer 1995).
where c = (9 + a2 )/(1 + a2 ). Note that in the limit of
V = W = S = 0, this system of equations reduces to
that describing the 3D Lorenz model (see Eqs 11, 12 and
13).
Since the three additional equations for V , W and S
are coupled to the Lorenz’s equations, Kennamer concluded that this 6D model was a new HD Lorenz model.
His studies of the onset of chaos demonstrated that the
model transitions to chaos at r = 39.14 and to fully developed chaos at r = 41.54, which means that the additional
vertical modes of the 6D system make it more resistant
to the onset of chaos than the 3D Lorenz model. Otherwise, the 6D system is very similar to the Lorenz model.
Projections of the strange attractor of the 6D model on
the XY Z and V W S phase space are plotted in Figs 2
and 3, respectively. Comparison of these phase portraits
to the Lorenz chaotic attractor (see Fig. 1) shows smilarities between the attractors of both systems. Plots of two
largest Lyapunov exponents versus the control parame-
10
FIG. 4: Two largest Lyapunov exponents for the 6D Lorenz
model (after Kennamer 1995).
FIG. 3: The same as Fig. 2 but for r = 50.0 (after Kennamer
1995).
ter r are shown in Fig. 4. According to these results, the
system’s route to chaos is the same as that observed in
the 3D Lorenz system.
Studies of this 6D model showed that it has bounded
solutions and it conserves energy in the dissipationless
limit (Musielak et al. 2005a). However, there is a problem with the model, namely, the modes Ψ1 (1, 2) and
Θ2 (1, 2) were omitted in the mode selection process without giving any physical justification. Moreover, the selection of the Θ2 (0, 4) mode is also inconsistent with the
general principle of selecting these modes formulated by
Thieffault and Horton (1996).
Because of the above problems with the 6D model constructed by Kennamer (1995), and the fact that neither
7D nor 8D models (developed by the method of vertical
truncation) are new HD Lorenz models, one may consider adding more higher-order modes and obtain a 9D
model. Such model was constructed by Roy and Musielak
(2007b) who added Ψ1 (1, 2), Ψ1 (1, 3), Θ2 (1, 2), Θ2 (1, 3),
Θ2 (0, 4) and Θ2 (0, 6) to the three Lorenz modes (see Table 1), and derived a set of nine first-order, nonlinear
differential equations. The obtained equations show that
all modes are well-coupled in this new 9D system, the solutions are bounded, the system conserves energy in the
dissipationless limit, and that the system’s route to chaos
is the same as that observed in the 3D Lorenz system.
Therefore, it was concluded that the 9D Lorenz model
was the lowest-order HD Lorenz system that can be constructed by using the method of vertical mode truncation.
The above results demonstrate that the 5D model constructed by using the method of horizontal mode truncation, and the 9D model developed by the method of
vertical mode truncation are the lowest-order HD Lorenz
models obtained by each method, respectively; note that
both models have bounded solutions and that they conserve energy in the dissipationless limit. The large difference in dimension of these models clearly implies that
the horizontal modes are more efficiently coupled to the
3D Lorenz system than the vertical modes.
Obviously, the methods of adding either vertical or
horizontal modes are very limited and the obtained models can only be used to investigate the efficiency of the
mode coupling in these systems. However, in order to
construct more realistic HD Lorenz systems, the methods
must be combined so that both vertical and horizontal
modes can be selected. Such models are described in the
following section, including the lowest-order HD Lorenz
system that can be constructed by using the method of
vertical-horizontal mode truncation.
D.
More realistic models
We now describe HD Lorenz models that are constructed by using the method of vertical-horizontal mode
11
Selected Modes
Ref.
3D
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
[151]
5D∗
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (2, 1), Θ2 (2, 1)
[220]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ2 (1, 1), Θ1 (1, 1)
[38]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (1, 3), Θ2 (1, 3), Θ2 (0, 4)
[130]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (1, 2), Ψ1 (2, 1), Θ2 (2, 1)
[113]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (0, 1), Ψ1 (1, 2), Θ2 (1, 2)
[106]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (0, 1), Ψ1 (1, 2), Θ2 (1, 2)
Θ2 (0, 4)
[260]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (1, 2), Ψ1 (2, 1), Θ2 (1, 2)
Θ2 (2, 1), Θ2 (0, 4)
[222]
Ψ1 (1, 1), Θ2 (1, 1), Θ2 (0, 2)
Ψ1 (1, 2), Ψ1 (1, 3), Θ2 (1, 2)
Θ2 (1, 3), Θ2 (0, 4), Θ2 (0, 6)
[221]
Model
5D
6D∗∗
6D
6D
FIG. 5: Chaotic attractor of a 6D Lorenz system developed
by Humi (2004). The attractor is plotted for r = 30 (after
Humi 2004).
truncation. The models developed by this method are
more realistic than those described in the previous section because the mode selection should not be limited to
only one type of modes.
Humi (2004) used this combined method and added
the following modes (see Table 1) to the original Lorenz
modes: Ψ1 (1, 2), Ψ1 (2, 1) and Θ2 (1, 2). As a result, he
constructed a 6D Lorenz model and showed that the system has bounded solutions and transitions to fully developed chaos at r = 28.45. The system’s chaotic attractor
does not show any similarity to the Lorenz strange attractor (compare the results of Figs 5 and 1), and the
system’s route to chaos is the period-doubling cascade.
As shown by Roy & Musielak (2007b), this 6D system
does not conserve energy in the dissipationless limit.
There is also a 14D model constructed by Curry (1978),
who selected six Ψ1 modes and six Θ2 modes with 1 ≤
m ≤ 3 and 1 ≤ n ≤ 4, and in addition the Θ2 (0, 2)
and Θ2 (0, 6) modes (see Table 1). It was shown that
the system’s route to chaos is via bifurcation by a family
of invariant two tori. Curry (1979) also demonstrated
that the system has bounded solutions. However, this
14D model does not conserve energy in the dissipationless
limit (Roy & Musielak 2007b).
To construct a HD system that has bounded solutions
and conserves energy in the dissipationless limit, Roy and
Musielak (2007c) selected (see Table 1) the two horizontal modes Ψ1 (2, 1) and Θ2 (2, 1), the two vertical modes
Ψ1 (1, 2) and Θ2 (1, 2), and also added the mode Θ2 (0, 4)
as required by the energy conservation principle (see
Thiffeault and Horton 1996). With these modes and the
Lorenz modes, the Saltzman’s equations (see Eqs 7 and
8) can be reduced to a set of eight first-order, nonlinear
differential equations describing the system. Introducing X(τ ) = Ψ1 (1, 1), Y (τ ) = Θ2 (1, 1), Z(τ ) = Θ2 (0, 2),
X1 (t) = Ψ1 (2, 1), Y1 (t) = Θ2 (2, 1), Z1 (t) = Θ2 (0, 4),
7D
8D
9D∗∗
14D
Ψ1 (1, 1),
Ψ1 (1, 3),
Ψ1 (3, 3),
Θ2 (2, 2),
Θ2 (1, 1),
Ψ1 (2, 2),
Ψ1 (2, 4),
Θ1 (3, 1),
Θ2 (2, 4),
Θ2 (0, 2)
Ψ1 (3, 1)
Θ2 (1, 3)
Θ2 (3, 3)
Θ2 (0, 4)
[50]
TABLE I: Fourier modes selected to construct the original
3D Lorenz system and HD Lorenz systems. The models labeled by ∗ and ∗∗ were constructed by using the method of
horizontal and vertical mode truncation, respectively.
X2 (t) = Ψ1 (1, 2) and Y2 (t) = Θ2 (1, 2), one obtains
dX
= −σX + σY − c1 X1 X2 ,
dτ
(25)
dY
= −XZ + rX − Y + c1 (X2 Y1 + Y2 X1 + X2 Y2 ) , (26)
dτ
dZ
= XY − bZ + 2X1 Y1 ,
dτ
(27)
σ
dX1
= −c2 σX1 + 2 Y1 + c1 XX2 ,
dτ
c2
(28)
dY1
= −2X1 Z + 2rX1 − c2 Y1 − c1 (X2 Y + X1 Y2 ) , (29)
dτ
12
dZ1
= 2X2 Y2 − 4bZ1 ,
dτ
(30)
dX2
σ
= −c3 σX2 + Y2 − c1 XX1 ,
dτ
c3
(31)
dY2
= −c3 Y2 + 2rX2 − c1 (XY1 + X1 Y ) − 2X2 Z1 , (32)
dτ
where τ = π 2 (1 + a2 )t∗ , a = h/L is the aspect ratio, h is the thickness of a convection region, L is
the characteristic scale, t∗ is the dimensionless time,
b = 4/(1 + a2 ), r = R/Rc ,√R is the Rayleigh number,
Rc = π 4 (1 + a2 )3 /a2 , c1 = 3 2/4, c2 = (1 + 4a2)/(1 + a2 )
and c3 = (4 + a2 )/(1 + a2 ).
A simple inspection of this system of equations clearly
indicates that the selected higher-order modes are wellcoupled to the original modes of the 3D Lorenz system.
Since the selected two horizontal modes and three vertical
modes represent the minimum number of modes for each
method, the developed 8D system is the lowest-order HD
Lorenz model that can be constructed by the combined
method of the vertical-horizontal mode truncations.
We now use Eqs (14) and (15) to obtain
K∗ =
FIG. 6: Three leading Lyapunov exponents are plotted for an
8D Lorenz system developed by Roy & Musielak (2007).
1 2
δ (1, 1)X 2 + δ 2 (2, 1)X12 + δ 2 (1, 2)X22 , (33)
2
and
U∗ = −
1σ
Y 2 + Y12 + Y22 + Z 2 + Z12 ,
2R
(34)
where δ 2 (1, 1), δ 2 (2, 1) and δ 2 (1, 2) are parameters. To
verify that the total energy, E ∗ = K ∗ + U ∗ , is conserved,
we write
dE ∗
= δ 2 (1, 1)ẊX + δ 2 (2, 1)Ẋ1 X1 + δ 2 (1, 2)Ẋ2 X2
dτ
−
σ
(Ẏ Y + Ẏ1 Y1 + Ẏ2 Y2 + ŻZ + Ż1 Z1 ) .
R
(35)
Using the RHS of Eqs (25) through (31) and taking the
dissipationless limit σ → 0, we obtain
lim
σ→0
dE ∗
=0,
dτ
(36)
which gives E = K ∗ + U ∗ = const. Hence, the 8D
system conserves energy in the dissipationless limit. According to Roy and Musielak (2007c), the condition for
the bounded solutions (see Eq. 18) was found to be always satisfied for this 8D model.
The onset of chaos and route to chaos in the 8D model
are determined by solving numerically the set of Eqs (25)
through (32). The parameters b = 8/3 and σ = 10 are
fixed in the calculations, and the control parameter r is
varied over the range 0 ≤ r ≤ 40. The numerical simulations were used to determine the value of rmin , for which
∗
FIG. 7: Phase portraits of the chaotic attractor of the 8D
Lorenz system constructed by Roy & Musielak (2007c). The
phase plots of the variables (X, Y , Z) and (X1 , Y1 , Z1 ) are
given in the upper and lower panel, respectively. The results
are obtained for r = 38.5 (after Roy & Musielak 2007c).
the onset of chaos is observed in the system, and the
value of rmax , for which the system exhibits fully developed chaos. The route to chaos is determined by studying the behavior of Lyapunov exponents in the range
rmin ≤ r ≤ rmax .
Three leading Lyapunov exponents of this system are
plotted in Fig. 6, which shows that for r = 32.5 − 34.5 all
13
three exponents become zero, an indicator of the quasiperiodic behavior and formation of 3-frequency torus in
the phase space. By increasing the value of r, we find out
that the system enters a chaotic regime at rmin = 35.6
and then becomes fully chaotic when rmax = 38.5. Projections of the system’s chaotic attractor on two different
sets of the phase space variables (X, Y , Z) and (X1 , Y1 ,
Z1 ) are shown in Fig. 7; note that the results are obtained for r = 38.5. For other phase portraits of this
system see Musielak & Roy (2007).
Power spectra obtained for the 8D system with four
different values of r are presented in Fig. 8. The upper
left panel of this figure shows that only one characteristic
frequency can be identified in the power spectrum when
r = 28.50. By increasing r to 29.25, two different frequencies become dominant (see the upper right panel).
The formation of a 3-frequency torus is observed in the
lower left panel for which r = 32.50. As already indicated by the plots of Lyapunov exponents (see Fig. 6),
the torus decays into chaos via the quasi-periodic route,
which is a different route to chaos than the chaotic transients observed in the 3D Lorenz model (see Sec. II.C).
To explain this change in the route to chaos, one must
compare the two models. In the 3D Lorenz model, there
are two nonlinear terms that couple the X and Z modes,
and the X and Y modes. The Y and Z modes remain
uncoupled and there is one equation without nonlinear
term. The situation is significantly different in the 8D
system, which shows 16 nonlinear terms with at least one
nonlinear term in each equation (see Eqs. 25 through 32).
Most modes are well-coupled and there are two equations
(see Eqs. 29 and 32), which have 3 mode coupling terms,
and one equation (see Eq. 26), which has 4 nonlinear
terms.
This abundance of mode coupling terms in the 8D system is responsible for its different behavior when compared to other systems. Therefore, one may conclude
that the main physical reason for changing the route to
chaos in the 8D system is the large number (twice the
number of degrees of freedom) of mode coupling terms,
and the fact that each equation has at least one nonlinear
term.
Comparison of the 8D model with the 6D model constructed by Humi (2004) shows that the latter is a subset
of the former (see Table 1). However, Humi’s model cannot be considered as the lowest-order HD Lorenz system
because it does not conserve energy in the dissipationless limit. Hence, the 8D system is the lowest-order HD
Lorenz model that can be constructed by the method of
vertical-horizontal mode truncations.
E.
Other models
All HD Lorenz models described in the last two
sections were constructed based on Saltzman’s boundary conditions that exclude all Ψ2 (m, n) and Θ1 (m, n)
modes, and his assumption that shear flows are not con-
sidered, which means that all Ψ1 modes with m = 0
must be zero (see Sec. III.A). Here, we present several
HD models that are not limited by the above boundary
conditions and assumptions.
One such example is a 5D model constructed by Chen
& Price (2006), who added the modes Ψ2 (1, 1) and
Θ1 (1, 1) to the three Lorenz modes (see Table 1). According to the authors, their model can account for the fact
that steady-state solutions of the 2D Rayleigh-Bénard
convection always exist in circles, whereas solutions of
the 3D Lorenz system always exist in pairs. However,
this 5D model does not conserve energy in the dissipationless limit (Roy & Musielak 2007b).
Another example is a 6D model developed by Howard
and Krishnamurti (1986), who selected the following
modes Ψ1 (0, 1), Ψ1 (1, 2) and Θ2 (1, 2) in addition to the
three modes chosen by Lorenz (see Table 1). Since one
of the selected modes is Ψ1 (0, 1), this 6D model accounts
for shear flows in the 2D Rayleigh-Bénard convection.
Thiffeault and Horton (1996) demonstrated that the 6D
model does not conserve energy in the dissipationless
limit. In order to fix the problem they added the mode
Θ2 (0, 4) and constructed a 7D model (see Table 1). This
is an interesting model considered to be the lowest-order
HD Lorenz model that accounts for shear flows; detailed
discussion of this 7D model is given by Thiffeault (1995).
All work described above, as well as the HD Lorenz
models discussed in Sec. III.C and D, concern 2D
Rayleigh-Bénard convection. However, there are also
many published papers dealing with 3D Rayleigh-Bénard
convection. Full references to those papers are given by
Tong and Gluhovsky (2002).
Also of interest is a 9D model of 3D convection developed by Reiterer et al. (1998). To describe 3D square
convection cells, the researchers expanded the x, y and
z components of a vector potential A, with V = ∇ × A,
and the temperature variations θ into triple Fourier expansions, with l, m and n representing the modes in
the y, x and z direction (see Eqs 9 and 10), respectively. These Fourier expansions were truncated up to
the second order and the following Fourier modes were
used: X(t) = A1 (0, 2, 2), X1 (t) = A1 (1, 1, 1), X2 (t) =
A2 (2, 0, 2), Y (t) = A2 (1, 1, 1), Y1 (t) = A3 (2, 2, 0),
Y2 (t) = Θ(0, 0, 2), Z(t) = Θ(0, 2, 2), Z1 (t) = Θ(2, 0, 2)
and Z2 (t) = Θ(1, 1, 1), where A1 , A2 , A3 and Θ are the
Fourier coefficients of the Ax , Ay , Az and θ expansions,
respectively. This selection of Fourier modes leads to the
following set of equations that model a 9D system:
dX
= −σb1 X − X1 Y + b4 Y 2 + b3 X2 Y1 − σb2 Z , (37)
dτ
1
dX1
= −σX1 + XY − X1 Y1 + Y Y1 − σZ2 ,
dτ
2
(38)
dX2
= −σb1 X2 + X1 Y − b4 X12 − b3 XY1 + σb2 Z1 , (39)
dτ
14
FIG. 8: Power spectra for an 8D Lorenz system developed by Roy & Musielak (2007c). The results presented in the upper left,
upper right, lower left and lower right panels correspond to r = 28.50, r = 29.25, r = 32.50 and r = 38.50, respectively (after
Roy & Musielak 2007c).
dY
1
= −σY − X1 X2 − X1 Y1 + Y Y1 + σZ2 ,
dτ
2
(40)
1
1
dY1
= −σb5 Y1 + X 2 − Y 2 ,
dτ
2
2
(41)
dY2
= −b6 Y2 + X1 Z2 − Y Z2 ,
dτ
(42)
dZ
= −b1 Z − rX + 2Y1 Z1 − Y Z2 ,
dτ
(43)
dZ1
= −b1 Z1 + rX2 − 2Y1 Z + X1 Z2 ,
dτ
(44)
dZ2
= −Z2 − rX1 + rY − 2X1 Y2 + 2Y Y2 + Y Z − X1 Z1 ,
dτ
(45)
where τ = (1 + 2k 2 )t, with k = kx = ky , b1 = 4(1 +
k 2 )/(1+2k 2 ), b2 = (1+k 2 )/2(1+k 2 ), b3 = 2(1−k 2 )/(1+
k 2 ), b4 = k 2 /(1 + k 2 ), b5 = 8k 2 /(1 + 2k 2 ) and b6 =
4/(1 + 2k 2 ).
Detailed numerical studies of this 9D model were done
by Reiterer et al. (1998), who showed that the system
had bounded solutions and that its route to chaos was
via the period-doubling cascade. Two plane projections
of the chaotic attractor of this system are plotted in Fig.
9. Since the second positive Lyapunov exponent becomes
positive when the value of r is greater than 43.3, the
results presented in Fig. 9 correspond to the range of
parameters for which the model exhibits hyperchaos.
To check whether this system conserves energy in the
dissipationless limit, we use Eqs (14) and (15), and extend them by adding summation over l. Since the potential energy approaches zero as σ → 0, we only calculate
the kinetic energy of the system and obtain
K=
1
C1 X 2 + C2 X12 + C3 X22 + C4 Y 2 + C5 Y12 (46)
2
where C1 , C2 , C3 , C4 and C5 are parameters that depend
on l, m and n (see Eq. 14). By taking the derivative with
15
FIG. 9: Two plane projections of the strange attractor generated by the 9D Lorenz system are plotted for σ = 0.5 and the
control parameter r = 45.0 (after Reiterer et al. 1998).
respect to time and using Eqs (37) through (41), one
finds that there are several terms on the RHS of these
equations, which do not explicitly depend on σ; those
terms will remain non-zero and they cannot cancel each
other when the limit σ → 0 is applied. Hence, we write
dE
6= 0 ,
(47)
dτ
where E = K + U . This shows that the total energy
of this 9D system is not conserved and, therefore, the
results on the onset of chaos and route to chaos obtained
for this system may not be valid.
Despite the above limitation of the model, the existence of hyperchaos in this model is a significant result
as it shows that hyperchaos can only be observed in 3D
convection but not in 2D convection; so far none of the
HD Lorenz models describing 2D convection was found
to exhibit hyperchaos (see Sec. III.C and D).
lim
σ→0
F.
Coupled 3D Lorenz models
Another way to construct HD Lorenz models is to
couple together several 3D Lorenz systems. Three such
systems were coupled together and modeled by electrical circuits that were used to collect experimental data
(Sanchez & Matias 1988, 1999). Motivated by the data,
Pazó et al. (2001) and Pazó & Matias (2005) developed
a theoretical model that consists of three 3D Lorenz systems coupled together unidirectionally; it is described by
the following set of nine first-order differential equations
dXj
= −σXj + σYj ,
dτ
(48)
dYj
= −Xj Zj + rX̄j − Yj ,
dτ
(49)
dZj
= Xj Yj − bZj ,
dτ
(50)
where j = 1, 2 and 3, X̄j = Xj−1 but with j 6= 1, and
the boundary condition X̄1 = X3 that is responsible for
the coupling; other notation is the same as that used in
the Lorenz equations (see Eqs 11, 12 and 13).
Numerical studies of this set of equations showed that
the system exhibits a high-dimensional strange attractor
whose dimension is greater than 4, and that the system’s
route to chaos resembles that observed in the 3D Lorenz
model. In addition, the system shows only one positive
Lyapunov exponent, which means that the origin of the
chaotic attractor is not associated with hyperchaos (see
Pazó & Matias 2005).
G.
Summary
The above results clearly show that HD Lorenz systems can be constructed by adding higher-order modes
to the original 3D Lorenz system. However, the mode selection must be guided by the requirement that the constructed systems have bounded solutions (Curry 1978),
and that they conserve energy in the dissipationless limit
(Treve & Manley 1982; Thiffeault & Horton 1996). In
general, there are three different methods of constructing HD Lorenz systems, namely, the method of vertical,
horizontal, and vertical-horizontal mode truncations.
The lowest-order HD Lorenz systems constructed by
each method range from 5D and 9D systems for the horizontal and vertical mode truncations, respectively, to
an 8D system for the third method (Roy & Musielak
2007a,b,c). This clearly demonstrates that the higherorder horizontal modes are more efficiently coupled to
the original modes selected by Lorenz (1963) than the
16
higher-order vertical modes. Moreover, both 9D and 8D
are more resistant to the onset of chaos than the 3D
Lorenz model; however, the 5D model is not. While the
route to chaos in the 5D and 9D models remains the
same as in the 3D Lorenz model, it changes from chaotic
transients to quasi-periodicity in the 8D model.
Among other discussed HD Lorenz models (see Table
1), four models do not conserve energy in the dissipationless limit, which means that these results must be taken
with caution.
There is also an interesting class of HD Lorenz models
that can be constructed by including shear flows in the
2D Rayleigh-Bénard convection. Among the models that
include shear flows (see Table 1), a 7D model developed
by Thiffeault (1995) and Thiffeault & Horton (1996) has
bounded solutions and it conserves energy in the dissipationless limit. Hence, this 7D model seems to be the
lowest-order HD Lorenz model of this class.
An important result is that all HD Lorenz models describing the 2D Rayleigh-Benard convection show only
one positive Lyapunov exponent. However, the two positive Lyapunov exponents observed in the 9D Lorenz system developed by Reiterer et al. (1998) to describe 3D
Rayleigh-Benard convection are indicators of hyperchaos
in this system. Again, this result must be taken with
caution because the 9D system does not conserve energy
in the dissipationless limit.
Another method of constructing HD Lorenz models is
to couple together several 3D Lorenz systems. A specific
model of three such systems coupled together was developed and studied both experimentally and numerically.
The obtained results showed that the model exhibits a
high-dimensional strange attractor, but its origin is not
associated with hyperchaos.
The HD Lorenz models constructed so far are important in investigating the effects caused by increasing the
number of degrees of freedom. The HD models show a
variety of phenomena typical for dissipative and driven
HD dynamical systems exhibiting chaos and, therefore,
they play a significant role in studying high-dimensional
chaos.
IV.
HIGH-DIMENSIONAL DUFFING SYSTEMS
A.
Duffing equation and HD oscillators
The general form of the Duffing equation (Duffing
1918) is
ẍ + αẋ + βx + γx3 = Bcos(ωt) ,
(51)
where α is the damping coefficient, β and γ are constant
coefficients of the nonlinear restoring force, and B and ω
are the amplitude and frequency of the periodic driving
force, respectively. A simplified version of this equation
can be considered by taking β = 0 (e.g., Ueda 1979;
Ueda et al. 1993; Benner 1997). In general, the systems
described by this equation with γ > 0 and either β > 0 or
β < 0 are called either single-well or double-well Duffing
oscillators (e.g., Moon 1987).
Studies of systems governed by the Duffing equation
showed that a number of long term stable and chaotic
motions exist, and that the period-doubling cascade is
the main route to chaos in these systems (e.g., Ueda
1980, 1991; Holmes 1979; Holmes & Holmes 1981; Novak
& Frehlich 1982; Thompson & Stewart 1986; Andronov
et al. 1987; Jordan & Smith 1990; Kahn 1990; Moon
1992; Murakami et al. 2003; Trueba et al. 2003; see
also Ivancevic & Ivancevic 2007, and references therein).
Recently, a complex Duffing system with random excitation was also considered and its relation to the nonlinear
Schrödinger’s equation was discussed (Xu et al. 2008).
To construct HD Duffing oscillators, single LD oscillators described by Eq. (51) are coupled together and
the resulting HD oscillators are investigated to determine
their onset of chaos, routes to chaos and regimes of fully
developed chaos. Many different systems have been studied, including coupled single-well and double-well Duffing
oscillators (e.g., Kozlowski et al. 1995; Savi & Pacheco
2002; Kenfack 2003), symmetrically and asymmetrically
coupled Duffing systems (e.g., Benner 1997; Musielak et
al. 2005a), and chains of coupled Duffing oscillators (e.g.,
Dressler 1988; Umberger et al. 1989; Dressler & Lauterborn 1990; Kapitaniak 1993). In the following subsections, the obtained results are described and discussed
with a special emphasis given to two, three, four, five
and six coupled Duffing oscillators.
B.
Two coupled oscillators
Two identical coupled single-well Duffing oscillators
driven by a periodic force were investigated by Kunick
and Steeb (1985), Kozlowski et al. (1995), Stagliano et
al. (1996), Benner (1997), Paul Raj & Rajasekar (1997,
1999), Yin & Dai (1998), Xu & Jing (1999), Kenfack
(2003), and Musielak et al. (2005). Different stable and
chaotic regimes were explored by changing the strength
of the driving force, and the routes to chaos were determined.
Comprehensive studies of two coupled, periodically
driven, single-well Duffing oscillators were performed by
Kozlowski et al. (1995), who used bifurcation and phase
diagrams to investigate the behavior of this system in
both stable and chaotic regimes, and they also determined the system’s route to chaos. The amplitude and
frequency of the driving force were used as control parameters. An interesting result is that a high-dimensional
chaotic attractor was found for relatively small amplitudes of the driving force. In addition, the perioddoubling cascade and quasi-periodicity were identified as
two main routes to chaos for this coupled Duffing system. The values of the control parameters for which one
of these routes dominates were established.
Two identical coupled double-well Duffing oscillators
17
driven by a periodic force were investigated by Kenfack
(2003). Taking β = −1 and γ = +1 in Eq. (51), and
introducing the coupling constant kc , the equations of
motion of these oscillators are given by
ẍ1 + αẋ1 − x1 + x31 − kc (x2 − x1 ) = Bcos(ωt) ,
ẍ2 + αẋ2 − x2 + x32 + kc (x2 − x1 ) = 0 ,
(52)
(53)
where both α and kc are fixed, and the amplitude B and
frequency ω of the driving force are used as the control
parameters.
The bifurcation structure of this system was studied by
varying the control parameters. The main result is that
the shape of the potential in the system has a profound
influence on the routes to chaos. In contrast to the results obtained for the single-well system, the double-well
system transitions to chaos suddenly, which may be identified as the crisis route to chaos (see below); this route to
chaos appears for a certain range of control parameters;
however, for other values of B and ω the period-doubling
cascade and quasi-periodicity are observed. According
to Kenfack (2003), the double-well system also exhibits a
great abundance of Hopf bifurcations and, interestingly,
many of them occur more than once for a given ω.
In studies performed by Benner (1997), two coupled
periodically-driven Duffing oscillators were considered. It
is assumed that the parameter β in Eq. (51) is zero,
which means that the system is described by a set of
equations similar to those given by Eqs (52) and (53),
except that the terms with x and y are not included. The
resulting system of autonomous equations (with y1 = x1 ,
y4 = x2 and y3 = ωt) is
ẏ1 = y2 ,
ẏ2 = −αy2 − y13 + kc (y4 − y1 ) + Bcos(y3 ) ,
(54)
(55)
ẏ3 = ω ,
(56)
ẏ4 = y5 ,
(57)
ẏ5 = −αy5 − y43 − kc (y4 − y1 ) ,
(58)
where ẏi = dyi /dt with i = 1, 2, 3, 4 and 5.
This system of equations was investigated by using phase portraits, power spectra, Poincaré sections,
and Lyapunov exponents (Benner 1997; Musielak et al.
2005a). Fixed values of α = 0.2, kc = 10, and ω = 1
were considered, and the range of the control parameter B was assumed to be 0 < B ≤ 150. The calculated
Lyapunov exponents were used to establish two chaotic
regions for the system. The first region is bounded by
B = 14.5 and B = 19, while the second region is limited
by B = 23 and B = 25.5. The system enters chaos in
FIG. 10: Poincaré sections for the variables (y1 −y2 ) and (y5 −
y4 ) are plotted in the upper and lower panels, respectively, for
the control parameter B = 19.0 (after Benner 1997).
the first region by the period-doubling cascade and resumes periodic motion when B becomes larger than 19.
The route out of chaos is crisis, as the strange attractor
collides with one of the unstable fixed points shown in
Fig. 10. The figure presents two different views of the
system’s chaotic attractor in Poincaré sections for the
variables (y1 − y2 ) and (y5 − y4 ) obtained for the control
parameter B = 19.0; the unstable points are marked by
’X’ in the lower panel of Fig. 10.
After taking the system out of chaos for 19 < B < 23,
it is also crisis that causes the system to enter the second
region at B = 23. The system remains chaotic until
B = 25.5 is reached. At this value, the return points of
the strange attractor begin to intersect with one of the
unstable fixed points and the system becomes periodic.
Hence, it is crisis again that allows the system to attain
periodic behavior. These results clearly show that the
crisis route to chaos is the dominant route for the two
coupled Duffing oscillators driven by a periodic force.
In the above system, the oscillators are coupled together by the linear coupling parameter kc and driven by
one periodic force whose amplitude B is the control parameter in numerical simulations. A more general system
of two Duffing oscillators coupled together by a nonlinear
coupling and driven by two periodic forces was consid-
18
ered by Savi & Pacheco (2002). The authors analyzed
the onset of chaos in this system by varying the amplitudes of the driving force and changing the parameter of
nonlinear coupling. Their results demonstrate the effects
of different nonlinear couplings and the transmissibility
of motion between the two oscillators.
The system of coupled periodically-driven Duffing oscillators considered by Benner (1997) and Musielak et
al. (2005a) was also modified by increasing kc from 10
to 1000, and adding one more periodic force acting on
another mass. It was assumed that the forces may have
different relative phasing and different frequencies. The
range of relative phasing was from 90o to 270o , and the
two frequency ratios 0.2 and 0.7 were considered. Although the increase of kc does not change the system’s
routes to chaos, it does make the chaotic regions bigger. In addition, the coupling strength of cubic order
causes an increase in the number of chaotic regions and
makes crisis to be the dominant route to chaos for these
regions. On the other hand, the system with the two
driving forces shows no chaotic behavior for the considered range of phasing and frequency ratios.
Similar studies of two coupled double-well Duffing oscillators driven by two periodic forces were performed by
Yin et al. (1998). Their investigation of the effects of
phase difference between the forces show that the phase
difference destroys the synchronization of the oscillators,
and that large phase differences prevent the system from
becoming chaotic. In addition, the authors studied the
effects of different coupling parameters kc on the system’s behavior. They demonstrated that large values of
kc do not lead to changes in the dynamics of the system. Clearly, these results are in agreement with those
discussed above (see also Benner 1997).
C.
Chains of coupled oscillators
The studies described in the previous subsection show
that the two coupled periodically-driven Duffing oscillators behave differently than a single Duffing oscillator
driven by a periodic force. To explore the behavior of HD
Duffing oscillators with even higher number of degrees
of freedom, chains of coupled Duffing oscillators were
considered (e.g., Dressler 1988; Umberger et al. 1989;
Dressler & Lauterborn 1990; Kapitaniak 1993; Benner
1997; Wei-Qi et al. 1999; Musielak et al. 2005a). The
main result is that chains of coupled Duffing oscillators
exhibit high-dimensional strange attractors and hyperchaos, and that their routes to chaos are also different
than those found in lower-order (two coupled) HD Duffing oscillators.
A HD system of fifteen coupled Duffing oscillators
driven by a Van der Pol oscillator was considered by
Dressler (1988), who demonstrated how to generalize
Hamiltonian dynamics to encompass this kind of dissipative and driven system. In the work of Dressler & Lauterborn (1990), a ring of coupled periodically-driven Duffing
oscillators was studied and Ruelle’s rotation number (Ruelle 1985) was calculated. It was shown that the system
preserves the essential features of Hamiltonian dynamics.
Another interesting chain of forced coupled Duffing oscillators was constructed by Umberger et al. (1989); in
the continuum limit, this system represents a nonlinear,
dissipative and driven wave equation. According to the
authors, this HD system shows the extreme intermittency
in its temporal behavior.
In the paper by Kapitaniak (1993), a chain of arbitrary number of unidirectionally coupled Duffing oscillators driven by a periodic force is used to investigate a
transition from chaos to hyperchaos. He shows that depending on the control parameters, the system’s behavior
can be either periodic or chaotic, or even hyperchaotic.
In addition, the obtained results demonstrate that the
clear distinction between the chaotic and hyperchaotic
regimes can only be made by using the correlation dimension; however, the Lyapunov dimension cannot be
used for this purpose (see Sec. II.B).
Comprehensive analysis of multiple coupled periodically driven Duffing oscillators was performed by Benner
(1997) and Musielak et al. (2005). In these studies, single
Duffing oscillators described by Eq. (51) with β = 0 were
linearly coupled to form chains of three, four, five and
six coupled oscillators. For each system, the phase portraits, Poincaré sections, power spectra, and Lyapunov
exponents were calculated for the control parameter B
ranging from 0 to 150.; all calculations were performed
with α = 0.2, kc = 10 and ω = 1. The value of B that
corresponds to the onset of chaos, the range of B for
which fully developed chaos is observed, and the system’s
routes to chaos were determined. We briefly describe the
obtained results.
The system of three linearly coupled Duffing oscillators
is asymmetric with respect to the location of the linear
coupling element (see Fig. 11). Introducing V /m = 1,
Kl /m = kc , c/m = α and P/m = B, the system is
described by the following set of autonomous equations
ẏ1 = y2 ,
(59)
ẏ2 = −αy2 −y13 +kc (y5 −y2 )+(y4 −y1 )3 +Bcos(y3 ) (60)
ẏ3 = ω ,
(61)
ẏ4 = y5 ,
(62)
ẏ5 = −α(y5 − y2 ) − (y4 − y1 )3 + kc (y6 − y4 ) ,
(63)
ẏ6 = y7 ,
(64)
ẏ7 = −αy7 − y63 − kc (y6 − y4 ) .
(65)
The above set of equations was solved numerically,
and the calculated Lyapunov exponents (see Fig. 12)
19
FIG. 11: Three linearly coupled periodically driven Duffing
oscillators of mass m. In addition, V is the coefficient of the
cubic stiffness, Kl is the linear coupling parameter, c is the
damping parameter, and P and ω is the amplitude and frequency of the driving force, respectively (after Benner 1997).
show that there are two chaotic regions and two regions
where quasi-periodic motion with three independent frequencies dominates. The first region is defined in the
range B = 25.5 − 28 where the system is not chaotic,
but rather stable quasi-periodic with three independent
frequencies; the motion is represented in phase space by
a stable 7D torus. In the second region, B varies from
63 to 72 and chaotic behavior is observed. The system
transitions to chaos via the crisis route and becomes periodic through the decay of an unstable three-frequency
torus, which is the quasi-periodic route to chaos. The
third region is spanned by B = 74.5 − 77 where the system is chaotic; it transitions to chaos via crisis, and the
crisis bifurcation events allow the system to return to
its periodic behavior. The fourth region extends from
B = 142.5 to B = 146.5, where the system undergoes a
three-frequency quasi-periodic motion.
A system of four coupled Duffing oscillators is symmetric with respect to the location of the linear coupling
element. According to Benner (1997) and Musielak et
al. (2005a), three intervals of B are important for this
system. The first interval is 14.5 − 17.5 where the system’s motion is quasi-periodic. In the second interval
the value of B ranges from 40.5 to 42. This small chaotic
region is created by an interrupted period-doubling cascade, and a crisis causes the period doubling sequence
to be discontinued. Crisis is also the route that allows
the system to return to periodic motion. The third interval is defined by the range B = 82 − 124 where there
are two 3 frequency torii that lead to chaos. They are
located on either side of the chaotic window that exists
for B = 91 − 102. An interesting result is that both torii
decay through a period that is three times the length of
the forcing period, which is consistent with the quasiperiodic scenario.
Similar studies were also performed for the five and
six coupled oscillator systems, which are asymmetric and
symmetric systems, respectively (Benner 1997; Musielak
et al. 2005a). For the asymmetric system, the primary
region of interest is B = 75 − 105. Fully developed chaos
is observed for B ≥ 85.7 with quasi-periodicity being the
route to chaos. The decay of the torus via periodic mo-
FIG. 12: Plots of Lyapunov exponents for the system of three
linearly coupled Duffing oscillators (after Benner 1997).
tion with periods two and four times of that of the forcing
function for this system are shown in Fig. 13. The presented results were obtained for the control parameter B
ranging from 80.5 to 85.7. In many aspects, the behavior
of this system is similar to that observed in the system
of three coupled oscillators.
Now, the symmetric system of six coupled Duffing
oscillators has one interesting region that is between
B = 72 and B = 95. In this region, a stable twofrequency torus forms at B = 73. The torus forms at
B = 73.0 decays through a period three times that of the
forcing period when B is increased to 74.15 (see Fig. 14);
chaotic motion commences at 76.6. This phenomenon is
similar to that observed in the four coupled oscillators
system. Hence, there are similarities between the behavior of three and five, and four and six oscillators; and
in general some similarities also exist between the asymmetric and symmetric systems (see below). However, as
shown in Figs 13 and 14 main differences remain in the
way the torus decays in these two types of systems. Since
both asymmetric and symmetric systems enter chaos via
quasi-periodicity, the two different processes that lead to
the decay of torus make clear distinction between these
two systems.
D.
Summary
Different HD Duffing oscillators were constructed by
coupling (linearly or nonlinearly) together Duffing oscillators and assuming that they are driven either by one or
two periodic forces. Typically, the amplitude of the driving force was used as a control parameter in analyzing
the behavior of the system. The main result is that the
number of chaotic regions in the HD Duffing systems is
greatly reduced when compared to the original 3D Duff-
20
FIG. 13: Poincaré sections showing the decay of the torus
via periodic motion with periods two and four times of that
of the forcing function for the system of five coupled Duffing
oscillators; the control parameter B ranges from 80.5 in the
most upper panel to 85.7 in the lowest panel (after Benner
1997).
ing system, and that crisis becomes the dominant route
to chaos when the number of degrees of freedom is increased. The latter is especially important for the three,
four, five and six coupled oscillators.
Extensive analysis of a system of two coupled periodically driven Duffing oscillators shows that the perioddoubling cascade and quasi-periodicity are their main
routes to chaos. The role of the crisis route was also
explored and it was demonstrated that this route is responsible for the system to return from its chaotic behavior to periodic. Studies of the system with two driving
forces of different phases and frequencies, and different
coupling parameters were also performed. The obtained
results show that the excitation forces of different phases
and different frequency ratios may lead to elimination of
chaos. On the other hand, an increase of the coupling
constant makes the chaotic regions bigger; however, it
does not change the system’s route to chaos. It must
also be noted that the coupling strength of cubic order
causes an increase in the number of chaotic regions, and
that crisis is the dominant route to chaos in these regions.
A common feature of both asymmetric (three and five)
and symmetric (four and six) HD systems is that crisis
is the dominant route to chaos for these Duffing cou-
FIG. 14: Poincaré sections showing the decay of the torus via
periodic motion with period three times of that of the forcing
function for the system of six coupled Duffing oscillators; the
control parameter B ranges from 73.0 in the most upper panel
to 76.6 in the lowest panel (after Benner 1997).
pled oscillators. In addition, for each system there is one
chaotic region for which quasi-periodicity is the route to
chaos; both asymmetric and symmetric systems form a
three-frequency torus. Now, an important difference between these two classes of dynamical systems is the way
the torus decays. For the asymmetric systems, the torus
always decays through a period two times that of the
forcing period; however, for the symmetric systems, the
torus always decays through a period three times that of
the forcing period. Obviously, additional studies of HD
Duffing oscillators with more degrees of freedom are necessary before these results can be generalized to chains
with many coupled Duffing oscillators.
The main result obtained so far for chains of coupled
Duffing oscillators is that high-dimensional strange attractors are formed and hyperchaos is observed. Actually, the behavior of the system can be either periodic or
chaotic, or even hyperchaotic depending on the values of
the control parameters. Another interesting result is that
for long chains of coupled Duffing oscillators, the routes
to chaos can be different than those found in lower-order
HD Duffing systems.
21
V.
OTHER HIGH-DIMENSIONAL SYSTEMS
A.
HD Rössler systems
The first hyperchaotic system with two positive Lyapunov exponents was proposed by Rössler (1979), who
modified his original 3D system (Rössler 1976) by adding
one extra variable; both 3D and 4D models describe particular chemical reactions (e.g., Kapitaniak et al. 2000).
Mathematically, the 4D system can be represented by the
following set of equations
ẋ = −y − z ,
(66)
ẏ = x + ay + w ,
(67)
ż = b + xz ,
(68)
ẇ = −cz + dw ,
(69)
where x, y and z are the variables of the Rössler 3D
system, w is the new variable, and a, b, c and d are
system’s parameters. For the fixed parameters a = 0.25,
b = 3.00, c = 0.50 and d = 0.05, the values of Lyapunov
exponents are λ1 = 0.112, λ2 = 0.019, λ3 = 0.0 and
λ4 = −25.188; since there are two positive exponents,
the system shows hyperchaos.
To construct other HD Rössler models, one may couple
two or more 3D Rössler systems (e.g., Harrison & Lai
1999, 2000; Yanchuk & Kapitaniak 2001; Lai et al. 2003)
or couple the 3D Rössler system to either the Van der
Pol oscillator or to the Lorenz model (e.g., Belykh et
al. 2008). In this section, we briefly describe the results
obtained for the former systems; however, we postpone
our discussion of the two latter systems to Sec. VII.
Two coupled non-identical Rössler systems were investigated by Harrison & Lai (1999, 2000), who computed all
six Lyapunov exponents and demonstrated that the second largest Lyapunov exponent passes through zero continuously. In addition, the authors showed that changes
of the Lyapunov dimension (see Sec. II.C) when the system transitions from chaos to hyperchaos are smooth.
A mathematical model of two coupled identical Rössler
systems was also developed by Yanchuk & Kapitaniak
(2001), using the following set of six first-order differential equations
FIG. 15: The variation of three largest Lyapunov exponents
versus the control parameter a plotted for the two coupled
Rössler systems (Yanchuk & Kapitaniak 2001).
ẏ2 = y1 + ay2 ,
(74)
ẏ3 = b + y3 (y1 − c) + d(y3 − x3 ) ,
(75)
where the sets of variables (x1 , x2 , x3 ) and (y1 , y2 , y3 ) describe the two Rössler systems, a, b and c are system
parameters, and d is the coupling coefficient.
The authors solved the above equations numerically
with b = 2.0, c = 4.0 and d = 0.25, and used a as a
control parameter. They showed that for a = 0.363 the
system has only one positive Lyapunov exponent (λ1 ≈
0.039); however, for a = 0.368 the system has two positive Lyapunov exponents (λ1 ≈ 0.040 and λ2 ≈ 0.002).
In both cases, the system’s attractor is chaotic and its 2D
projections are different. The three largest Lyapunov exponents are plotted as a function of a in Fig. 15, which
shows that there is a smooth transition from chaos to
hyperchaos at a ≈ 0.3673; this is in agreement with the
previous results obtained by Harrison & Lai (1999, 2000).
Extension of the above models to even higher dimensions was done by Lai et al. (2003). In this work,
general rules of constructing chains of coupled identical
chaotic oscillators were presented; chains of non-identical
chaotic oscillators were also discussed. Specifically, the
authors considered a chain of N-coupled Rössler systems
described by the following equations
ẋ1 = −x2 − x3 ,
(70)
ẋ2 = x1 + ax2 ,
(71)
ẋ3 = b + x3 (x1 − c) + d(y3 − x3 ) ,
(72)
ẏi = ωi xi + ayi ,
(77)
ẏ1 = −y2 − y3 ,
(73)
żi = b + zi (xi − c) ,
(78)
ẋi = −ωi yi − zi + ε(xi+1 + xi−1 − 2xi ) ,
(76)
22
where i = 1, 2, 3, ..., N , ωi is the mean frequency of the
ith system, ε is the coupling parameter, and a, b and c
are the parameters of the individual Rössler model.
In the work by Lai et al. (2005), the specific model of
three coupled Rössler systems (N = 3) was considered
by taking a = 0.165, b = 0.2, c = 10.0 and ωi = ω = 1.0,
and by varying ε. For this set of parameters, each individual Rössler system is chaotic and has one positive
Lyapunov exponent. Hence, if the coupling is weak with
a very small ε, then the model has three positive Lyapunov exponents and shows HD chaos. However, when
the coupling is increased so that ε becomes larger than
εc ≈ 0.065, then the model exhibits chaotic behavior with
only one positive Lyapunov exponent. It is also shown
that the onset of chaos for ε ≥ εc coincides with the
onset of stable synchronization among the three original
Rössler models (see Sec. VII).
B.
Other coupled oscillators
Among many well-known nonlinear oscillators, the Van
der Pol oscillator (see Van der Pol 1922) is one of the
most important because of its numerous applications in
mechanics, electrical engineering, and in studies of the
human heart (e.g., Hayashi 1985; Thompson & Stewart
1986; Lakshmanan & Murali 1996). The Van der Pol
system is considered to be one of the prototypes of all LD
systems that exhibit low-dimensional chaos. Our interest
here is in HDC, so we describe the results for two coupled
Van der Pol oscillators obtained by Poliashenko et al.
(1991a, b) and Poliashenko & McKay (1992), and the
results for coupled Van der Pol and Duffing oscillators
presented by Woafo et al. (1996).
Following Poliashenko & McKay (1992), we write equations of motion for two coupled oscillators in the following
general form
ẍ1 − µ1 F1 (x1 , x2 )ẋ1 + ω12 G1 (x1 , x2 )x1 = Kx2 ,
(79)
ẍ2 − µ2 F2 (x1 , x2 )ẋ2 + ω22 G2 (x1 , x2 )x2 = Kx1 ,
(80)
where Fi (x1 , x2 ) = 1 − νi x21 − ρi x22 and Gi (x1 , x2 ) =
1 − δi x21 − σi x22 with i = 1 and 2. In addition, νi , ρi ,
δi , σi and ωi are parameters, and K is the coupling coefficient. Note that the equations account for the dependence of oscillation frequency on amplitudes, the socalled nonisochronism; however, the system described by
these equations is not driven. In order to obtain the two
independent original Van der Pol oscillators, one must
take ρi = δi = σi = 0, ωi = 1 and K = 0.
An interesting method to study the HD system described by the above equations was presented by Poliashenko & McKay (1992). The authors assumed that
the oscillators are weakly coupled and weakly nonlinear,
and followed Van der Pol (see Van der Pol 1922) to write
solutions of Eqs (79) and (80) as
x1 (t) = a1 (t) cos[ωt + φ1 (t)] ,
(81)
x2 (t) = a2 (t) cos[ωt + φ2 (t)] ,
(82)
where a1 (t) and a2 (t) are the amplitudes and φ1 (t) and
φ2 (t) are the phases of the two oscillators, and ω = (ω1 +
ω2 )/2. Using these solutions and the first-order averaging
methods (e.g., Guckenheimer & Holmes 1984), Eqs (79)
and (80) can be written as
ȧ1 = (α1 − γ1 a21 − η1 a22 )a1 + ka2 sinψ ,
(83)
ȧ2 = (α2 − γ2 a22 − η2 a21 )a2 − ka1 sinψ ,
(84)
ψ̇ = ∆ + βa21 − κa22 + k
a2
a1
−
a1
a2
cosψ ,
(85)
where αi = µi /2 and γi = µi αi /4 are parameters of the
linear and nonlinear damping, respectively, ηi represents
the nonresonant coupling, k = K/2ω is the coupling coefficient, β and κ describe nonisochronous features of the
oscillators, ∆ = ω1 − ω2 is the detuning of the oscillator’s
frequencies, and the phase difference ψ = φ2 − φ1 , which
is responsible for the resonant coupling.
Different dynamical properties of the coupled Van der
Pol oscillators can be investigated by using Eqs (83), (84)
and (85). Analytical and numerical studies of these equations by Poliashenko & McKay (1992) and Poliashenko
et al. (1991a, b) determined sets of the system’s parameters that lead to chaotic behavior. The shape of the
resulting strange attractor was established, and it was
shown that this attractor is developed by the so-called
Shilnikov mechanism (Shilnikov 1970). In this mechanism, a strange attractor is formed by a countable set
of periodic trajectories that are located in the vicinity
of a homoclinic orbit. The obtained results also demonstrate that for some range of the parameters the system
exhibits three-frequency oscillations, which are not expected in this type of weakly-coupled systems.
A special system without nonresonant coupling, η1 =
η2 = 0, was considered by Poliashenko et al. (1991b),
who solved Eqs (83), (84) and (85) by taking α1 = −1.3,
α2 = 1.5, γ1 = γ2 = 0.3, β = 10, κ = 0.3, k = 6.5
and two different values of ∆, namely, 17.0 and 17.5.
Two phase portraits corresponding to the above values
of the parameters are presented in Fig. 16, which shows
that for ∆ = 17.0 the system is chaotic; however, for
∆ = 17.5 the system is characterized by a four-cycle motion. Additional increase of ∆ transforms the four-cycle
into another chaotic attractor. The process is repeated
for higher values of ∆, so that a sequence of transformations from chaotic attractors to three and four-cycles, and
vice versa, is formed. Most of these transformations are
caused by the period-doubling cascade, but sometimes
intermittency is also observed.
Another HD oscillator can be constructed by coupling
together Van der Pol and Duffing oscillators. As shown
by Woafo et al. (1996), such system is described by the
following set of equations
ẍ − ε1 F (x)ẋ + G(x) = c1 y + c2 ẏ ,
(86)
23
rameters. The route to chaos for this system was not
formally determined; however, the obtained results indicate that chaos arises suddenly, which may imply that
crisis is the route to chaos.
C.
HD systems with new states
Another way to construct a HD system is to add a new
state to one of the known 3D systems. An important
example of such a HD model is the 4D Rössler model
introduced by Rössler (1979). In this model, a new state
represented by the variable w is added to the original 3D
Rössler model (see Eqs 66 through 69) and, as a result,
the system exhibits hyperchaos (see Sec. V.A).
The same method was used to construct several other
HD systems. Here, we briefly describe studies of a generalized system (Celikovsky & Chen 2002; Li et al. 2005),
a modified canonical Chua’s circuit (Chua 1992, 1994;
Chua & Lin 1990, 1991; Chua et al. 1993; Thamilmaran
et al. 2004), and a non-autonomous system with continuous periodic switch (Lu & Wu 2004; Wu et al. 2007).
According to Li et al. (2005), a generalized 3D system
is described by the following equations
 
 

ẋ
x
a11 a12 0
 ẏ  =  a21 a22 0   y 
ż
z
0 0 a33

 
0 0 0
x
+ x  0 0 −1   y  ,
0 1 0
z
FIG. 16: The phase portraits for the system of two coupled
Van der Pol oscillators without nonresonant coupling; (a) ∆ =
17.0 and (b)∆ = 17.0 (after Poliashenko et al. 1991b).
ÿ − ε2 G(y)ẏ = c1 x + c2 ẋ ,
(87)
where F (x) = 1 − x2 , G(x) = ω12 x + α1 x3 and H(y) =
ω22 y+α2 y 3 . In addition, ω1 and ω2 are natural frequencies
of the oscillators, α1 and α2 are nonlinearity coefficients,
and c1 and c2 are the elastic and damping coupling coefficients, respectively.
The authors used the multiple scale technique (Nayfeh
& Mook 1979) to obtain analytical solutions for the resonant and nonresonant cases. These solutions show that
the coupling generates modulations of the amplitudes.
Shilnikov’s theorem (see Shilnikov 1970) was used to determine the range of the coupling parameters that leads
to chaotic behavior. Verification of this theoretical prediction by numerical simulations demonstrated that there
is a difference between the theory, which is based on small
values of the system’s parameters, and the numerical results that were obtained without any limits on the pa-
(88)
where aij are constants. The system is called generalized because by choosing the appropriate constants, the
above set of equations can describe either the Lorenz
model (Lorenz 1963), the Chen system (Chen & Ueta
1999), the Lu system (Lu & Chen 2002), or some other
3D dynamical systems.
Studies of the above equations by Li et al. (2005)
showed that a chaotic attractor is obtained when a11 =
−a12 = −35, a21 = 7, a22 = 12 and a33 = −3. Since this
generalized system is low-dimensional, there is only one
positive Lyapunov exponent; detailed discusion of the onset of chaos and chaotic behavior of the system is given
by Celikovsky & Chen (2002).
Any generalized 3D system described by Eq. (88) can
be extended to higher dimensions by adding new states.
Li et al. (2005) added one new state denoted as u and
obtained a 4D generalized system described by
 

 
x
a11 a12 0 0
ẋ
 a21 a22 0 1   y 
 ẏ 
 ż  =  0 0 a 0   z 
33
u
−k 0 0 0
u̇
0
0
+x 
0
0

0
0
1
0
0
−1
0
0
 
0
x
0  y 
,
0  z 
0
u
(89)
24
FIG. 17: Theoretical (the left six panels) and experimental (the right six panels) phase portraits of a hyperchaotic system (see
the main text). For the theoretical results: (a) the x − z plane, (b) the y − z plane, (c) the x − y plane, (d) the x − u plane, (e)
the z − u plane, and (f) the x − z − u space. For the experimental results, the same as for and the theoretical results except
(e) the y − u plane and (f) the z − u plane (after Li et al. 2005).
where k is the control parameter.
Numerical studies of this system were performed for
the values of a11 = −a12 = −35, a21 = 7, a22 = 12,
a33 = −3, and the following range of the control parameter 0 < k ≤ 50. It was shown that the system exhibits
hyperchaos when 0 < k ≤ 21.84 and 37.44 ≥ k ≤ 41.84.
In addition, there are four chaotic regions, two quasiperiodic regions and four periodic regions (for details,
see Li et al. 2002). Examples of selected phase portraits
computed for this system are presented in Fig. 17.
To verify their numerical results, Li et al. (2005) designed an electrical circuit that corresponds to the system described by Eq. (89) and produced experimental
phase portraits. Some of these portraits are given in Fig.
17, and they can be compared to the theoretical phase
portraits presented in the same figure. The comparison
shows many similarities between the theoretical and experimental phase portraits, and thereby it verifies the
numerical results.
We now discuss another class of HD systems that are
constructed by using Chua’s circuit, which is an important 3D system originally introduced by Leon O. Chua in
1983 (see Chua et al. 1986; Chua 1992). The Chua circuit
is a modified RLC circuit (Chua 1994, 2005; Chua & Lin
1990, 1991; Chua et al. 1993), which consists of a linear
inductor L, a linear resistor R, two linear capacitors C1
and C2 , and a nonlinear resistor NR ; this is the simplest
electronic circuit that exhibits chaos. There are three
Chua’s equations, and their numerical studies as well as
experimental verification are described in many papers
(e.g., Matsumoto 1984; Zhong & Ayrom 1985; Chua et
al. 1986; Chua & Lin 1991; Chua et al. 1993; Kyprianidis et al. 1995; Cafagna & Grassi 2003; Chua 2005).
Two 3D Chua’s circuits were coupled together to form a
HD circuit whose chaotic and non-chaotic behavior was
explored by Kapitaniak & Chua (1994). Another HD
system was considered by Kyprianidis et al. (2000), who
developed a 4D model of a nonlinear electrical circuit.
Here, our interest is in the so-called modified canonical
Chua’s circuit introduced by Thamilmaran et al. (2004).
To construct this circuit, a new state representing an
additional linear inductor was added to the 3D Chua circuit. Hence, the modified 4D Chua’s circuit consists of
two linear inductors, L1 and L2 , in addition to all other
elements of the original 3D Chua circuit. The modified
circuit is also called canonical because it is capable of
25
resistor R was used as the control parameter. The fact
that this HD circuit exhibits both chaotic and hyperchaotic behavior is shown by the experimental results
presented in Fig. 18. These results are consistent with
those obtained from numerical simulations. The main observation is that the system transitions to hyperchaos via
period-three doubling bifurcations; the effects of quasiperiodicity and intermittency were also noted.
We conclude this subsection with the non-autonomous
system with continuous periodic switch proposed by Lu
& Wu (2004) and then considered by Wu et al. (2007).
This system is described by the following set of the firstorder differential equations
ẋ = [25sin2 (ωt) + 10](y − x) ,
(95)
ẏ = [28 − 35sin2 (ωt)]x − xz + [29sin2 (ωt) − 1]y , (96)
1
ż = xy − [sin2 (ωt) + 8]z ,
3
FIG. 18: Experimentally observed phase portraits (the uppermost panels), Poincaré sections (the middle panels) and power
spectra (the lowest panels) for modified canonical Chua’s circuit; the left and right panels present the system’s chaotic
and hyperchaotic attractor, respectively (after Thamilmaran
et al. 2004).
realizing the behavior of every member of the Chua’s
family. The equations describing this 4D circuit are
C1 V̇1 = IL1 − G(V1 ) ,
(90)
C2 V̇2 = G1 V2 − IL1 − IL2 ,
(91)
L1 I˙L1 = V2 − V1 − RIL1 ,
(92)
L2 I˙L2 = V2 ,
(93)
where ω is an adjustable parameter. Despite only three
differential equations, this non-autonomous system is actually equivalent to a 4D autonomous system. Hence,
the system has four Lyapunov exponents and possibility for hyperchaos. It has also another interesting property, namely, when t is increased the system continuously
switches among different chaotic models, including the
Lorenz, Chen and Lu systems (Lorenz 1963; Chen & Ueta
1999; Lu & Chen 2002).
Studies of this system demonstrated that it exhibits
hyperchaos when the control parameter ω ranges from
0.05 to 2.7; for other values of ω, the system is either
chaotic or periodic. Routes to chaos and hyperchaos were
not studied for this system because of its complicated behavior caused by the continuous switching between different chaotic systems. According to the authors, this
system has great potential to find broad applications in
various chaos-based information systems.
D.
where G1 is negative conductance, V1 and V2 are voltages across the capacitors C1 and C2 , and IL1 and IL2
denote the currents through the inductances L1 and L2 ,
respectively. In addition, the function G(V1 ) represents
the nonlinear resistance and it is given by
G(V1 ) = Gb V1 +0.5(Ga −Gb )[|V1 +Bp |−|V1 −Bp |] , (94)
where Ga , Gb and Bp are coefficients of the nonlinear
resistance.
For numerical and experimental studies, the following
system parameters were chosen: C1 = 7.0 nF, C2 = 15.0
nF, L1 = 31.0 mH, L2 = 795.0 mH, G1 = −0.45 mS,
Ga = −0.105 mS, Gb = 7.0 mS, and Bp = 0.68 V. The
(97)
HD maps
The main focus of this review paper is on dissipative
and driven HD dynamical systems that are described by
differential equations. Clearly, 1D and 2D maps are not
such systems. Nevertheless, as discussed in Sec. II, they
have played important roles in establishing the field of
nonlinear dynamics and chaos, and some of these maps
may become Poincaré sections of known 3D dynamical
systems. The 2D Hénon map is such an example, as it is a
Poincaré section of the 3D Lorenz system (e.g., Benedicks
& Carleson 1991). It remains to be determined whether
any of HD maps discussed below may play similar role
for dissipative and driven HD dynamical systems.
As stated in Sec. II, discrete 1D and 2D maps may
show low-dimensional chaos. The most important examples of such LD systems are the 1D logistic map (May
26
FIG. 19: Lyapunov exponents of the two linearly coupled logistic maps are plotted versus the control parameter r1 (after
Harrison & Lai 2000).
1976, 1980) and the 2D Hénon map (Hénon 1969, 1976).
To construct HD maps, one has to couple together at
least two 1D maps or two higher-order maps. This means
that two coupled logistic maps will form a 2D system
that will be considered here a HD map. Similarly, by
adding a new variable to the 2D Hénon map one obtains
a 3D Hénon map, which is considered here to be a HD
system. Several interesting HD maps were constructed
and their chaotic and hyperchaotic behaviors were investigated (e.g., Ikeda 1979; Hammel et al. 1985; Kaneko
1992; Baier & Klein 1990; Stefanski 1998; Lai 1999; Harrison & Lai 2000; Sprott 2006; Ivancevic & Ivancevic
2007).
We begin with two linearly coupled logistic maps studied by Kaneko (1992) and described by the following
equations
xn+1 = r1 xn (1 − xn ) + εyn ,
(98)
yn+1 = r2 yn (1 − yn ) + εxn ,
(99)
where r2 = 3.5, ε = 0.05, and r1 is the control parameter. This system transitions to chaos at r1 ≈ 3.175 and
to high-dimensional chaos (HDC) at r1 ≈ 3.41 (Harrison
& Lai 2000). As seen in Fig. 19, the transition to chaos
is not smooth (the first Lyapunov exponent is not continuous); however, the transition to HDC is smooth and the
second Lyapunov exponent is continuous. The smoothness of the transition to HDC is also shown by plotting
the Lyapunov dimension versus the control parameter r1
(see Harrison & Lai 2000).
A generalized Hénon map given by
xn+1 = a − yn2 − bzn ,
(100)
yn+1 = xn ,
(101)
zn+1 = yn ,
(102)
was introduced by Baier & Klein (1990), who demonstrated that the map has two positive Lyapunov exponents for a = 3.4 and b = 0.1. Other values of the
parameters for which the map exhibits hyperchaos and
chaos were also determined.
Other studies of generalized Hénon maps include the
work done by Stefanski (1998), who designed two such
maps. One map was designed in such a way that it produces a chaotic attractor with topological dimension 1,
and the other map was required to have a chaotic attractor with topological dimension 2. As a result, the first
map has one positive Lyapunov exponent and exhibits
chaos, while the second map with two positive Lyapunov
exponents exhibits hyperchaos. Both maps are 3D Hénon
maps, the first one with expansion in 1 and contraction
in 2 dimensions, and the second map with contraction
in 1 and expansion in 2 dimensions. The obtained results show that despite obvious differences between the
chaotic and hyperchaotic attractors, there are some inherent similarities between chaos and hyperchaos, at least
for 3D maps of this kind.
The time-delayed Hénon map was investigated by
Sprott (2006). The map is described by the following
equation
xn = 1 − ax2n−1 + bxn−d .
(103)
where a and b are parameters. Note that for d = 2 this
map becomes the 2D dissipative Hénon map that shows
chaos for a = 1.4 and b = 0.3. In addition, for d = 1 the
map is equivalent to the 1D logistic map.
The time-delayed Hénon map was studied by Sprott
(2006), who choose b = 0.1, used a as the control parameter, and considered two different cases with d = 2
and d = 100; while the former represents a LD system,
the latter clearly corresponds to a HD map. For both
maps, regions of different dynamical behavior and the
corresponding attractors were established, and routes to
chaos were also determined. The results presented in Fig.
20 show that the LD map has only one positive Lyapunov exponent, and that two positive exponents exist
for the HD map. As expected, the LD map transitions
to chaos via the period-doubling cascade; however, the
quasi-periodic route seems to dominate in the HD map.
The time-delayed Hénon map provided an opportunity
to investigate global bifurcations and multiple attractors.
A method for identifying such bifurcations was described
and used to demonstrate the existence of a large number
of coexisting attractors near the system’s onset to chaos,
which may be a new route to chaos in HD maps (Sprott
2006).
In the coupled HD maps discussed above the coupling
was linear. To study the effects of nonlinear coupling, Lai
(1999) and Harrison & Lai (2000) considered two nonlinearly coupled Ikeda maps. Since the Ikeda map describes
27
FIG. 20: Lyapunov exponents of the time-delayed Hénon map are plotted versus the control parameter a for d = 2 and d = 100
in the left and right panels, respectively (after Sprott 2006).
the dynamics of an idealized optical ring cavity (Ikeda
1979; Hammel et al. 1985), the coupled Ikeda maps represent two mutually coupled optical cavities. The discrete equations describing the two nonlinearly coupled
Ikeda maps are
(1)
xn+1 (1) = a + b[xn (1)cosφ(1)
n − yn (1)sinφn ] ,
(104)
(1)
yn+1 (1) = b[xn (1)sinφ(1)
n + yn (1)cosφn ] ,
(105)
(2)
xn+1 (2) = a + b[xn (2)cosφ(2)
n − yn (2)sinφn ] ,
(106)
(2)
yn+1 (2) = b[xn (2)sinφ(2)
n + yn (2)cosφn ] ,
(107)
with
p
φ(1)
n = k−
1+
φ(2)
n = k−
p
2πε[xn (1) − xn (2)] , (109)
1 + x2n (1) + yn2 (1)
x2n (1)
+ yn2 (1)
2πε[xn (2) − xn (1)] , (108)
and where a, b, k and p are parameters of the Ikeda map,
and ε is the modulation strength.
The system was studied numerically by choosing the
following parameter setting: a = 0.85, b = 0.9, k =
0.4 and p = 5.8. The range of ε, which is the control
parameter, is 0 ≤ ε ≤ 0.5. The obtained results show
that the system transitions from chaos to hyperchaos at
ε ≈ 0.185 (Harrison & Lai 2000).
As the final example of a HD map described in this
subsection, we consider the Zaslavsky map (see Zaslavsky
1978) coupled to the logistic map through the following
set of equations
1 − e−a
yn mod(2π) ,
(110)
xn+1 = xn +
a
yn+1 = e−a yn + k sin(xn+1 + 2πzn ) ,
zn+1 = rzn (1 − zn ) + εyn ,
(111)
(112)
where a, k and r are parameters. Note that the original
Zaslavsky map is obtained when one takes zn = 0 in
Eqs (110) and (111); studies of the Zaslavsky map with
zn being a random variable were performed by Yu et al.
(1990, 1991).
The above set of equations was originally introduced
by Harrison & Lai (2000), who refer to this system as the
chaotically driven Zaslavsky map. They solved the equations numerically for k = 0.5, ε = 0.01, 0.05 ≤ r ≤ 0.30
and 3.55 ≤ r ≤ 3.90. It was shown that the system
exhibits periodic, chaotic and hypechaotic behavior depending on the range of the parameters a and r (see
Fig. 21). The authors conclude that the system can be
viewed as being composed of two subsystems; once one of
these subsystems becomes chaotic via one of the routes
to chaos known for the LD systems, then the second subsystem becomes also chaotic as a result of the driving
chaotic force imposed by the first subsystem. This scenario can be considered as a new route to chaos in HD
systems.
E.
Summary
An important method to construct HD systems was
introduced by Rössler (1979), who added a new state to
his original 3D model and constructed a 4D system that
exhibits hyperchaos. Other HD systems that were constructed by using this method include the HD Lorenz,
Chua, Chen and Lu systems. Numerical, and in some
cases experimental, studies of these HD systems show
28
FIG. 21: (a) Two Lyapunov exponents λ1 and λ2 of the Zaslavsky map coupled to the logistic map are plotted versus the
parameter p that is given in terms of a and r; (b) The Lyapunov exponent λ2 is plotted as a function of p, with p being in the
range for which transition to high-dimensional chaos is observed (after Harrison & Lai 2000).
a clear distinction between chaotic and hyperchaotic attractors. Interesting results regarding routes to chaos in
HD systems were obtained for a modified canonical Chua
circuit. It was shown that this system transitions to hyperchaos via period-three doubling cascade, and some
effects of quasi-periodicity and intermittency were also
observed.
Using the above method, one may add a continuous
periodic switch, which is a new state, to a 3D dynamical
system. Because the switch depends explicitly on time,
the 3D system becomes non-autonomous, which means
that it is equivalent to a 4D autonomous system. An interesting property of this system is that it continuously
switches between many chaotic systems, including the
Lorenz, Chen and Lu systems. There is a range of control parameters for which the system exhibits hyperchaos;
however, for other ranges the system is either chaotic or
periodic.
Generalized Hénon maps represent another class of HD
systems constructed by adding new states to the original
Hénon map. Studies of these maps show that despite
obvious differences between chaotic and hyperchaotic attractors for these systems, there are also some inherent
similarities between chaos and hyperchaos. Another example is the time-delayed Hénon map, which shows hyperchaos and transitions to its hyperchaotic behavior via
quasi-periodicity. The map also displays a large number
of coexisting attractors just before the onset to chaos;
this may be a new route to chaos in HD maps.
All other HD systems described in Sec. V are constructed by coupling together different LD systems. Specific models included two and three linearly coupled 3D
Rössler models, two coupled Van der Pol oscillators, coupled Van der Pol and Duffing oscillators, two coupled
together logistic maps, two nonlinearly coupled Ikeda
maps, and coupled together Zaslavsky and logistic maps.
Studies of HD Rössler systems show that there is always a range of control parameters for which the systems
exhibit hyperchaos, and that the transition from chaos to
hyperchaos is always smooth. Analytical and numerical
studies of two weakly nonlinear and weakly coupled Van
der Pol oscillators demonstrate that the system’s chaotic
attractor develops by the Shilnikov mechanism, and that
there is a sequence of transformations from chaotic attractors to three and four-cycles, and vice versa. For the
coupled Van der Pol-Duffing oscillators, crisis seems to
be the route to chaos.
Hyperchaos is also displayed by the Zaslavsky map
coupled to the logistic map, where the latter was used
as a chaotic driver. For each map, the value of the control parameter at which the system transitions from chaos
to hyperchaos was established. Plots of Lyapunov exponents for this system demonstrate that the transition is
always smooth. Furthermore, detailed numerical studies of the coupled system indicate that it can be viewed
as being composed of two subsystems. Once one of these
subsystems becomes chaotic via one of the routes to chaos
known for LD systems, then this already chaotic subsystem drives the other one to become chaotic as well; this
scenario can be considered as a new route to chaos in HD
systems.
VI. GENERAL RULES OF
HIGH-DIMENSIONAL CHAOS
One of the most challenging problems in the field of
nonlinear dynamics and chaos is to formulate general
rules that govern high-dimensional chaos (HDC). We now
address this problem by making comparison between HD
and LD systems.
The comparison is justified by the fact that all HD systems discussed here were constructed by either adding
additional modes or new states to the LD systems, or
coupling together identical or different LD systems. Since
each HD system constructed this way has the corresponding LD system as its subset, the comparison between HD
and LD systems can be made and the resulting differ-
29
ences can be used as a guide to formulate general rules
for HDC. However, it must be kept in mind that these
rules are only valid for dissipative and driven dynamical systems and that by no means they are final. It is
likely that these rules will require modification or even
re-formulation once more complicated HD systems are
constructed and their chaotic behavior is investigated.
We begin with two important rules for the constructing
HD systems from LD systems. These rules are:
(i) Any HD system constructed either by adding higherorder modes or new states to a LD system, or by coupling
LD systems together, must obey the laws of physics and
its solutions must be bounded;
(ii) Any HD Lorenz system constructed by adding higherorder modes to the original Lorenz model must conserve
energy in the dissipationless limit.
The above validity criteria must be obeyed by all HD
systems. Rule (i) must be obeyed by all systems that
describe real physical processes. However, this rule may
not apply to models of economic or social systems for
which behavior laws are not known. As already stated in
Sec. III.B, the rule (ii) is critical in constructing physically meaningful HD Lorenz systems. Several HD Lorenz
systems that do not obey rule (ii) were described and discussed in Sec. III. C, D and E, where we emphasized that
the results obtained must be taken with some reservation.
All HD systems discussed in this paper can be divided
into two groups. One group consists of the HD systems with one positive Lyapunov exponent and a highdimensional chaotic attractor. The HD systems with
both chaotic (one positive Lyapunov exponent) and hyperchaotic (two or more positive Lyapunov exponents)
behaviors belong to the other group. The distinction
between these two groups is taken into account in the
following list of general rules of HDC:
(1) HD systems are more resistant to the onset of chaos
than their corresponding LD systems;
(2) The number of chaotic regions in a HD system can be
greatly reduced when compared to the number of chaotic
regions observed in the corresponding LD system;
(3) The chaotic behavior of a HD system can be eliminated by selecting the excitation forces of different phases
and different frequency ratios;
(4) HD systems transition to fully developed chaos via
routes that may be different than those observed in LD
systems;
(5) Different routes to chaos are observed in HD systems
with different number of degrees of freedom;
(6) HD systems with dimensions as low as 4D may exhibit
hyperchaos;
(7) The number of positive Lyapunov exponents increases
when the system’s dimension is increased;
(8) Transition from chaos to hyperchaos is smooth in HD
dynamical systems and HD maps;
(9) Routes to chaos and hyperchaos are usually different.
(10) Chaos and hyperchaos become dynamically persistent in HD systems with sufficiently high dimensions.
The results presented in Sections III, IV and V demonstrate that the values of control parameters for which the
onset of chaos is observed in HD systems are typically
higher than those required for the corresponding LD systems. This resistance to the onset of chaos in HD systems
is caused by their higher number of physical degrees of
freedom. For the same reason, the number of chaotic regions in a HD system is much smaller than that observed
in the LD system. Moreover, the chaotic behavior of a
HD system can be eliminated by the excitation forces of
certain phases and frequencies of certain ratios (see Sec.
IV.B). The basic five routes to chaos identified for the LD
systems (see Sec. II.C) are also observed in HD systems.
Let us now concentrate on the HD systems characterized by one positive Lyapunov exponent and a highdimensional chaotic attractor with multiple zero Lyapunov exponents. In a typical scenario, the routes to
chaos of a HD system and its corresponding LD system are not the same. In general, different routes to
chaos are observed in HD systems with different number
of degrees of freedom. Examples of these systems include the 8D Lorenz system described in Sec. III. D, and
the symmetrically and asymmetrically coupled Duffing
oscillators discussed in Sec. IV.C; for these HD systems
quasi-periodicity and crisis are the dominant routes to
HDC.
High-dimensional systems that exhibit hyperchaos
range from the 9D Lorenz model describing the 3D
Rayleigh-Benard convection (see Sec. III.E) and chains
of coupled Duffing oscillators (see Sec. IV.C) to the 4D
Rössler system and other HD dynamical systems and HD
maps discussed in Sec. V. The obtained results show that
transition from chaos to hyperchaos is smooth in these
systems, and that the number of positive Lyapunov exponents increases when the system’s dimension is increased.
The proposed routes to HDC in HD systems with
hyperchaos include a period-three doubling cascade
(Thamilmaran et al. 2004), a bifurcation that is mediated by an infinite number of unstable periodic orbits
(Kapitaniak et al. 2000), coexistence of many attractors
just before the onset of chaos (Sprott 2006), a sequence
of transformations from chaotic attractors to three and
four-cycles and vice versa (Poliashenko et al. 1991b),
formation of a 2D torus and then a 3D torus whose decay through quasi-periodicity triggers HDC (Pazo et al.
2001), and a two-step general route that first takes a HD
system from its regular behavior to chaos and then from
chaos to hyperchaos (Harrison & Lai 1999).
Another route to HDC was suggested by Harrison &
Lai (2000), who studied the Zaslavsky map coupled to
the logistic map, and concluded that this system can be
viewed as being composed of two subsystems. Once one
of these subsystems becomes chaotic via one of the routes
to chaos known for LD systems, then this already chaotic
subsystem drives the other one to become chaotic as well.
According to the authors, this scenario could be a new
30
route to HDC in HD systems.
From a theoretical point of view, new routes to chaos
in HD systems are expected and theoretical ideas range
from the so-called direct transition to hyperchaos without
intermediate chaos (e.g., Pazo & Mathias 2005) to routes
as exotic as torus-doubling (instead of period-doubling)
originally proposed by Yang (2000). However, a classification of routes to chaos in HD systems has yet to be
established.
Interesting studies of dynamical persistence in HD systems were performed by Albers et al. (2005), who defined dynamical persistence as the system’s behavior that
does not change with functional perturbation or parameter variation. The main result of an extensive statistical survey is that chaos is dynamically persistent in HD
systems if their dimensions become sufficiently high. In
other words, the parameter space in which chaos and hyperchaos become persistent increases in size when the dimension of HD systems is increased. Another important
effect is that the number of positive Lyapunov exponents
also increases monotonically with increasing the dimension of HD systems.
VII.
A.
CONTROL AND SYNCHRONIZATION OF
HIGH-DIMENSIONAL CHAOS
Overview of basic ideas and techniques
In the theoretical and experimental work on nonlinear HD systems described so far, the main objective
was to discover the chaotic and hyperchaotic behavior in
these systems, establish a range of physical parameters
for which this behavior is observed, and determine their
routes to chaos and hyperchaos. Now, it is also interesting to address how or if these chaotic and hyperchaotic
HD systems can be controlled.
The main purpose of this section is to give an overview
of chaos control in HD systems. One form of this control
is typically achieved by coupling together LD systems
and establishing a range of control parameters for which
the systems become synchronized. Since this range is different than that required for the system’s chaotic and hyperchaotic behavior, studies of synchronization are very
important, as they reveal different aspects of the behavior
of HD systems than those described in Sec. III through
VI.
Although chaos is unpredictable over long periods of
time, its deterministic nature can be used to our advantage. That is, the chaotic behavior of many nonlinear dynamical systems can be controlled. Chaos control
refers to purposefully manipulating the chaotic behavior
of nonlinear dynamical systems. This chaotic behavior is
represented in phase space by a chaotic attractor, which
consists of an infinite number of unstable periodic orbits
(e.g., Zoldi & Greenside 1998). The basic idea of chaos
control is to stabilize one of these unstable periodic orbits
by means of small perturbations applied to the chaotic
system, so its dynamics is substituted by a periodic one
(e.g., González-Miranda 2004).
The idea of chaos control was first addressed by Ott,
Grebogi and York (1990). They introduced an effective
technique to control chaos by taking advantage of the
sensitivity to small disturbances of a LD chaotic system
to stabilize it in the neighborhood of a desirable unstable periodic orbit naturally embedded in the chaotic motion. This approach, also known as the OGY method
or Linearization of the Poincaré Map, has been applied
to several experimental systems including the stabilization pattern dynamics in a Taylor vortex flow with hourglass geometry (e.g., Wiener, et al. 1999), and to control
chaotic Taylor-Couette flow (e.g., Lüthje et al. 2001).
A more efficient technique to control chaos was later
proposed by Pygaras (1992, 2002). His technique uses
a time-delayed feedback to some dynamical variables of
a LD system. Pygaras approach to chaos control is also
known as Time-Delayed Auto-Synchronization (TDAS)
method because it uses continuous time-delayed feedback (e.g., Pygaras et al. 2004). TDAS was shown
experimentally to be an efficient method for controlling
chaotic electronic oscillators (e.g., Pyragas & Tamasiavicius 1993), mechanical pendulums (e.g., Hikihara &
Kawagoshi 1996), lasers (e.g., Bielawski, et al. 1994),
and chemical reactions (e.g., Parmanada, et al. 1999),
including the Belousov-Zhabotinsky (BZ) reaction (e.g.,
Petrov et al. 1993).
In addition to the OGY and TDAS methods, several
other feedback techniques were developed and tested using both numerical and experimental tools. These techniques include Singer’s method of stabilizing unstable periodic orbits (Singer et al. 1991), adaptive control algorithm (e.g., Fradkov & Pogromsky 2008), and nonlinear
control algorithm (e.g., Macau & Gregobi 2008; Fradkov
& Pogromsky 2008).
There are also non-feedback control techniques that
make use of a small perturbing external force such as a
small driving force, a small noise term, a small constant
bias or a weak modulation to some system parameter
(e.g., Lima & Pettini 1990; Kapitaniak 1992; Mascolo
& Grassi 1997; Ramesh & Narayanan 1999; Belhaq &
Houssni 2000; Mohamoud et al. 2001; Wang et al. 2006;
Laoye et al. 2007). These methods weakly modify the
underlying chaotic dynamical system, so that stable solutions appear. In this category of chaos control methods
we find: (i) Parametric perturbation method; (ii) Addition of a weak periodic signal, constant bias, or noise;
(iii) Entrainment-open loop control; and (iv) Oscillator
absorber method.
Detailed description of different feedback and nonfeedback methods of chaos control and discussion of relevant examples of controlled dynamical systems can be
found in books by González-Miranda (2004), Ivancevic
& Ivancevic (2007), Lakshmanan & Murali (1996), Lakshmanan & Rajasekar (2003), and in review papers by
Ding et al. (1997), Andrievskii & Fradkov (2003), and
Macau & Grebogi (2008).
31
The concept of synchronization was known since 1665
when Huygens made his historical observation of the synchronized oscillations in pendulum clocks. As many phenomena studied by nonlinear dynamics, synchronization
was observed and shown to play an important role in
many problems of a most diverse nature. The discovery
of deterministic chaos introduced a new kind of oscillating system, a chaotic generator. Chaotic oscillations are
found in many dynamical systems of various origins. The
behavior of such systems is characterized by limited predictability in time (see Sec. II.B). When two or more
chaotic systems are coupled, they may exhibit synchronized chaotic oscillations.
As an alternative way of control, the idea of chaos synchronization was introduced by Pecora & Carroll (1990).
The basic idea is to take two identical chaotic systems
that start from different initial conditions and link them
with a common signal or signals in such a way that both
systems follow the same trajectory; when the latter is
achieved, the systems are synchronized. The approach
may be counter-intuitive because of the exponentially diverging nearby trajectories of chaotic systems. However,
Pecora & Carroll demonstrated that their method works
by applying it successfully to the synchronization of mutually coupled Lorenz and Rössler oscillators. They also
showed that a system is synchronized when the sign of
all its Lyapunov exponents is negative.
Chaos synchronization is the process of making two or
more chaotic systems to oscillate in a synchronized manner. There are several different ways to achieve it and
all of them require that the chaotic systems are somehow coupled together. One way is to link the systems
together so they have common signals (Pecora & Carroll
1990). Another way is to have a chaotic system that is
driven by a chaotic signal, which may be produced by
a different chaotic system. One may also consider two,
three or more chaotic systems mutually coupled together
by some coupling parameters (Pecora & Carroll 1997,
and references therein). Synchronization requires large
enough coupling parameters as only then the coherent
dynamical states are observed; the latter means that the
systems evolve in synchrony. However, for lesser coupling the systems may develop a complicated behavior,
which is dynamically far away from synchronization (e.g.,
González-Miranda 2004).
Chaos synchronization is very important in practical
applications such as biological and chemical systems,
electronic circuits, secure communications, and fluid turbulence. The recognized potential for applying it to secure communication systems has driven chaos synchronization to become a distinct and important subfield of
nonlinear dynamics. Extensive discussions of diverse applications of synchronization of chaotic systems to different fields of science and engineering can be found in
papers by Pecora & Carroll (1991a, b), Carroll & Pecora (1991, 1993, 1995), Cuomo & Oppenheim (1993),
Cuomo et al. (1993), Ding & Ott (1994), Carroll (1995),
Pygaras (1996), Pecora et al. (1997), Ding et al. (1997),
Lee et al. (1998), Tang et al. (1998), Thaherion &
Lai (1999), Krawiecki & Sukiennicki (2000), Voss (2000,
2001), González-Miranda (2002a, b, c), Kurths et al.
(2003), Heisler et al. (2003), Tang & Liu (2003), Qi et
al. (2005), Laoye et al. (2007), Belykh et al. (2008),
and books (see Lakshmanan & Rajasekar 2003, GonzálezMiranda 2004, and Ivancevic & Ivancevic 2007, and references therein).
Chaos control and synchronization have evolved into
an important field of research in nonlinear science. Over
the last eighteen years various effective methods have
been proposed and utilized to achieve the control and
synchronization of chaotic systems, and some of these
methods were extended to high dimensional (HD) chaotic
systems. In the next subsection, we review chaos control
and synchronization of selected HD chaotic systems.
B.
Control and synchronization of chaotic HD
systems
Most chaos control techniques described above have
been formulated and applied to low-dimensional (LD)
chaotic systems. The fact that these techniques cannot
be directly applied to HD systems was already recognized
by Auerbach et al. (1992), who introduced a feedback
control mechanism for controlling HD systems directly
from time series data. Other HD control algorithms and
HD synchronization methods were developed and applied
to different HD chaotic systems (e.g., Romeiras et al.
1992; Hu & He 1993; Reyl et al. 1993; Sepulchre &
Babloyantz 1993; Colet et al. 1994; Hu & Qu 1994;
Schwartz & Triandaf 1994; Namajunas et al. 1995; So
& Ott 1995; Bleich & Socolar 1996; Christini et al. 1996;
Ding et al. 1997; Mascolo & Grassi 1997; Mirus & Sprott
1999; Blakely & Gauthier 2000; Hu et al. 2000; Wei et
al. 2002; Heisler et al. 2003; González-Miranda 2004;
Ivancevic & Ivancevic 2007; Laoye et al. 2007; Belykh et
al. 2008). We now describe some of these techniques and
present several illustrative examples.
High-dimensional chaos control methods incorporating
the dynamics of the system’s control parameter were developed by Romeiras et al. (1992) and modified by So
& Ott (1995). They introduced the notion of a systemplus-parameter dynamical system, and thereby generalized the previous approach. Further modifications were
incorporated by Ding et al. (1997). The methods require that the periodic orbit’s position and the system’s
control parameters are continually updated. A technique
dealing with this problem is called the adaptive control
of chaos and it was pioneered by Schwartz & Triandaf
(1994), who also applied it to HD systems. A method
of controlling chaos in LD and HD systems with periodic
parametric perturbations was proposed and examined by
Mirus & Sprott (1999). Their results show that small
perturbation frequencies near the natural frequencies of
unstable periodic orbits of HD chaotic systems lead to
limits cycles, and that they may reduce the dimension of
32
the systems.
A multimode laser model was considered by Colet et al.
(1994), who demonstrated that the system exhibits hyperchaos, with three positive Lyapunov exponents, and
that this hyperchaos can be controlled by using a single
feedback parameter. The reason is that the controlled
periodic orbits are locally unstable but only in one coordinate. A model-independent quasi-continuous chaos
control technique, known as the local control method
(Hübinger et al. 1993, 1994) was extended by Christini et al. (1996), who used the method to stabilize a
HD chaotic system. Their test HD model was a driven
double pendulum, and they demonstrated that a delay
coordinate extension of the local control method is sufficient to control chaos in this system. Description of
other HD models and discussion of the obtained results
can be found in González-Miranda (2004) and Ivancevic
& Ivancevic (2007).
As an example of chaos control in a 4D chaotic system, we review the Lorenz-Stenflo system, which was
developed by Stenflo (1996) as a low-frequency shortwavelength limit of the gravity wave equation. Numerical
studies of the Lorenz-Stenflo equations were performed
by Zhaou et al. (1997), Liu (2000), and Banerjee et al.
(2001, 2004).
To address the drawbacks of many chaos control methods, Laoye et al. (2007) extended the original work of
Mascolo & Grassi (1997) and developed a backsteppingbased scheme for controlling 4D chaotic systems. They
applied the method to the Lorenz-Stenflo system to the
4D system proposed by Qi et al. (2005). For the LorenzStenflo model, Laoye et al. (2007) used the the following
set of equations:
ẋ1 = σ(x2 − x1 ) + γx4 ,
(113)
ẋ2 = x1 (r − x3 ) − x2 ,
(114)
ẋ3 = x1 x2 − bx3 + u(t) ,
(115)
ẋ4 = −x1 − σx4 ,
(116)
where x1 , x2 and x3 correspond to the original Lorenz
variables X, Y and Z, respectively (see Sec. III.A). In addition, the parameters σ, r and b have the same meaning
as those in the Lorenz system, the variable x4 describes
the flow rotation, γ represents the rotation number, and
u(t) is the control function.
With u(t) = 0, the above set of equations reduces to
the Lorenz-Stenflo equations and, if in addition x4 = 0,
then the original Lorenz equations are obtained (see Eqs
11, 12 and 13).
Laoye et al. (2007) developed a backstepping nonlinear
control scheme, which is a recursive method that interlaces the choice of a Lyapunov function with the control
function (e.g., Harb 2004; Harb & Harb 2004; Zhang et
al. 2005). The basic idea of this technique is to define
FIG. 22: Uncontrolled state of the Lorenz-Stenflo system: (a)
the chaotic phase portrait, (b) time series of the variable x1 ,
and time series of the variable x4 (after Laoye et al. 2007).
error states e1 = x1 − x1d , e2 = x2 − x2d , e3 = x3 − x3d
and e4 = x4 − x4d , where x1d , x2d , x3d and x4d are
desired states. Substituting these error states into Eqs
(113) through (116), a set of error dynamic equations
is obtained. A Lyapunov function is defined and its first
derivative is calculated (for details see Laoye et al. 2007).
To make this derivative to be negative positive, it is required that the control function is given by
u(t) = b(e2 + e3 ) + [e1 (r − e2 − e3 ) − e2 ] − e1 e2 . (117)
For the numerical simulations, Laoye et al. (2007) used
σ = 1.0, b = 0.7, r = 26.0 and γ = 1.5, so that the system
is chaotic. Typical chaotic orbits of this system are shown
in Fig. 22, which also includes two time series data. The
results obtained with the activated control function u(t)
are presented in Fig. 23, which shows that the control
law given by Eq. (117) is effective.
Chaos synchronization is typically achieved by coupling together either identical or non-identical LD systems and establishing a range of control parameters for
which these systems become synchronized. Synchronization of two coupled Lorenz models, two and more coupled
Rössler systems, coupled Lorenz-Rössler systems, coupled Van der Pol-Rössler systems, coupled lasers, and
others have been extensively studied in the literature
(e.g., Roy & Thornbury 1994; Pygaras 1996; Pecora et
al. 1997; Ding et al. 1997; Lee et al. 1998; Tang et al.
1998; Thaherion & Lai 1999; Blakely & Guthier 2000;
Hu et al. 2000; Krawiecki & Sukiennicki 2000; Voss 2000;
33
may include exact synchronization for identical coupled
oscillators, partial synchronization for globally coupled
oscillators, and phase or lag synchronization for nonidentical chaotic oscillators. Moreover, by increasing the
coupling parameter, the system may transition from HD
desynchronous chaos to LD synchronous chaos (e.g., Hu
et al. 2000).
Two specific examples of chaos synchronization are described below. One example involves two mutually coupled Lorenz models studied by González-Miranda (2004),
and the other deals with coupled Rössler-Lorenz oscillators investigated by Belykh et al. (2008). The purpose
of these examples is to illustrate the synchronization process for two identical and two nonidentical HD systems.
According to González-Miranda (2004), two mutually
coupled Lorenz models form a 6D system that is described by the following equations
dX1,2
= σ(Y1,2 − X1,2 ) + C(X2,1 − X1,2 ) ,
dτ
(118)
dY1,2
= X1,2 (r − Z1,2 ) − Y1,2 + C(Y2,1 − Y1,2 ) , (119)
dτ
dZ1,2
= X1,2 Y1,2 − bZ1,2 + C(Z2,1 − Z1,2 ) ,
dτ
FIG. 23: Controlled state of the Lorenz-Stenflo system after
the control function u(t) is activated: (a) the controlled phase
portrait, (b) time series of the variable x1 , and time series of
the variable x4 (after Laoye et al. 2007).
González-Miranda 2002a, b, c;, Heisler et al. 2003; Tang
& Liu 2003; González-Miranda 2004; Ivancevic & Ivancevic 2007; and Belykh et al. 2008).
Based on our definition of HD systems, two or more
coupled oscillators must be considered as high dimensional, and many such systems are described in Sec. III
through V. Studies relevant to chaos synchronization are
described in Sec. IV.B and V.A. The results of numerical
simulations of two coupled oscillators with two driving
forces of different phases and frequencies, and different
coupling parameters by Benner (1997), Yin et al. (1998),
Musielak et al. (2005a) demonstrated similar effects to
those observed in synchronized systems (see Sec. IV.B).
In the work by Lai et al. (2005), three 3D Rössler oscillators are coupled together and the effects of the coupling
strength on the system’s behavior is investigated. The
obtained results give a range of the coupling parameter for which the onset of stable synchronization among
the three Rössler models is observed; for details, see Sec.
V.A.
In general, depending on the strength of the coupling
parameter, a HD system of two coupled oscillators may
display a complicated behavior (see Sec. IV and V) or
exhibit a variety of synchronization behaviors; the latter
(120)
where C is the coupling parameter, the subscripts 1 and 2
label two different Lorenz models, and all other variables
and parameters are the same as in the original Lorenz
model (see Eqs 11, 12 and 13).
For this system to be in chaotic state, the following
values of the parameters were used σ = 10, b = 8/3
and r = 60. It was assumed that the control parameter
C varies from 0.0 to 1.5. Numerical studies show that
for C < CT ≈ 0.71 two Lyapunov exponents are positive, which means that the system exhibits hyperchaos.
For C = CT , the system transitions from hyperchaos to
chaos and only one of its Lyapunov exponents is positive.
The transition signals the onset of synchronized motion,
which is observed for all values of C greater than CT
(see Fig. 24); note that the system remains chaotic in
the synchronized state.
The basic feedback control principles for phase synchronization are described by Belykh et al. (2008), who
also present several interesting examples. One of them is
a HD system of coupled Rössler-Lorenz models given by
the following equations
ẋ = τ [−ω(1 + αR u)y − z] ,
(121)
ẏ = τ [ω(1 + αR u)x + aR y] ,
(122)
ż = bR − cR z + xz ,
(123)
Ẋ = σ(Y − X) ,
(124)
34
FIG. 25: Phase synchronization of coupled Rössler and Lorenz
systems is presented by plotting the difference of the mean
observed frequencies ΩR − ΩL versus the control parameter α
(after Belykh et al. 2008).
C.
FIG. 24: Lyapunov exponents of the systems of two coupled
Lorenz models are plotted versus the coupling parameter C in
panels (a) and (b). Transition to synchronization is demonstrated by parametric plots of x2 as a function of x1 given
in panels (c) and (d) for C = 0.6 and C = 0.8, respectively
(after González-Miranda 2004).
Ẏ = rX − Y − XZ ,
(125)
Ż = (1 − αL u)(−bL Z + XY ) ,
(126)
u̇ = −γu + βxZ ,
(127)
where (x, y, z) and (X, Y, Z) are the sets of variables of
the Rössler and Lorenz system, respectively, and u is the
control variable. In addition, aR , bR , cR , αR , τ and ω
are parameters of the Rössler system, σ, r, αL and bL
are parameters of the Lorenz system, and γ and β are
parameters of the control variable u. Note that by setting
u = 0 and τ = ω = 1 and c = 0, the original Rössler and
Lorenz models are obtained (see Eqs 66, 67 and 68, and
Eqs 11, 12 and 13).
To synchronize this system, Belykh et al. (2008)
choose aR = 0.15, bR = 0.1, cR = 8.5, τ = 8.3,
ω = 0.98, σ = 10, r = 28, bL = 8/3, γ = β = 1, and
αR = −αL = α, so that the fixed set of parameters gives
a chaotic system. The difference of the mean observed
frequencies ΩR − ΩL is plotted versus the control parameter α in Fig. 25, which clearly shows an interval of α
where phase synchronization occurs.
Summary
To achieve the desired type of behavior in chaotic systems, different chaos control techniques have been proposed. Many techniques are based on the original OGY
and TDAS methods and, in general, the chaos control
methods can be classified as feedback and non-feedback
control techniques, and both types have been successfully applied to many different LD systems. Since most
of these techniques cannot be directly used to control
chaos in HD systems, other HD control algorithms have
been developed.
One class of HD chaos control is based on the idea of
incorporating the dynamics of the system’s control parameter, which is also known as a system-plus-parameter
dynamical system. Several versions of this method have
been applied to different HD systems. Other classes consist of techniques that use a single feedback parameter,
the adaptive control of HD chaos, periodic parametric
perturbations, a delay coordinate extension of the local control method, and recursive backstepping nonlinear
controller. These control techniques have been applied
to HD systems ranging from a multimode laser model
and a driven double pendulum to the 4D Lorenz-Stenflo
system, which describes the behavior of low-frequency
short-wavelength gravity waves. The Lorenz-Stenflo system has been used here as an example of chaos control
by using the recursive backstepping method.
Synchronization is an alternative way to obtain the desired type of behavior in chaotic HD systems. Chaos synchronization is typically achieved by coupling together
either identical or non-identical LD systems and establishing a range of control parameters for which these systems become synchronized. There are three types of synchronization, namely, exact synchronization observed in
identical coupled oscillators, partial synchronization of
globally coupled oscillators, and phase or lag synchronization observed in non-identical chaotic oscillators. To
study these types of synchronization, two coupled Lorenz
oscillators, two and more coupled Rössler models, cou-
35
pled Lorenz with Rössler systems, coupled Van der Pol
with Rössler systems, coupled lasers, and other coupled
systems have been considered.
Two specific examples of chaos synchronization are described in this review paper, namely, two mutually coupled Lorenz models and coupled Rössler-Lorenz models. The purpose of these examples is to illustrate the
synchronization process for two identical and two nonidentical HD systems. In the first example, the range
of the coupling parameter responsible for the system’s
transition from hyperchaos to chaos synchronization is
determined. In the second example phase synchronization is discussed and the range of the control parameter
for which this synchronization occurs in the system is
established.
VIII.
CONCLUSIONS
differences are observed in the onset of chaos, routes to
chaos, and the persistence of chaos in the systems.
We used these differences to formulate ten general rules
of HDC. According to these rules, HD systems are more
resistant to the onset of chaos than their corresponding LD systems, and the routes to chaos are different
in these systems. In addition, different routes to chaos
are observed for HD systems with different dimensions.
Several new routes to chaos were already discovered and
others were proposed (see Sec. VI); some of the proposed
routes are as exotic as torus-doubling (instead of perioddoubling). However, a classification of routes to chaos in
HD systems has yet to be established.
To achieve the desired type of behavior in chaotic HD
systems, several chaos control and synchronization techniques have been proposed. Applications of these control methods to specific systems such as the LorenzStenflo system, mutually coupled Lorenz models, coupled Rössler and Lorenz systems, and several others were
proven to be successful. However, HD systems with hyperchaos are still poorly understood and there are no
satisfactory general solutions for their control, synchronization, and prediction. We expect that these problems
will be addressed in future studies of chaotic HD systems.
As already emphasized in this paper, there are numerous practical applications of the results obtained for
chaotic HD systems, and these applications range from
science and mathematics to engineering, economics, life
sciences and medicine. Since chaotic HD systems have
complex time signals that are not as vulnerable to wellknown unmasking procedures as chaotic LD systems,
they have great potential to improve secure communications. Future studies of chaotic HD systems will determine their best practical applications and, hopefully,
the results will help us to describe the real world in much
better mathematical terms
We reviewed methods to construct, investigate, and
control nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos (HDC). The
dynamical systems whose phase space dimension exceeds
3D are called here high-dimensional (HD) systems, and
their chaotic behavior is referred to as HDC when these
systems exhibit either chaos (with one positive Lyapunov
exponent), or hyperchaos (with more than one positive
Lyapunov exponent), and when the non-integer dimension of their chaotic attractors are greater than three.
Thus, cases with multiple zero Lyapunov exponents are
also included.
Two general methods to construct HD systems from
the already known low-dimensional (LD) systems were
presented, and the validity criteria for these methods
were introduced. One method involves adding higherorder modes, or new states, to the original three-modes
systems, and in the other method, the physical degrees of
freedom are added by coupling together the LD systems.
Since each HD system constructed with these methods
has the corresponding LD system as its subset, detailed
comparison between the behavior of the LD and HD systems can be made.
Specific examples of HD systems described in this paper include HD Lorenz, Duffing, Rössler, and Van der Pol
oscillators, modified canonical Chua’s circuits, and other
HD dynamical systems and HD maps. The presented results were obtained by many researchers working in this
new field. Using these results, we were able to demonstrate that there are significant qualitative differences between HD and LD systems, and that the most prominent
This review paper is an extended version of a 3-hour
tutorial on ’High-Dimensional Chaos’ presented at the
IASTED International Conference on Modern Nonlinear
Theory: Bifurcations and Chaos in Montreal, Canada,
in May 2007. The authors are grateful to Dr. Ahmad
M. Harb, Conference Chair, and the conference organizers for the invitation to present the tutorial. Z.E.M. acknowledges financial support of this work by the Alexander von Humboldt Foundation.
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