Lyapunov Exponents and Strange Attractors in Discrete and
Continuous Dynamical Systems
Jo Bovy
[email protected]
Theoretical Physics Project
September 11, 2004
Contents
1 Introduction
2
2 Overview
2
3 Discrete and continuous dynamical
tems
3.1 The Logistic Map . . . . . . . . . .
3.2 The Hénon Map . . . . . . . . . .
3.3 The Lozi Map . . . . . . . . . . . .
3.4 The Zaslavskii Map . . . . . . . . .
3.5 The Lorenz System . . . . . . . . .
3.6 The Rössler System . . . . . . . .
5.1.2
5.1.3
5.1.4
5.1.5
5.1.6
sys.
.
.
.
.
.
.
.
.
.
.
.
4 Lyapunov Exponents
4.1 Definition and basic properties . . . .
4.2 Constraints on the Lyapunov exponents
4.3 Calculating the largest Lyapunov exponent - method 1 . . . . . . . . . . .
4.4 Calculating the largest Lyapunov exponent - method 2 . . . . . . . . . . .
4.4.1 Maps . . . . . . . . . . . . . .
4.4.2 Continuous systems . . . . . .
4.5 Calculating the other Lyapunov exponents . . . . . . . . . . . . . . . . . . .
4.6 Numerical Results . . . . . . . . . . .
5 Strange Attractors
5.1 Dimensions and definition of a strange
attractor . . . . . . . . . . . . . . . . .
5.1.1 Topological dimension . . . . .
5.2
Box-counting dimension . . . .
Correlation dimension . . . . .
Kaplan-Yorke dimension . . . .
Definition of a strange attractor
Concerning dimensions from
experimental data . . . . . . .
Algorithms for calculating dimensions
5.2.1 Box-counting dimension . . . .
5.2.2 Correlation dimension . . . . .
Results . . . . . . . . . . . . . . . . . .
12
13
13
14
14
14
14
15
15
2
2
3
5.3
4
4 6 Conclusion
17
4
5 A Oseledec’s multiplicative ergodic theorem
18
A.1 The theorem . . . . . . . . . . . . . . 18
5
A.2 A measure on the attractors . . . . . . 19
6
7
B Computer Programs
19
7
8
8
8
9
9
12
12
12
1
3
1
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Introduction
2
same algoritmes can be used on a large variety of
dynamical systems. Therefore I shall use the techHaving already introduced a chaotic system (the niques that I develop later in this text not only on
Lorenz system) in a previous paper [23] and hav- the Lorenz system, but on a selection of the most iming already studied chaos qualitatively in that paper, portant and famous non-linear dynamical systems. In
I will, in this paper, try to obtain some quantita- this section I will introduce these dynamical systems
tive results concerning chaos. Therefor I will intro- and list there basic properties. I begin by describing
duce some more dynamical systems, discrete systems some discrete systems, i.e. two-dimensional maps. I
(meaning they have discrete time steps) as well as will also introduce a second three-dimensional concontinuous systems. Discrete systems are a lot eas- tinuous dynamical system in addition to the Lorenz
ier to handle than continuous systems. They can be system: the Rössler system.
chaotic even in less than two dimensions (which is
precluded for continuous systems by the PoincaréBendixon theorem). And their solution doesn’t in- 3.1 The Logistic Map
volve solving differential equations.
The Logistic map is obtained by replacing the LogisI will then study two aspects of these chaotic sys- tic equation for population growth with a quadratic
tems. Firstly I will concentrate on measuring the recurrence relation. This model was first published
chaos in the system. This is done by introducing by Pierre Verhulst [27, 28].
Lyapunov exponents, which measure the exponential
The Logistic equation is
divergence of nearby trajectories. When a Lyapunov
dx
exponents is positive, we will say that the system is
= rx(1 − x)
(1)
chaotic.
dt
All these systems also show a strange attractor for
The Logistic map is given by
certain parameter values. We will calculate the dimensions of these attractors and see that the dimenxn+1 = rxn (1 − xn )
(2)
sions don’t have to be an integer. This fact is the
reason why we call them strange.
with r a positive constant sometimes known as the
“biotic potential”.
We can understand this equation as follows [5]: as2 Overview
suming constant conditions every year, the population (of insects for example) at year n uniquely deterFirstly I will describe some dynamical systems, both mines the population at year n + 1. Therefor we have
discrete as continuous, and give their basic proper- a one-dimensional map. Say that there are zn insects
ties. Then I will introduce the Lyapunov exponents at year n and that every insect lays, on the average, r
and give methods for their calculation. Next I shall eggs, each of which hatches at year n + 1. This would
move on to strange attractors, give their definition yield a population at year n + 1 of zn+1 = rzn . When
and explain how I used these definitions to write pro- r > 1, this yields an exponentially increasing populagrams that calculate their dimension.
tion. When the population is too large however, the
insects will exhaust there food supply as they eat and
grow, and not all insects will reach maturity. Hence
3 Discrete and continuous dy- the average number of eggs laid will become less than
r as zn is increased. A simple assumption is then that
namical systems
the number of eggs laid per insect decreases linearly
The techniques for calculating Lyapunov exponents with the insect population, r[1 − (zn /z)], where z
and dimensions of strange attractors are not specific is the population at which the insects exhaust their
to the dynamical system under investigation. The food supply. We then have the one-dimensional map
3
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
3
zn+1 = rzn [1 − (zn /z)]. Dividing both sides by z and The Hénon equations are
letting x = z/z, we obtain the Logistic map (2).
In general, this equation cannot be solved in closed
xn+1 = 1 − ax2n + yn
(7)
form and it is capable of very complicated behavior.
yn+1 = bxn
(8)
Many aspects of this equation and it’s chaotic behavior can however be studied exactly. We’ll take
This map is invertible, with the inverted map given
r ∈ [0, 4] to keep all xn in the interval [0, 1] [10]. The
by
Jacobian is
yn+1
dxn+1 xn =
(9)
=| r(1 − 2xn ) |
(3)
J = b
dxn 2
y
(10)
yn = xn+1 − 1 + a n+1
b2
The map is stable at a point x0 if J(x0 ) < 1.
To find the fixed points of the map we have to
This makes the calculation of basins of attraction
solve the equation (dropping the subscripts on x for
much simpler. We shall however not look at these
convenience)
basins of attraction (see for instance [8, 14]).
x = rx(1 − x)
(4)
The Jacobian matrix J of the Hénon map is
(1)
(1)
so the fixed points are x1 = 0 and x2 = 1 − 1r . The
Jacobian evaluated at these points gives is
(1)
J(x1 ) =| r |
−2ax 1
b
0
(5)
The determinant of this Jacobian is b. By a basic
(6) result of linear algebra, the factor by which an area
grows under a linear transformation is given by the
so the first fixed point is stable until r = 1, where absolute value of the determinant of the matrix repreit becomes unstable. The second fixed point is un- senting the transformation. Locally we can linearize
stable for r < 1 and stable for 1 < r < 3. At the Hénon transformation, so a small area near a
r = 3 both fixed points are unstable. At this value point P = (x, y) is reduced by the factor given by
for r period doubling occurs. The system begins a the absolute value of the determinant of the derivaperiod-doubling route to chaos that is completed at tive (the Jacobian matrix) of the transformation at
r = 3.5699456. At this value the system becomes that point. This gives |det(J)| = |b|, a constant value,
chaotic. This is the value we will use to calculate the not dependent on the location of P .
Lyapunov exponent and dimension of the strange atThe fixed points of the Hénon map are
tractor that occurs.
√
−1 + b ± 1 − 2b + b2 + 4a
(x, y) =
(1, b)
(11)
2a
3.2 The Hénon Map
(1)
J(x2 )
=| 2 − r |
Hénon introduced this map as a simplified version
of the Poincaré map of the Lorenz system [25]. The
Poincaré map of a system is the map which relates
the coordinates of one point at which the trajectory
crosses a Poincaré plane to the coordinate of the next
(in time) crossing point. The existence of this map is
assumed to be the consequence of the uniqueness of
the solution of the equation of the dynamical system.
When we calculate the eigenvalues of the Jacobian
matrix J, we can see when these fixed points are stable. The eigenvalues are
λ(1,2) = −ax ±
p
a2 x2 + b
(12)
For a = 1.4 and b = 0.3 (the values for which the
Hénon map has a strange attractor), these eigenval-
3
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
ues for the fixed points are
4
(again a = 1.7, b = 0.5) are
(1)
λ1 = −0.092029562
(1)
λ1
(1)
=
λ2 = 0.6915136742
(1)
λ2 =
(2)
λ1 = −0.8079567198
(2)
λ2 = 0.1559463222
(2)
λ1 =
so both fixed points are not-stable.
3.3
(2)
λ2 =
The Lozi Map
√
489
−17
−
20
20
√
489
−17
+
20 √ 20
489
17
−
20 √20
489
17
−
20
20
= −1.95566722
=
0.25566722
= −0.25566722
=
1.95566722
so both points are not-stable.
The Lozi map is a simplification of the Hénon map.
The quadratic term in xn (−ax2n ) is replaced by 3.4 The Zaslavskii Map
−a|xn |. The Lozi map is then given by
The Zaslavksii map [13] is given by the following
xn+1 = 1 − a|xn | + yn
(13) equations
yn+1 = bxn
(14)
xn+1 = xn + ν + ayn+1 (mod1)
(19)
−r
yn+1 = cos(2πxn ) + e yn
(20)
We can write the Jacobian matrix using the Heaviside step-function
To compute the Jacobian, we write the equations
as
follows
0 x<0
Θ(x) =
(15)
1 x≥0
xn+1 = xn + ν + a cos(2πxn ) + e−r yn (mod1)
(21)
The Jacobian J then becomes
−r
yn+1 = cos(2πxn ) + e yn
(22)
−a 2Θ(x) − 1 1
b
0
The Jacobian is then
The determinant is still b.
1 − 2πa sin(2πxn ) ae−r
This map has a strange attractor we shall study
−2π sin(2πxn )
e−r
for the parameter values a = 1.7, b = 0.5. The fixed
points for these parameter values are
The determinant is equal to e−r , so areas shrink by
a factor of e−r every iteration (r > 0).
5 5
The Zaslavskii map shows a strange attractor for
(x, y)1 = ( , )
(16)
11 22
ν = 400, r = 3, a = 12.6695.
−5 −5
(x, y)2 = (
,
)
(17)
6 12
3.5
The Lorenz System
The eigenvalues of the Jacobian matrix are
I have already studied the Lorenz system [7] extensively in a previous project [23]. I studied the basic
λ(1,2)
(18) properties of the Lorenz system (fixed points, stability and basins of attraction). So the reader should
and the eigenvalues evaluated at the fixed points consult this for an overview of the basic properties.
√
−a 2Θ(x) − 1 ± a2 + 4b
=
2
4
LYAPUNOV EXPONENTS
5
Other good references are [15, 31]. I will just state
the Lorenz equations
Ẋ = p(Y − X)
Ẏ = rX − XZ − Y
(23)
Ż = XY − bZ
I shall study the Lorenz strange attractor for the
parameter values p = 10, b = 8/3, r = 28.
3.6
The Rössler System
Setting the parameter c = 5.7 (we shall study the
strange attractor that occurs at this parameter value)
the eigenvalues are
λ1 = 0.0970008560175134871 + i0.995193491034748634
λ2 = 0.0970008560175134871 − i0.995193491034748634
λ3 = −5.68697550703502762
and
λ1 = −0.459615167119897806e − 5 + i5.42802593149083901
λ2 = −0.459615167119897806e − 5 − i5.42802593149083901
λ3 = 0.192982987303353531
The Rössler system [26] is one of the most simple so both fixed points are not-stable.
chaotic continuous systems. It is artificially designed
solely with the purpose of creating a simple model for
a chaotic strange attractor. The system of differential 4
Lyapunov Exponents
equations is
As with everything else in Physics, and especially in
ẋ = −(y + z)
(24) Theoretical Physics, we don’t content ourselves with
ẏ = x + ay
(25) just a qualitative picture of chaos. In this section
I’ll introduce a quantitative measure of chaos, the
ż = b + xz − cz
(26)
celebrated Lyapunov Exponents.
Introducing a quantitative measure of chaos is imwith a,b and c adjustable paramters of which a and
portant for several reasons. Most importantly it alb are usely fixed at a = 0.2, b = 0.2. This system is
lows us to define exactly what we mean by chaos.
obviously simpler than the Lorenz system, because it
When we only have a qualitative picture of chaos,
contains only one non-linear term. It is however not
everybody has a different opinion of what chaos is.
the simplest chaotic flow, since the simplest dissipa...
One just looks for instance at the picture in phase
2
tive flow has been argued to be x +Aẍ−ẋ +x = 0 [17]
space and decides wether or not he finds it chaotic.
(This equation can easily be converted to a system of
Science would not have come as far as it is now if
first order differential equations using the following
we had contented ourselves with such qualitative picsubstitutions: ẋ = y, ẍ = z).
tures. So introducing a measure of chaos allows us to
The divergence of the flow, ∇·f (writing the system
rigourously define what we mean by chaos.
∂
ẏ
∂ ż
as ẏ = f ) = ∂∂xẋ + ∂y + ∂z
= a + x − c. Note that this
Having this measure of chaos allows us to go furis not a constant, hence the shrinking/expansion of ther and compare different systems. We can define
volumes is not uniform over phase space.
what we mean by saying that one system is more
The fixed points are
chaotic than another system. Thus we can compare
the chaoticness of a system with different parameter
√
c ± c2 − 4ab
(x, y, z) =
(a, −1, 1)
(27) values, or the chaoticness of two completely different
2a
systems.
In the first subsection of this section I will define
The Jacobian matrix is
and discuss the Lyapunov exponents. In the next


subsections I will discuss how I calculated the Lya0 −1 −1
 1 a
punov exponents and state the results I got for the
0 
maps described in the previous section.
z 0 x−c
4
LYAPUNOV EXPONENTS
4.1
6
Definition and basic properties
decomposed in the orthonormal eigenvectors ej of Hn
n
Since we want to measure chaotic behavior, which we
X
u0 =
aj ej
(31)
intuitively defined as sensitivity on initial conditions,
j=1
we shall look at the evolution of a small displacement
of a initial condition x0 (with corresponding orbit xn
and the Lyapunov exponent of such an initial disn = 0, 1, 2, . . .). First take a map M. If we consider
placement can be written in function of the eigenvalan infinitesimal displacement from x0 in the direction
ues of Hn (x0 ), since
of a tangent vector y0 , the evolution of the tangent
n
vector, given by
X
u†0 · Hn (x0 ) · u0 =
a2j exp[2nλjn (x0 )]
(32)
yn+1 = DM(xn ) · yn
(28)
j=1
determines the evolution of the infinitesimal displacement of the orbit from the unperturbed orbit xn
(DM is just the Jacobian matrix). So |yn |/|y0 | is the
factor by which the infinitesimal displacement grows
or shrinks. From (28) we have yn = DMn (x0 ) · y0 ,
where DMn (x0 ) = DM(xn−1 ) · DM(xn−2 ) · . . . ·
DM(x0 ). We then define the Lyapunov exponent1
for initial condition x0 and initial orientation of the
infinitesimal displacement given by u0 = y0 /|y0 | as
1
ln(|yn |/|y0 |)
n
1
= lim ln|DMn (x0 ) · u0 |
n→∞ n
λ(x0 , u0 ) = lim
n→∞
(29)
An important question now is wether the limits
in (29) exist. This existence is guaranteed by Oseledec’s multiplicative ergodic theorem under very
general circumstances. This theorem is stated in appendix A with some of its consequences. One of its
consequences is that the Lyapunov exponents are the
same for almost every x0 with respect to the natural measure on the attractor (this natural measure is
also described in appendix A).
Now I will move on to define Lyapunov exponents
in continuous time systems. All the above considerations remain valid when we replace Eq. (29) by
1
ln(|yn |/|y0 |)
n
1
= lim ln|O(x0 , t) · u0 |
n→∞ n
λ(x0 , u0 ) = lim
n→∞
(33)
Note that these exponents are (for now) dependent
on the initial condition.
If the dimension of the map is N , then there will
dy(t)
be N Lyapunov exponents, since there are N inde- where dt = F(x(t)) · y(t), x0 = x(0), y0 = y(0),
u0 = y(0)/|y(0)|, and O(x0 , t) is the matrix solution
pendent initial displacement directions. Since
of the equation
1
λ(x0 , u0 ) u λn (x0 , u0 ) ≡ ln|DMn (x0 ) · u0 |
dO/dt = DF(x(t)) · O
(34)
n
1
=
ln|u†0 · Hn (x0 ) · u0 |
with initial condition
2n
(30)
O(x0 , 0) = I
where Hn = [DMn ]† DMn , and † denotes the This equation is called the variational equation. To
transpose, we get N eigenvalues of Hn (Hjn ) each calculate the Lyapunov exponents we’ll have to solve
corresponding to one Lyapunov exponent (λjn = an additional system of differential equations.
(2n)−1 ln Hjn ). Every initial displacement u0 can be
Now that we have defined Lyapunov exponents, we
can define what we mean by saying that a system is
1 In the literature one encounters often reference to Lyapunov numbers µ. These are given in terms of the Lyapunov chaotic. We say that a system is chaotic when it
exponents λ by µ = exp(λ).
has at least one strictly positive Lyapunov exponent.
4
LYAPUNOV EXPONENTS
When a system has a positive Lyapunov exponent,
small disturbances will give rise to exponential divergence. So this definition is in accordance with the
qualitative picture we had.
Now how can we represent what these different
Lyapunov exponents mean. When we consider a
small ball (say infinitesimal, radius dr) of initial
conditions around an initial condition x0 , and then
evolve every initial condition inside this ball for n iterates. The ball will have evolved into an ellipsoid
. In the limit of large time the Lyapunov exponents
give the time rate of exponential growth or shrinking
of the principal axes of the evolving ellipsoid. The
axes will be (approximately) given by the expressions
a1 = exp(nλ1 )dr and a2 = exp(nλ2 )dr.
4.2
Constraints on the Lyapunov exponents
We can deduce some constraints on the Lyapunov
exponents of a dynamical system. Using these constraints we should only calculate the largest Lyapunov exponent for all but one of the dynamical systems I described in section 3. I shall however in the
following sections calculate all Lyapunov exponents
and use them to check the validity of these relations.
In systems where the area-reducing factor (or volume reducing factor in three dimensional systems) is
constant, we can derive a relation
P between the Lyapunov exponents. Since exp( i λi ) is equal to the
area-reduction, this must be equal to the determinant of the Jacobian (for maps) or to exp(∇ · f ) (∇ · f
is the divergence of the flow for continuous dynamical
systems).
Applying this to the Hénon system, we see that
λ1 + λ2 = ln(|b|). We could only calculate the largest
exponent and derive the second from this relation.
For a continuous dynamical system we can look at
disturbances in the direction of the velocity vector
of the trajectory (see page 718 in [14]). Consider a
trajectory (x(t), y(t), z(t)) and a perturbed trajectory
(x̃(t), ỹ(t), z̃(t)) with initial condition
x̃(0) = x(δ), ỹ(0) = y(δ), z̃(0) = z(δ)
7
tonomous, it will follow the same trajectory as the
not-disturbed trajectory. These trajectories clearly
do not diverge from each other, on the average they
will keep at the same distance of each other. So in
this direction the Lyapunov exponent is zero. So in
every continuous dynamical system one of the Lyapunov exponents will be zero (except for systems with
a fixed point).
Applying this to the Lorenz system, which has a
constant divergence equal to −(1 + b + p), we have
the following two relations
λ1 + λ2 + λ3 = −(1 + b + p)
λ2 = 0
(35)
(36)
so again we could only calculate the largest Lyapunov
exponent (λ1 ), and deduce the other two from these
two relations. We can’t use this on the Rössler system
however because its divergence isn’t a constant.
4.3
Calculating the largest Lyapunov
exponent - method 1
As pointed out in appendix A, the largest Lyapunov
exponent is the easiest one to calculate. If we start
by choosing a direction vector randomly, we can decompose it as in (31). It will most probably have
a non-zero component in the direction of the eigenvector of the largest exponent. Since in the cases
we consider only one Lyapunov exponent is positive,
evolution will be dominated by the largest exponent,
as seen from (32).
We can then calculate this exponent by just
straightforwardly using our qualitative understanding (in the next subsection I will give a better method
based on the differential map). We have to calculate
λ1 = lim lim
n→∞ E0 →0
1
ln(|En |/|E0 |)
n
(37)
with E0 an initial disturbance. We can write
|En |
|En | |En−1 |
|E1 |
=
···
|E0 |
|En−1 | |En−2 |
|E0 |
(38)
substituting (38) in (37), we get
n
with δ > 0. So the trajectory starts on the reference
trajectory, and, if we assume the system to be au-
1X
|Ek |
ln(
)
n→∞ E0 →0 n
|Ek−1 |
λ1 = lim lim
k=1
(39)
4
LYAPUNOV EXPONENTS
8
We will however in our algorithm renormalize our
error to some chosen value . For one-dimensional
maps such as the Logistic map, we essentially haven’t
got any choice of direction of the error. For twoor three-dimensional systems we have to choose the
direction of the error. For two-dimensional maps we’ll
define an error of size on (x, y)
(x̃, ỹ) = (x + cos φ, y + sin φ)
(40)
with φ an arbitrary angle. For three-dimensional systems this becomes on (x, y, x)
(x̃, ỹ, z̃) = (x+ sin φ cos σ, y + sin φ sin σ, z + cos φ)
(41)
The algorithm2 now works as follows (it is described for maps, but it is easily adjusted to continuous systems):
1. We choose an initial point an let the map iterate
this for say 100 times. We do this to let transients die out.
2. We compute the perturbed point according to
(40) or (41).
3. We iterate both points and compute the distance
d between them
4.4.1
Maps
For maps this is not such a hard task. We just calculate the Jacobian matrix, and this gives us the differential map. This map tells us how an infinitesimal
small error gets amplified. The algorithm is then as
follows
1. We choose an initial point an let the map iterate
this for say 100 times. We do this to let transients die out.
2. We consider for an arbitrary angle φ the direction (cos φ, sin φ) (generalizing this to three dimensions is obvious).
3. We compute the next point of the map H(x, y)
and the transformed error E(x, y) using the differential map. So we get the point E(x, y) =
DM(x, y) · (cos φ, sin φ)† .
4. The error has increased by d = kE(x, y)k. We
add log d to an accumulator.
5. We renormalize the direction vector to a unit
vector (E 0 (x, y) = E(x, y)/d). Then we go back
to step 3 using the new point and the new error.
6. The result is once again the accumulator divided
by the number of iterations.
4. We add log d/ to an accumulator
5. We renormalize the error
4.4.2
Continuous systems
The algorithm is essentially the same as the one described in the previous section. We only have to ad7. The result is the average of the log di /, thus the just the manner of computing the transformed direcaccumulator divide by the number of iterations tion vector. To find this transformed direction vector, we have to solve the variational equation. Also
in continuous systems, we have to choose the time
4.4 Calculating the largest Lyapunov interval over which each iteration reaches. When we
consider discrete maps, we have always iterated the
exponent - method 2
map one discrete time step. When considering conThe program outlined in the previous subsection isn’t tinuous maps we choose to let the system evolve for
quite accurate, because we should actually take the one timestep (such that t
n+1 = tn + 1). So when we
limit → 0. We can take this limit by considering choose an integration step of 1/N in our numerical
the differential map.
solver, we have iterate this for N times. After these
2 this algorithm and the ones in the following section are
N times we look at our new direction vector, and
written along the lines given in [14] pg.710
measure its amplification.
6. We iterate steps 3-5
4
LYAPUNOV EXPONENTS
4.5
9
Calculating the other Lyapunov have to be careful about the renormalization however. Because the first Lyapunov is the largest, the
exponents
Because of theorem 2 in appendix A.1, when calculating the second (or third or higher) Lyapunov exponent, we should start with a vector that is orthogonal
to the eigenvector of the first Lyapunov exponent (or
in the case of higher Lyapunov exponents, we should
start with a vector orthogonal to all eigenvectors, belonging to the Lyapunov exponents bigger than the
one we want to calculate). This is however not an
easy task. For continuous systems, which we have to
solve numerically, the situation is even worse because
due to numerical integration errors, components from
the eigenvector of the largest Lyapunov exponent will
become not-zero, and dominate the further evolution.
So we have to think of another way of calculating
the remaining Lyapunov exponents. I will give the
description of this method for the second Lyapunov
exponent, but one can easily see from this exposition
how the algorithm works for arbitrary Lyapunov exponents. The idea is to compute not the second Lyapunov exponent, but the sum of the first and second
Lyapunov exponent, and then compute the second
out of our knowledge of the first (for the third for instance we shall have to compute the sum of the first
three exponents and then calculate the third from our
knowledge of the first and the second).
How are we going to calculate the sum of the first
and second exponent? By looking at the evolution
of two orthonormal error directions, and look at the
amplification of the area they span (we shall immediately use the differential map to calculate the transformed errors). When assuming that the original vectors have components in the directions of the eigenvectors of the first and the second eigenvalue, the
area will behave as An ≈ exp[n(λ1 + λ2 )]A0 (when
the vectors also have components in the direction of
the other eigenvectors, the largest two will dominate
the growth). Thus we have
area will shrink very quickly, and the parallellogram
spanned will become more and more a line in the direction of the first eigenvector. Thus we shall renormalize the two transformed vectors by Gram-Schmidt
orthogonalization, so that they once again form a orthonormal pair.
I shall give an overview of the algorithm I used for
a two-dimensional map
1. We choose an initial point an let the map iterate
this for say 100 times. We do this to let transients die out.
2. We consider for an arbitrary angle φ the
direction (cos φ, sin φ) and the direction
(− sin φ, cos φ) (generalizing this to three
dimensions is obvious).
3. We compute the next point of the map H(x, y)
and the transformed errors E1 (x, y), E2 (x, y)
using the differential map. So we get the
points E1 (x, y) = DM(x, y) · (cos φ, sin φ)† and
E2 (x, y) = DM(x, y) · (− sin φ, cos φ)† .
4. The area has increased/shrunk by d =
det |E1 (x, y) E2 (x, y)|. We add log d to an accumulator.
5. We renormalize the directions using GramSchmidt orthogonalization. Then we go back to
step 3 using the new point and the new errors.
6. The result is once again the accumulator divided
by the number of iterations.
The adjustments necessary for continuous systems
should be clear.
4.6
Numerical Results
I calculated all Lyapunov exponents of the systems
(42) from section 3 using 10000 iterations and using the
differential map method. The results are stated in
So the algorithm doesn’t change much. We have to table 1. I have also included graphs showing the
calculate two transformed directions, and the area convergence towards the Lyapunov exponent for the
that they span will be the amplification factor. We largest Lyapunov exponent of the Hénon system and
1
ln(An /A0 )
n→∞ n
λ1 + λ2 = lim
4
LYAPUNOV EXPONENTS
the Lorenz system. In both cases we see that the
convergence is very quick.
We immediately see that all results are in agreement with the relations we discussed in section 4.2.
The small deviations on the exponents that should be
zero (for instance for the Lorenz system) can actually
be used to estimate the error we make in calculating
the Lyapunov exponents. So using this and taking in
consideration the fluctuations we see when we zoom
in on the convergence-figure 1, I would estimate the
error for the Lorenz system to be ±0.0005. For the
Rössler system the error will be of the same magnitude. And since we don’t make numerical integration
errors in the two-dimensional maps, we expect the
value for the Lorenz system to be an upper bound on
the error of the two-dimensional maps. So I shall use
this value for all systems.
I will now shortly comment on the individual results.
Logistic map The Logistic map is not chaotic for
the parameter value r = 3.5699456. We see that it is
on the verge of getting chaotic and this is in agreement with the value one finds at which the perioddoubling ends. Setting r = 3.9, we see that for this
value the Logistic map is chaotic.
Hénon map We see that the Hénon map is chaotic.
The sum of its two Lyapunov exponents (−1.204) is
in very good agreement with its area reducing factor
(0.3, we should take the logarithm of this number
in order to get the sum of the exponents, ln 0.3 =
−1.20397).
Lozi map The Lozi map is also chaotic, and it
is even more chaotic than the Hénon map. The
sum of its Lyapunov exponents (-0.6931) is in very
good agreement with its theoretical value (ln 0.5 =
−0.693147).
Zaslavskii map The Zaslavskii map is very chaotic
(largest Lyapunov exponent 3.6865). The sum of the
Lyapunov exponents should equal the parameter r,
and it does so very nicely.
10
Lorenz system The Lorenz system is chaotic for
two of the parameter values I studied. For r = 148 it
is not chaotic. In my previous project [23] we saw
that the Lorenz system has a periodical attractor
at this parameter value. The calculated Lyapunov
exponents confirm this : the largest is zero for the
same reason as stated in section 4.2, along the periodical orbit there is on the average no divergence
or convergence of nearby trajectories, so one of the
exponents has to be zero. Only when we have a
fixed point, none of the exponents will be zero. So
this orbit must be a periodical orbit (since we have
only three possibilities: fixed point, periodical orbit or chaotic attractor). In all three cases under
study the three Lyapunov exponents add up nicely
to −(1 + p + b) = −41/3 = −13.6666 . . ..
Rössler system We see that the Rössler system
is only slightly chaotic. Adding the Lyapunov exponents has no use, since the divergence of the flow isn’t
a constant. The sum however does agree nicely with
the average of the divergence over the attractor.
4
LYAPUNOV EXPONENTS
Logistic
Logistic (r = 3.9)
Hénon
Lozi
Zaslavskii
Lorenz (r=28)
Lorenz (r=148)
Lorenz (r=142)
Rössler
11
λ1 ± 0.0005
-0.001
0.4945
0.4189
0.4721
3.6865
0.9051
2.39E-04
1.2533
0.0696
λ2 ± 0.0005
-1.6229
-1.1652
-6.6865
8.12E-05
-0.4271
9.20E-05
-2.11E-04
λ3 ± 0.0005
-14.5718
-13.2398
-14.9201
-5.3928
P
i
λi
-1.204
-0.6931
-3
-13.667
-13.667
-13.667
-5.3234
Table 1: Lyapunov Exponents. Parameters are as in section 3, unless otherwise indicated.
(a) Hénon
(b) Lorenz (r = 28)
Figure 1: Convergence of first Lyapunov exponent.
5
5
STRANGE ATTRACTORS
Strange Attractors
The systems described in section 3 all show a strange
attractor for certain parameter values. I will define
what a strange attractor is in the next subsection. In
investigating the properties of these strange attractors, I will focus on their dimension. We shall shortly
see that dimension is a broader notion then one might
think. The dimension doesn’t have to be an integer
for example.
Now what is the meaning of the dimension of a
strange attractor? In a dissipative system, almost
all initial values will eventually settle on an attractor. When we know the dimension of this attractor, we could say that the degrees of freedom the
dynamical system has, is essentially this dimension,
and this could be significantly less than the dimension of the underlying phase space. In the systems
under study here this isn’t very spectacular (twodimensional phase space with 1.5-dimensional attractor for example), but we must not forget that these
systems are idealizations which don’t occur in real
life. Processes in real life can for example be described by a system of partial differential equations,
which typically has a infinite number of degrees of
freedom. When we see that these systems exhibit a
finite-dimensional strange attractor, we can look for
a small set of variables which describe the system in
its attractor state. Also in experimental situations,
one often doesn’t know the dimension of the phase
space he’s working in, since one doesn’t know exactly
all the variables contributing to a phenomenon. So
the dimension of the strange attractor a experimental
system may show, is the only thing one knows about
the degrees of freedom of the system.
12
5.1
5.1.1
Dimensions and definition of a
strange attractor
Topological dimension
The dimension we are all familiar with is called the
topological dimension. A set of disconnected points
(by this I mean that they are not infinitely close together as on a line, which is also just a set of points)
has dimension zero. Curves have dimension one, surfaces have dimension two. The topological dimension
of an object is always an integer. As a non-trivial example I mention the Cantor set (which one gets by
dividing the unit interval in three equal pieces, throwing away the middle piece and iterate this procedure
on the two remaining pieces). This set is just a set of
points (however uncountable) and thus has a topological dimension zero.
5.1.2
Box-counting dimension
One can define for every object something called its
Hausdorff-Besicovitch dimension. The definition of
this dimension is however rather intricate. It can be
found in the appendix of [1]. For most systems it
coincides with the Box-counting dimension. This dimension is defined as follows: Consider ’boxes’ (in
one dimension this would be intervals, in two dimensions squares, in three dimensions cubes, and so on)
of side R, then cover the object with these boxes, and
then we count the number N (R) of boxes necessary
to contain all points of the object. As we let the
size of these boxes get smaller, we expect the N (R)
to increase. The box-counting is then defined as the
number Df that satisfies
N (R) = lim kR−Df
R→0
(43)
where k is a proportionality constant. We find Db
For all these reasons, strange attractors have made
thus by taking the logarithm of both sides, before
their way into the whole off the scientific world. They
taking the limit
are used to describe a large variety of systems, for
n log N (R)
log k o
example in immunology [32] or biology [30].
Df = lim −
+
(44)
R→0
log R
log R
Firstly I shall give some explanation on the dimensions involved and then use these dimensions to define Since k is a constant the last term will go to zero,
a strange attractor. Then I will describe how I used when R goes to zero, so actually we have
these definitions to calculate the dimensions of the
log N (R)
Df = − lim
(45)
systems under study and state my results.
R→0
log R
5
STRANGE ATTRACTORS
In practice we shall verify this law (43) by computing N (R) for an appropriate scaling region, and then
plotting − log N (R) against log R. By using linear
regression we can see how well this law is satisfied
and the Df is then given by the slope of the fitted
line.
Is this a good definition for a dimension? Well it is
easy to see that for all Euclidean objects (a point, a
line, a plane, . . .) it just gives the topological dimension. I.e. we always need one box to cover a point, so
the limit (45) will be zero. In case of a line segment
of length L we need N (R) = L/R boxes to cover it.
So the limit will be one.
Consider however the set { n1 }∞
n=2 . This is just a
countable set of disconnected points, so one would
expect its dimension to be zero. However, I calculated the box-counting dimension in the following
1
way. Consider intervals of length R(R−1)
. This rep1
1
resents the distance between the points R and R−1
.
1
All points i for which i < R will be separated from
1
each other by an amount greater than R(R−1)
. So we
need for each of these points an individual interval.
This gives us already R − 2 intervals. To cover the
1/R
= R − 1 intervals.
remaining points we need 1/R(R−1)
Thus N (R) = 2R − 3 and we get
13
1982 Grassberger and Procaccia suggested a new way
to define a dimension [29].
Start with a long-time series on the attractor
~ i } N ≡ {X(t
~ + iτ )} N where τ is an arbitrary
{X
i=1
i=1
but fixed time increment. We then define the correlation to be
1
N →∞ N (N − 1)
C(r) = lim
N
X
~i−X
~ j |)
Θ(r − |X
i,j=1 i6=j
(47)
where Θ is the Heaviside function as defined in (15)
and we use for example the Euclidean norm (one
could also take the maximum norm to speed up the
calculations). We then assume that for small r C(r)
behaves as follows
C(r) ∝ rDc
(48)
Dc is then called the correlation dimension. We calculate it using the same procedure as in the previous
subsection, choosing an appropriate scaling interval
and fitting a line on the plot of log C(r) against log r.
This dimension is sensitive to the distribution of
the points on the attractor, since crowded regions will
yield an higher correlation. When the distribution of
points on the attractor is uniform, the correlation
log 2R − 3
1
log N (R)
dimension equals the box-counting dimension. Oth= lim
=
Df = − lim
1
R→0 log R(R − 1)
R→0 log
erwise it is smaller. We could compare box-counting
2
R(R−1)
(46) and correlation dimension with average and variance
which obviously isn’t zero. So this can be re- in statistics, the average also doesn’t care much about
3
garded as a failure of the definition of the box- the distribution of the values .
counting dimension. It can be shown however that
the Hausdorff-Besicovitch dimension of this set is
5.1.4 Kaplan-Yorke dimension
zero. So the Hausdorff-Besicovitch dimension hasn’t
got this problem.
Kaplan and Yorke proposed a dimension based on
the Lyapunov exponents of the system. Let us rank
the Lyapunov exponents from the largest λ1 to the
5.1.3 Correlation dimension
smallest λd . Let j be the largest integer such that
The box-counting dimension, however relatively easy λ1 + λ2 + . . . + λj > 0, then the Kaplan-Yorke dimento understand, isn’t easy to calculate. In higher di3 actually we should compare the correlation dimension with
mensional spaces there is so much more space, so we
the
second moment, since the first moment, the variance in
need much more boxes, and thus much more points
statistics, can be compared with yet another dimension, the
on our attractor, which are mostly calculated by a information dimension. This dimension is part of a bunch
computer or come from experimental data. And cal- of generalized dimensions which can be compared with the
culating these points can take a lot of time. So in arbitrary moments of a distribution in statistics.
5
STRANGE ATTRACTORS
sion is defined to be
14
Now there are theorems (Takens, 1981; Mañe, 1981
see for instance [21]) which state that we can embed
λi
DKY = j + i=1
(49) a one-dimensional time series in a high dimensional
−λj+1
space (typically twice the Hausdorff dimension), thus
obtaining a good projection of the attractor, which
Kaplan and Yorke conjectured that, for a two- has the same properties as the one which is under
dimensional mapping, the box-counting dimension investigation.
Df equals the Kaplan-Yorke dimension DKY [24].
We construct this from our one-dimensional time
This was subsequently proven to be true in 1982. A series {x } N as follows:
i i=1
later conjecture held that the Kaplan-Yorke dimension is generically equal to another dimension called
~xi = (xi , xi+tL , xi+2tL , . . . , xi+(d−1)tL )
the information dimension, which is also closely related to the box-counting dimension and the correla- where tL is some appropriate time lag (which is not
tion dimension. This conjecture is partially verified so easy to choose [16]) and d is the embedding dimenby Ledrappier (1981). (This paragraph partially from sion.
Calculating dimensions and also Lyapunov expo[9]).
nents from experimental data is however much more
difficult than from the systems we study. One should
5.1.5 Definition of a strange attractor
for example use enough points in order to get a corWe can now define what we mean by a strange attrac- rect result [22, 6].
tor. We shall say that an attractor is strange when
its Hausdorff dimension strictly exceeds the topolog5.2 Algorithms for calculating dimenical dimension [1] (in dynamical systems we speak of
sions
strange attractors, otherwise these objects are called
fractals). So a strange attractor is an attractor with 5.2.1 Box-counting dimension
a non-integer Hausdorff dimension.
For two-dimensional attractors the box-counting dimension is relatively easy to calculate. For three5.1.6 Concerning dimensions from experidimensional systems such as the Lorenz system, this
mental data
calculation already becomes much more complicated
Before turning to the description of the algorithms (for reasons stated in section 5.1.3).
used to calculate dimensions, I would like to say
As a first attempt I just made an algorithm purely
something about experimental situations. As I said in from the definition. I divided the region of phase
the introduction to strange attractors, one wants to space in squares (in the two-dimensional case) and
find the dimension of strange attractors observed in looked for every square wether a point of the attracan experimental situation. Now when we look at the tor was in it or not. Repeating this for smaller and
definitions of the dimensions, we see that we have to smaller squares, I could then find the box-counting
know in what phase space we’re working to be able dimension from linear regression described in section
to calculate the dimensions. When calculating the 5.1.2. Realizing that I was doing a lot of work that
box-counting dimension we have to use boxes. Now wasn’t necessary (in every step I checked for every
these boxes have the dimension of the phase space square wether a point was in it), I programmed a
we’re working in (this is also partially solved in the recursive algorithm. This recursive algorithm begins
Hausdorff-Besicovitch dimension). In calculating the with the region the attractor is contained within. If
correlation dimension we also use a time series made a point is in it (which obviously is true) it gets split
up of points in our phase space. In experimental sit- up in for smaller squares. Then the same algorithm
uations however we can only measure one, or a few does the same for these squares. So it checks for the
variables.
four squares wether a point is in it and if so it splits
Pj
5
STRANGE ATTRACTORS
it further up, until some bottom level, which can be
specified, is reached.
These algorithms are very slow. They also use
a enormous amount of memory: applying them on
the Lorenz system didn’t work because my computer
didn’t have enough working memory to hold the information of all the boxes.
I then came up with a better idea. Instead of looking at all the squares, I just looked at all the calculated points of the attractor. So I ‘hypothetically’
divided the region of phase space where the attractor
lies (by ‘hypothetically’ I mean that the computer
didn’t have to do this explicitly). Then I looked for
every point in which square it lies and then assigned
to that point a number, uniquely determined by the
square. Doing this for all the calculated points, I
then had an array of numbers. Of course when two
points are within the same square, they will get the
same number assigned to them. So this array contains doubles. Getting rid of these doubles leaves us
with an array which length is the number of squares
needed to cover the whole attractor. This program
works a thousand times as fast as the other (this is
just an estimation, not a rigourously checked statement, it isn’t however an exaggerated guess).
15
already calculated the Lyapunov exponents of all systems). For the two-dimensional maps I calculated the
box-counting dimension. For the three-dimensional
continuous systems I calculated the correlation dimension.
I can immediately note that all results are independent of the initial value. For the calculation of the
box-counting dimension I checked this for 160 or 200
different initial values. For the correlation dimension
I only checked five initial values, because of the computing time required. All the initial values gave the
same result, with, especially for the maps, very small
variance.
For the box-counting dimension of the twodimensional maps I would estimate the error to be
±0.01 since this is how much the dimension changes
when I change the scaling interval without getting a
bad fit.
For the Lorenz system and the Rössler system I
would estimate this error to be larger. The scaling
interval I used here is very small (for the Lorenz system 0.1:0.025:0.2). This gives the best fit, but changing the interval doesn’t make the fit much worse,
the slope however changes significantly. So the error should be at least ±0.05 and maybe even ±0.1.
All results (including the Lyapunov exponents) are
given
in table 2. In each case the Kaplan-Yorke di5.2.2 Correlation dimension
mension is in good agreement with our results. When
Calculating the correlation dimension was pretty consulting the literature [20, 29, 2, 3] we see that all
straightforward. I just implemented the definition dimensions are in rather good agreement with the
(47). The only thing to note here is that I ad- literature results.
justed the definition a little, to prevent autocorrelation. Calculating the points of a continuous system
yields that for small stepsize, successive points will
automatically be close together. So they will surely
be correlated, exaggerating the correlation. So instead of just i 6= j, I demanded |j − i| > k for some
suitable k (I used 50).
5.3
Results
Since the amount of computer time necessary to get
good dimension estimates was enormous, I calculated
for every strange attractor only one dimension (for
every system I also calculated the Kaplan-Yorke dimension, which obviously wasn’t much work since I
5
STRANGE ATTRACTORS
Hénon
(a=1.4, b=0.3)
Lozi
(a=1.7, b=0.5)
Logistic
(r = 3.5699456)
Logistic (r = 3.9)
Zaslavskii
(ν = 400, r =3,
a = 12.6695)
Lorenz
(p=10, b=8/3, r=28)
Lorenz
(p=10, b=8/3, r=148)
Lorenz
(p=10, b=8/3, r=142)
Rössler
(a=0.2, b=0.2, c=5.7)
16
Df ± 0.01
1.251
Dc ± 0.05
λ1
0.4189
λ2
-1.6229
1.365
0.4721
-1.1652
1.405166
0.5462
-0.001
0.9118
1.479
0.4945
3.6865
-6.6865
1.551335
0.9051
8.12E-05
-14.5718
2.39E-04
-0.4271
-13.2398
1.2533
9.20E-05
-14.9201
0.0696
-2.11E-04
-5.3928
2.0519
1.9729
Table 2: Results
λ3
DKY
1.258118
2.062
2.013
REFERENCES
6
Conclusion
17
[5] Edward Ott. Chaos in dynamical systems. Cambridge University Press, 1993.
When calculating the Lyapunov exponents and di[6] Edward Ott, Tim Sauer, and James A. Yorke,
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editors. Coping With Chaos. Wiley Series in
introduced in section 3, we see that all methods work
Nonlinear
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[8] Eric W. Weisstein.
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From
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Mathworld–A Wolfram Web Resource. http:
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7
Mathworld–A
Wolfram
Web
Resource.
http://
ize some of these results. Investigating up to 4 ×10
mathworld.wolfram.com/LogisticMap.html.
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strange attractor. Commun. Math. Phys., 50:69– A.1 The theorem
77, 1976.
Theorem 1 (Continuous-time Multiplicative Er[26] O.E. Rössler. An equation for continuous chaos. godic Theorem). Let ρ be a probability measure on
a space M , and f : M → M a measure preserving
Physics Letters A, 57:397–398, 1976.
map such that ρ is ergodic. Let also T : M → the
[27] P.-F. Verhulst. Recherches mathématiques sur la m × m matrices be a measurable map such that
loi d’accroissement de la population. Nouv. mm.
Z
de l’Academie Royale des Sci. et Belles-Lettres
ρ(dx) log+ kT (x)k < ∞,
de Bruxelles, 18:1–41, 1845.
+
[28] P.-F. Verhulst. Deuxième mémoire sur la loi where log (u)=max(0,log(u)). Define the matrix
n
n−1
x) · · · T (f x)T (x). Then, for ρ-almost
d’accroissement de la population.
Mm. de Tx = T (f
all
x,
the
following
limit exists:
l’Academie Royale des Sci., des Lettres et des
Beaux-Arts de Belgique, 20:1–32, 1847.
lim (Txn∗ Txn )1/2n = Λx
(50)
n→∞
[29] Peter Grassberger and Itamar Procaccia. Characterisation of strange attractors. Physical Re- (We have denoted by Txn∗ the adjoint of Txn , and
taken the 2nth root of the positive matrix Txn∗ Txn )
view Letters, 50:346–349, 1983.
B
COMPUTER PROGRAMS
The logarithms of the eigenvalues of Λx are called
characteristic exponents. These are just the Lyapunov exponents as defined in (29). They are ρalmost everywhere constant. So the Lyapunov exponents are not dependent on the initial value, only
in a subset of ρ-measure zero can they be different.
Let λ(1) > λ(2) > · · · be the characteristic exponents, no longer repeated by multiplicity; we call m(i)
(i)
the multiplicity of λ(i) . Let Ex be the subspace of
Rm corresponding to the eigenvalues ≤ exp λ(i) of
(1)
(2)
Λx . Then Rm = Ex ⊃ Ex ⊃ · · · and the following
holds
19
frequencies the natural measures of the boxes. For
a typical x0 in the basin of attraction, the natural
measure of a typical box Ci is
η(Ci , x0 , T )
T →∞
T
µi = lim
(52)
where η(Ci , x0 , T ) is the amount of time the orbit
originating from x0 spends in Ci in the time interval
0 ≤ t ≤ T.
B
Computer Programs
The implementation of the algorithms given in the
text were all written as m-files in Matlab. The source
1
code for all these programs can be found on my weblim
log kTxn uk = λ(i)
(51)
site http://m0219684.kuleuven.be (click on Matlab
n→∞ n
Programs). Here you can also find some pictures of
(i+1)
(i)
. In particular, for all vectors u the strange attractors.
if u ∈ Ex \Ex
(2)
that are not in the subspace Ex , the limit is the
(1)
largest characteristic exponent λ .
Theorem 2 For ρ-almost all x,
When we randomly choose a vector u, it will
most probably not lie in one of the special subspaces
(i)
Ex i > 1. So this theorem then says that we will get
the largest Lyapunov exponent when calculating the
limit. To get the ith exponent, we have to carefully
(i+1)
(i)
. This
select a vector u such that u ∈ Ex \Ex
makes the computation of the other Lyapunov exponents very difficult.
A.2
A measure on the attractors
In order to apply these theorems to the strange attractors we encounter in the dynamical systems under
study in this project, we need to make sure that there
is a measure on these attractors. This measure can
be defined as follows.
We can cover the attractor with a grid of boxes
(as in computing the box-counting dimension) and
then look at the frequency with which typical orbits
visit the various boxes covering the attractor in the
limit that the orbit length goes to infinity. When
these frequencies are the same for all initial conditions
in the basin of attraction of the attractor except for
a set of Lebesgue measure zero, then we call these