Nonlinear Dynamics in Economic Models
Market Models: Monopoly and Duopoly
ELEC 507 Project Report
Eugenio Magistretti
The study of the evolution of dynamic systems in the form of x˙ = f (x,t) has been
subject of economic research for centuries now, with the goal of improving the
understanding of phenomena such as economic growth, cycles, and market analysis, in
order to anticipate and control their behavior. For example, the logistic population growth
model
!
N˙ (t) = aN(t)(1 " bN(t))
!
accounts for resource effects on population growth N(t).
Historically, such models were highly simplified in order to obtain linear approximations
that permitted their study. However, especially in the twentieth century the advances in
mathematics and computer simulation (and visualization) have permitted to return to the
study of the original complex models (and devise new ones) to obtain more accurate
descriptions of reality. For example, to analyze how income affects the population
growth, the Maltusian model above has been extended by Haavelmo as:
N(t)
N˙ (t) = aN(t)(1 " b
)
Y(t)
Y(t) = AN(t) #
where Y(t) is the total income of the population (i.e., the income per capita is Y(t)/N(t)) –
the examples are taken from [a].
!
A wide range of applications of nonlinear models has been devised in economics. In this
project, I aim to discuss several of the most important nonlinear Market Models, and to
delve into their behavior through simulations.
My main reference will be the recently published textbook by Tomu Puu [b], which
covers basic principles of nonlinear models and focuses on economic applications. The
state space approach adopted in this book favored its choice with respect to, e.g.,
Zhang’s classic [a].
Beside market models, relevant applications of nonlinear models to economics include
the topic of fractal markets, e.g., as treated in [c] where the author takes an alternative
look to time series diverging from the traditional ARCH/GARCH models in order to
present evidences of deterministic chaotic behaviors that go beyond the probabilistic
noise. Recently, the famous popular best-seller “Black Swan” by Taleb [d], disciple of
the famous Mandlebrot, has generated a wide attention toward the damage that chaotic
effects may have in an economic system were most monetary resources are concentrated
in the hand of few giant banks.
[a] W.-B. Zhang “Differential Equations, Bifurcations, and Chaos in Economics”, Series
on Advances in Mathematics for Applied Sciences, vol.68, World Scientific publisher,
2005.
[b] T. Puu, “Attractors, Bifurcatorions, and Chaos – Nonlinear Phoenomena in
Economics,” Springer 2000.
[c] E. Peters, “Fractal Market Analysis – Applying Chaos Theory to Investment and
Economics,” John Wiley and Sons, 1994.
[d] N.N. Taleb, “The Black Swan: The Impact of the Highly Improbable,” 2007.
Organization
This report is organized in one main chapter, and an extensive appendix. The chapter
discusses nonlinear market models. The appendix provides theoretical bases.
Chapter 1 introduces nonlinear market models.
Section 1.1 “Monopoly” discusses a nonlinear model for the monopoly market; the
analysis uses tools including bifurcation analysis, Lyapunov exponent, and the method of
critical lines.
Section 1.2 “Duopoly” discusses a preliminary study of the duopoly market according to
the model formulated by Cournot, and includes the bifurcation analysis.
Appendix A discusses the general theoretical bases of nonlinear analysis that are required
to understand the economic model, but that go beyond the subjects covered in the course.
Notice that several additional topics, such as Symbolic Dynamics, Sharkovsky’s
Theorem, Schwarzian Derivatives, Fractal Dimensions will not be covered because either
less important in this project, or because also discussed in class.
Specifically, Section A.1 “Stability of Iterated Maps” introduces the concepts of
nonlinear mapping, which is particularly important in economics since most of the
models are expressed in discrete terms.
Section A.2 “Chaos Identification” discusses tools devised to identify chaotic behavior,
such as the bifurcation diagram and the Lyapunov Exponent (in 1 and 2 dimensions). For
space reasons, the symbolic dynamics are not covered.
Section A.3 “Henon Model – simulative study” shows how to apply the tools above, with
an example based on the Henon model; simulation results obtained in Matlab are
included.
Section A.4 “Attractor Identification - Method of Critical Lines” covers the method of
critical lines used to draw the boundary of the chaotic attractor.
Tools
A number of relevant tools have been used in this project. Here, we provide references,
delaying a short description until needed.
[1]
XPP
AUTO
Bifurcation
Analysis
Tool
http://www.math.pitt.edu/~bard/bardware/tut/xppauto.html
[2] MATCONT Bifurcation Analysis Tool http://www.matcont.ugent.be/
[3] LET Lyapunov Exponent http://www.mathworks.com/matlabcentral/fileexchange/233
CHAPTER 1 NONLINEAR MARKET MODELS
This chapter discusses economic market models (specifically, the monopoly and the
duoopoly) that, while typically formulated in a linear framework, can be extended to
nonlinear for the sake of generality, as well as accuracy.
The premise to this work is the general economic framework of markets. Economists
represent market behavior in the quantity/price plane. Specifically, they identify a supply
curve (with positive slope) that represents the propensity of producers to produce a larger
quantity with the incentive of a higher price, and a demand curve (with negative slope)
that represents the propensity of consumers to buy a larger quantity under the incentive of
a lower price. The equilibrium point of competitive markets is the intersection of the
curves.
However, not all markets behave according to this rule. Other types of markets are easier
analyzed by considering two more quantities related to the production cost/benefits. The
marginal revenue is the revenue that one additional produced quantity of good brings to
the producers (this can be analytically determined by the demand curve), while the
marginal cost is the cost of producing such element. The marginal revenue curve
typically decreases with the quantity, while the marginal cost increases. Notice that in
most economies these values are not fixed and depend on the produced quantity.
1.1 Monopoly
In the monopoly markets the price of the goods is not determined by the intersection of
supply/demand curves; in fact, the monopolist effectively chooses the price by producing
the quantity of goods that permits to maximize its profit. Specifically, it can be easily
shown that the profit is maximal when the quantity of produced goods corresponds to the
intersection of marginal cost and marginal revenue curves.
The typical monopoly model considers marginal cost and marginal revenues linear (or at
least monotonic) curves in the quantity. However, there are several known instances
(Robinson, “The economics of imperfect competition,” Cambridge University Press,
1933) where such model diverges from reality, in particular because the demand curve
generates a non-monotonic marginal revenue curve. This may give rise to several
intersection points between marginal revenues and costs, where each intersection point is
a singular (equilibrium) point of the system.
Before going into the technical details, I would like to notice that the analysis carried
below brought to identify a number of incorrect derivations in the book (I denote this as
“Errata Corrige” in the text).
The setting The goal of this part of the report is to understand the behavior and the
equilibrium of the singular points of a common nonlinear monopoly model, where the
demand takes the form
p = A " Bx + Cx 2 " Dx 3
and generates a marginal revenue curve as
!
d
dp
( px) = p + x
= A " 2Bx + 3Cx 2 " 4Dx 3
dx
dx
The total cost is instead assumed to be convex, with marginal cost first decreasing with
increasing supply and eventually increasing
MC = E " 2Fx + 3Gx 2
MR =
!
!
The problem If the monopolist knew exactly the two curves, the monopolist would be
able to compute the intersection points. In practice, there are a number of reasons why the
monopolist cannot know the curves, e.g., market researches are difficult and expensive,
and the market of close substitutes may rapidly change. In that case, the simplest
algorithm is to estimate the difference of marginal costs and revenues from the last two
visited points, using the Newton-like method
# $ #t $1
qt +1 = qt + " t
qt $ qt $1
xt
where "t = "(x t ) =
% (MR
t
# MCt )d$ is the profit at time t, qt is the quantity of goods
0
!
!
!
!
!
!
produced at time t, and δ the step size. These Newtonian iterations may lead to any of the
singular points depending on the coefficients and δ. By following [b], we assume A=5.6,
B=2.7, C=0.62, D=0.05, E=2, F=0.3, and G=0.02.
!
It is clear that the second-order map above can be transformed into the system
x t +1 = y t
#(y t ) $ #(x t )
y t +1 = y t + "
yt $ xt
Replacing the coefficients above, we obtain
x t +1 = y t
(2.1.1)
y t +1 = y t + " (3.6 # 2.4(x t + y t ) + 0.6(x t2 + x t y t + y t2 ) # 0.05(x t3 + x t2 y t + x t y t2 + y t3 )
(Errata Corrige)
Fixed points First of all, we wish to calculate the equilibrium points of the map and their
stability. According to Section A.1, the fixed point of the system can be calculated by
replacing x t +1 = y t +1 = y t = x t in the above system of equations (2.1.1):
(3.6 " 4.8x t +1.8x t2 " 0.2x t3 ) = 0
The roots of this equation are x a = 3 and x b,c = 3 ± 3 , which are also the fixed points.
Notably, the same result could have been derived by solving "' (x) = MR # MC = 0 , since
!
we know that the fixed points must be the quantities for which marginal revenues equal
marginal costs.
!
!
! stability of the fixed points, we can
Stability of the fixed points In order to calculate the
rewrite the system of equations at the fixed point as
x t +1 = x t + " (3.6 # 2.4(x t + x t ) + 0.6(x t2 + x t x t + x t2 ) # 0.05(x t3 + x t2 x t + x t x t2 + x t3 )
and compute the Jacobian of the map by taking the derivative of the equation above
(Errata Corrige)
J = 1+ " (#4.8 + 3.6x # 0.6x 2 )
At the fixed point xa the absolute value of the Jacobian is
!
!
!
!
!
J(x a ) = 1+ 0.6"
which is always greater than 1 for positive δ; in conclusion, xa is unstable.
At the fixed points xb and xc the absolute value of the Jacobian is
J(x b ) = J(x c ) = 1 "1.2#
which is smaller than 1 for δ<5/3; in conclusion, xb and xc are stable for δ<5/3.
An identical conclusion can be reached by formally considering the map two-dimensional
and computing the Jacobian. In that case the characteristic equation would be:
p( ") = "2 # " [1+ $ * g(x, y)] # $ * g(y, x)
where
g(x, y) = "2.4 + 0.6x +1.2y " 0.15y 2 " 0.1xy " 0.05x 2
The characteristic equation has two solutions
(1+ #g(x, y)) ± (1+ #g(x, y)) 2 + 4#g(y, x)
"=
2
Consider g(x,y)=g(y,x)=K at the equilibrium, it is easy to show that
(1+ "K) + (1+ "K) 2 + 4"K
2(1+ "K)
>
>1
2
2
for K>0
On the other hand,
!
!
(1+ "K) ± (1+ "K) 2 + 4"K
<1
2
for K<0 and 1>1+δK>0, i.e., 0>δK>-1 (in fact, the “-” solution is decreasing over the
whole interval, with maximum of 1 for δK=0, while the “+” solution is decreasing on
[0,L], and increasing on [L,1], with maxima of 1 for δK=0 and δK=1).
The rest of the chapter is devoted to the study of the model via simulation. Section 2.1.1
includes a fundamental analysis via bifurcation tools, aimed to show the influence of the
parameter on the model behavior. Section 2.2.2 is devoted to the study of number and
shape of the attractors, and of their basin of attraction.
1.1.1 Fundamental Analysis
Bifurcation Analysis, Lyapunov Exponent, and State Space confirmation
Bifurcation Analysis Figure 1 shows the bifurcation graph for the model parameter δ,
obtained via manual simulation of the system. The methodology adopted to generate the
graph consists in simulating the model for 2000 iterations, discarding the first 1750, and
recording all points visited in the last 250. These are the points represented on the graph.
The procedure is iterated for 200 equally spaced values of δ, from 1 to 3.88. In order to
rule out numerical instabilities, the initial conditions of the model are varied in an
epsilon-ball around the stationary points; still, the practical dependence on initial
conditions cannot be excluded.
Figure 1 shows that the two steady solutions xb and xc coexist for δ<5/3 (in the plots,
these values are to the left of the first black vertical line). As δ exceeds that threshold, the
steady solutions are substituted by two coexistent cycles (first bifurcation point in the
figure).
Hopf Bifurcation: AUTO In order to exactly determine the position of the first
bifurcation point, the AUTO software has been used. The output of the program, as
shown in Figure 2, confirms that the system has a Hopf bifurcation (indicated as HB in
the figure) at δ~1.66 (i.e., 5/3). However, AUTO did not recognize any other bifurcation
in the range 1<δ<3.5, starting from the initial condition x b,c = 3 + 3 .
!
Figure 1 Bifurcation Graph
Figure 2 Bifurcation Analysis with AUTO
Further analysis of Figure 1 shows the definitive presence of chaotic behavior for values
of δ>2.7 (second black vertical line in the graph), where the points form continuous
sequences along the y-axis. The color plot also shows interesting conclusions about the
attractors. Specifically, for parameter value δ<2.83 (third black vertical line in the graph),
two expanding attractors co-exist, corresponding to starting conditions close to the either
of the singular points. For values of δ>2.83, the attractors immediately fully intermingle,
and a single attractor is generated. The state-space model simulations later in the chapter
(see Figure 6) nicely confirm these results.
Finally, further investigation should be devoted to the unclear sequences about 2.5<δ<2.7
(see also the study of the largest Lyapunov exponent below).
Lyapunov Exponent The largest Lyapunov exponent gives a good indication of chaotic
behavior, by representing the sensitivity of the model to the initial conditions.
Specifically, the largest Lyapunov exponent represents the exponential rate of trajectory
divergence. Our study investigates the largest exponent for the model, by following the
procedure delineated in Section A.2.3 in the Appendix. In this experiment, we simulate
the model for starting conditions (2,2), and for 100000 iterations, for 0.1<δ<3.5 in steps
of 0.01. Figure 3 shows that the system behaves consistently chaotically starting from
δ>2.7. The spike at δ=2.6 (enlarged in Figure 3.B(ottom) plot) reflects the analogous
behavior present in the bifurcation graph.
Figure 3 Largest Lyapunov Exponent. The B(ottom) image zooms into the
parameter range for which the exponent is positive.
Lyapunov Exponent: LET It is possible to assist manual simulation with automated tools
to confirm the results above. Among the number of automated tools available to perform
the study of the Lyapunov exponents, we adopted LET because it provides discrete
system analysis. Notably, LET shows the evolution of both Lyapunov exponents (and not
only the largest as studied above). The tool shows the temporal evolution of the
Lyapunov exponents, as the number of iterations increases, but only for a fixed value of
the parameter δ. Figure 4 shows the result we obtained for δ=3.0. We notice that the
result obtained for the largest exponent nicely matches the one obtained manually.
Figure 4 Lyapunov Exponents for δ=3.0 with LET
State Space Analysis of the Chaotic Regime Figure 5.T (resp. Figure 5.B) visualizes the
trajectory of the system for δ=2.3 (resp. δ=2.7). For the sake of clarity, the trajectories in
both figures are generated for 2000 iterations of the map; however, for clarity both
figures discard the first 900 iterations, so that the trajectories before the limit behavior do
not clutter the plot. Only trajectories corresponding to stable starting points are shown.
Figure 5.T shows that for δ=2.3 the stable solutions converge to two clearly defined limit
cycles. Figure 5.B shows that for δ=2.7 the solutions converge to two chaotic attractors.
Figure 5 T(op) – State space representation of the limit cycle behavior for δ=2.3;
B(ottom) Chaotic attractors for δ=2.7
Theoretically, the reference book shows that the model is expected to become chaotic for
δ>2.488. However, the results above show sporadic practical evidences of chaos only for
δ>2.6 (see Figure 3 – there is a small hardly perceptible positivity of the Lyapunov
exponent for 2.5<δ<2.6 – see the color plot), and consistent continuous systematic
evidences for δ>2.7. The reason for this is the “discontinuous chaoticity” (with respect to
the parameter δ) of the model for 2.488<δ<2.7 (and similarly the “discontinuous non-
chaoticity” for 2.8<δ<2.9), i.e., there is a discontinuous set of parameters for which the
system is chaotic.
1.1.2 The Attractors
Number, shape and basins
Number of coexisting attractors In the following, we use graphical tools to show the
behavior of the system for larger values of δ. Specifically, we study the basin of
attraction of the pairs of attractor " <˜ 2.8 (Figure 6.Top) and of the single attractor 2.85<δ
(Figure 6.Center and 6.Bottom), until complete instability.
!
Figure 6 – Merging of two chaotic attractors for δ=2.8 (Top) into a single attractor
for δ=2.85 (Center) and for δ=3.5 (Bottom)
The basin of attraction We explore the basin of attraction by simulating the temporal
evolution of the system from t=0 to t=300, for N initial points located in the range
[-7.5,12.5]x[-7.5,12.5]. We explore the solution for 1≤δ≤3.6 for N=2000 initial points,
and increase to N=10000 point for δ=3.9, i.e., once the stable initial conditions become
extremely sparse.
Figure 7 Basin of attraction (stable initial conditions denoted by red X marks)
Figure 7 shows the basin of attraction for the two extreme values δ=1 and δ=2.5; in this
range of the parameter, the basin of attraction is a connected ellipsis, which slightly
shrinks as the value of δ increases.
As the value of δ exceeds 3.4 (Figure 8L), the basin of attraction is no more convex, and
includes “holes” containing starting conditions within the ellipsis that lead to unstable
behaviors (see also Figure 6.9 in the book – the case therein closely resembles the plot for
δ=3.5, not reported here for brevity). As δ increases, the basin of attraction becomes more
and more perforated by unstable initial conditions (e.g., Figure 8R for δ=3.6).
Figure 8 Basin of attraction (stable initial conditions denoted by red X marks)
Figure 9 shows that when δ=3.9, the stable solutions become extremely sparse. For δ>4.0
the system becomes unstable for any choice of starting points.
Figure 9 Basin of attraction (stable initial conditions denoted by red X marks)
The shape of the attractor As discussed above, practically for δ>2.7 the system becomes
chaotic. For this range of δ, it is important to study the shape of the attractor and, in case
of multiple attractors, their basin of attraction. We have already performed a study of the
transition from the regime where two attractors coexist to the regime where the attractors
are merged (δ>2.83). In this part, we delve into the study of the shape of the single
attractor with the method of critical lines.
Method of critical lines The method of critical lines is a powerful tool to determine the
shape of the attractor. In order to apply it, we need to identify the folding lines of the
model at the first iteration. In general, the folding lines are denoted by the null derivative;
in multiple dimensions, the derivative is naturally replaced by the determinant of the
Jacobian.
J = "#Px = 3(x " 3) 2 + 2(x " 3)(y " 3) + (y " 3) 2 " 6
(24 " 2y) ± 48y " 8y 2
6
To apply the method, we select a subset of the denoted ellipse, i.e., for 2<y<4, with a step
of 10-3. By iterating the model (7 times in this implementation), the method delineates the
shape of the attractor. Figure 10 denotes the application of the method for δ=2.8 and
δ=3.5. The method obtains a good approximation of the attractor (see Figure 6 Center and
Bottom, where the attractor is obtained by simulation).
x=
!
!
Figure 10 Critical Lines for 7 iterations of the initial points denoted by the green
lines (Top δ=2.8, Bottom δ=3.5).
1.2. Duopoly
The basic widely studied market models are monopoly, where the monopolist decides the
price, and perfect competition, where all participants are price-takers. Duopoly markets
represent a first intermediate step, whose study is much more challenging than the
extremes. The two milestone models of duopoly are due to Cournot (1838) and
Stackelberg (1938); in this report, we focus on the first. It was not conjectured until 1978
(by David Rand) that Cournot model can generate chaos.
1.2.1 Cournot’s Model
!
In brief, in the Cournot model each duopolist assumes the last step of the competitor to
be the competitor’s last step, and optimizes its own reaction based on such assumption.
Assumptions and notation Assume isoelastic demand (i.e., changing as 1/p, where p is
the price of the good), and denote with x and y the quantity of goods the competitors
produce respectively. Thus, under the assumption that all supplies are consumed,
p=1/(x+y). Assume also that the duopolists produce with constant marginal costs, a and b
respectively (i.e., their total costs are ax and by respectively).
Model derivation and analysis Under the assumptions above, the profits of the firms are
x
U(x, y) =
" ax
x+y
y
V (x, y) =
" by
x+y
The first firm aims to maximize U(x,y) with respect to the variable it controls, i.e., the
quantity of produced goods x. Similarly, the second firm aims to maximize V(x,y) with
respect to y. Equating the partials to 0, we obtain
y
x=
"y
a
x
"x
b
which can be discretized as
yt
x t +1 =
" yt
a
y=
!
xt
" xt
b
The second order conditions show that these are indeed maxima. Unfortunately, it is well
possible that the factors under square root carry negative signs. Considering that x and y
are quantities of goods, i.e., positive and real, we introduce a correction inspired to the
one discussed in a conference paper of F. Tramontana, L. Gardini, T. Puu, “New
properties of the Cournot duopoly with isoelastic demand and constant unit costs”, i.e.,
y t +1 =
!
!
!
!
% y
1
' t
x t +1 = & a " y t y t # a
'( x + $
else
t
% x
1
' t
y t +1 = & b " x t x t # b
'( y + $
else
t
Solving for the singular points, we obtain
b
x0 =
(a + b) 2
a
y0 =
(a + b) 2
By equating the Jacobian to 1, it is also possible to show that the model is stable for
3 " 2 2 # a /b # 3 + 2 2
3 " 2 2 # b /a # 3 + 2 2
or
1.2.2 Fundamental Analysis of Cournot’s Model
Bifurcation
!
As preliminary step of our analysis, we perform the investigation of the bifurcation of the
model. Because the system symmetrically depends on the two parameters a and b, we
consider as bifurcation parameter the quantity a/b and fix b=1. Our procedure does not
lose any generality, since any other parameter setting will mirror the results we obtain. In
order to perform the analysis of the bifurcation we simulate the model for a range of
a
parameter 0.5 " " 6.25 , for initial conditions minimally perturbed with respect to the
b
singular points x0 and y0. With a procedure similar to the one we adopted in the
monopoly case, we iterate our model 3000 times, and we record the last 750 points
visited; then, we plot all state values obtained.
!
!
Figure 11 shows the bifurcation graph; again, without loss of generality, the state variable
we plot is x, but y would mirror the tren). As we can see x0 is stable for values of
a /b " 3 + 2 2 ; after that point, we see that the line splits. In order to better investigate
the model for a/b>5.85, we zoom in that region in Figure 12. The figure shows that the
system moves into progressive period doubling cycles of period 2/4/8 until, for a/b~6.17
chaotic behaviors appear. It is significant to plot the temporal behavior of the iterations
for a/b in the chaotic region. Figure 13 shows the evolution of the model for a/b=6.234
for 3000 iterations, and confirms the chaotic behavior.
As a conclusion to this part of the study, we notice that the investigation of the chaotic
behavior of the Cournot’s Model is considerably more challenging than the monopoly
model for a number of reasons, including the very limited parameter ranges for which the
chaotic behavior emerges.
Figure 11 Bifurcation graph of the Carnot’s Duopoly Model
Figure 12 Enlargement of the chaotic behavior of the Cournot’s Duopoly Model.
Figure 13 Temporal evolution of the X and Y variable for the Cournot’s Duopoly
Model
Conclusions
As acknowledged by recent advances in economics research, nonlinearity plays a major
role in economic phoenomena, and consequently in economic models. Specifically,
chaotic situations may emerge and severely affect the expected outcomes of control
operations, through the extreme sensitivity to initial conditions. The study in this report
aims to use the analytical tools studied in the class (with the addition of few more
reviewed in the appendix) to capture and understand the behavior of the monopoly and
duopoly markets. This permits to determine which parameter settings generate chaos, and
thus to be aware of potential dramatic inaccuracies in the predictions.
The most important tools identified are the following.
1) The bifurcation analysis emerges as the fundamental tool that permits to identify not
only chaotic parameter conditions, but also limit cycles and their periodicity. 2) The
analysis of the largest Lyapunov exponent is useful to determine how sensitive the
models are to the initial conditions. 3) Finally, the method of critical lines permits to
delineate bounds for the attractors, and thus for the potential trajectories, in case of
chaotic parameter settings.
APPENDIX A THEORETICAL FOUNDATIONS
A.1 Stability of Iterated Maps
Two reasons make the study of iterated maps particularly important. First, most relevant
economic models are formulated in discrete time. Second, the first return maps for points
on the Poincare’ sections is a primary tool of our analysis. In particular, the second aspect
is of primary importance to understand that maps are even more unstable than
differential equations. In fact, differential equations need to be at least three-dimensional
in order to display chaotic behaviors. The Poincare’ sections, having one dimension less
than the system they describe, seem to indicate that two-dimensional maps are needed to
obtain chaos; actually, maps can exhibit chaos even in one dimension.
For instance, for 3.57<µ<4 (i.e., between Feigenbaum point and instability) the logistic
map
x t +1 = µ(1 " x t )x t
behaves chaotically (to be more precise there are also infinite parameter choices between
3.57 and 4 for which the system is non chaotic – see below).
!
Fixed point
First, we rapidly notice that the notion of fixed points replaces that of a singular point for
maps. A fixed point is denoted by the map x t +1 = x t , i.e., for the map above
µ "1
x t = µ(1 " x t )x t # µ(1 " x t ) = 1 # x t =
and x t = 0
µ
Stability of a fixed point
!
!
!
It is important to remark immediately that !the conditions for the stability of a map are
different than for differential equations.
Theorem
A fixed point a is stable for the map x t +1 = f (x t ) if the absolute value of the derivative of
f at a (or of the absolute value of the eigenvalues of the Jacobian in multiple dimensions)
is strictly less than 1. The point is unstable if the derivative is strictly greater than 1.
Proof sketch: This theorem is easy to show informally by using the approximation
!
x " a f (x n ) " a f ' (a)(x n " a)
=
#
= f ' (a) . The
f (x) " f (a) + f ' (a)(x # a) , from which n +1
xn " a
xn " a
xn " a
derivative measures the rate at which successive iterates approach or diverge from a.
Tool 1: Box plot tool
! to study the conditions of the theorem is the box plot of
Graphically, an appropriate tool
the curve f(x) and the y=x line. The intersection point is of course a fixed point, whose
stability can be determined by inspecting the slope of the curve at the intersection (we
discussed similar curves in class, while studying the Poincare’ sections).
The application of this tool to the logistic map is a very interesting example (see section
4.2 in [b]).
A.2 Chaos Identification
A.2.1 Bifurcation Diagram
Since we have discussed such tool in class, it is necessary only to recall how it is possible
to use this tool to identify chaotic behavior. The bifurcation diagram shows the
relationship between a parameter and the fixed points (or periodic orbit values) of the
system. The bifurcation diagram shows chaos when entire intervals seem to be visited by
the plot; of course, we have never full intervals, but disjoint fractal point sets.
A.2.2 Lyapunov Exponent 1-D
!
!
One of the most common method to formally identify chaotic behavior is the Lyapunov
exponent, which measures the exponential rate of measurement error magnification.
The magnification of the increase in the divergence of initial conditions is measured by
the absolute value of the derivative of the mapping. The Lyapunov exponent measures
the separation over a long run of iterations.
1 n
"(x 0 ) = lim & ln f ' (x t %1 )
n#$ n
t =1
"(x 0 ) > 0 means that different initial conditions separate at an exponential pace, i.e. the
system is sentive to the initial conditions, and hence chaotic.
Observation 1 With respect to the logistic map example (Figure A.1 below), reference [b]
shows that the Lyapunov exponent crosses 0 at the Feigenbaum point. It is relevant to
notice that the Lyapunov exponent does not consistently exceed 0, but dips back to the
negative side infinitely many times, as the parameter belongs to the chaotic region (e.g.,
3.57<µ<4). This is because there are infinitely many values of the parameter inside what
we called the chaotic region that generate non chaotic behaviors.
Observation 2 Notice also that at the Feigenbaum point the chaotic behaviors involve
points still in the (0,1) range of the x variable. At µ=4 the chaotic behavior expands
outside this interval, and the system becomes sooner or later unstable (easy to see by
considering starting point outside such an interval in the map).
Figure A.1 – The logistic map: bifurcation and Lyapunov exponent (non original)
A.2.3 Lyapunov Exponent 2-D
In two dimensions the interpretation of the Lyapunov exponent becomes even more
interesting. Intuitively, there need to be a number of Lyapunov exponents equal to the
number of dimensions, and each exponent represent the evolution of the trajectories
along a given dimension (squeezing and elongation). In chaotic systems, not all
Lyapunov exponents can be positive (as they would indicate continuous expansions along
all dimensions and thus instability), but some of them will typically be positive
(expanding dimensions), while some of them will be negative (contracting dimensions).
In order to identify chaos, the largest Lyapunov exponent is typically the most relevant.
!
!
In two dimensions, the Lyapunov exponent needs to be adapted by replacing the
derivative with the Jacobian (and, technically speaking, keeping in mind the direction of
the computation). Formally,
1 n
" = lim & ln J t % v t
n#$ n
t =1
J #v
where v t = t "1 t "1 is the direction vector (an implementation of this in Matlab can be
J t "1 # v t "1
found in the next section). Notice that, after some iterations, the vt updating rule
described aligns vt with the direction of elongation of the system, and ends up providing
the largest Lyapunov exponent.
A.3 Henon Model – simulative study
The study of the Henon model is relevant from the point of view of understanding the
geometry of the transformations of a discrete map toward an attractor, and of the
Lyapunov exponent. The mapping is the following:
x t = 1 " ax t2"1 + y t "1
!
y t = bx t "1
Geometric transformations. It is possible to think this mapping is formed of three basic
operations operating on a rectangle: 1. Bending to a parabola (1-x2+y); 2. Squeezing in
the horizontal direction (bx); 3. Rotation (xày). Further iterations repeat such sequence,
by generating more and more complex objects that decrease in area (according to the
squeeze of step 2, by a factor bx), and elongates in one direction.
Figure A.2 shows the results of 6 iterations for a=1.4 and b=0.3 (typical parameter choice
for the Henon map) for 1000 initial points (in the black rectangle), where the mapping of
each iteration is represented in different colors (cyan, magenta, green, yellow, red and
blue). We observe that already at the 6th iteration the initial rectangle converges toward
the Henon attractor.
Figure A.2 – Henon Map: geometrical transformations of the black rectangle in
successive iterations
The next relevant step is to compute the (maximum given that we are working with a 2-D
map) Lyapunov exponent for the Henon system. In particular, its convergence is the
subject of our study. Figure A.3 shows the convergence of the Lyapunov exponent
resulting averaging the transformation of 22 points in a number of iterations from 10 to
1000. The figure also shows the computed theoretical value of 0.42 (i.e., positive, i.e.,
chaotic).
We observe that as the number of iterations grows the value converges to the theoretical.
Figure A.3 – Convergence of the Lyapunov exponent
for i=1:NUMITER
eval(['XK' int2str(i) '=YK' int2str(i-1) '+1-a*XK' int2str(i-1) '.^2;']);
eval(['YK' int2str(i) '=b*XK' int2str(i-1) ';' ]);
hold all;
vec=['d' color(mod(i-1,6)+1)];
eval(['plot(XK' int2str(i) ',YK' int2str(i) ', vec );']);
%Compute the Lyapunov exponent
eval(['elements = numel(XK' int2str(i) ');']);
eval(['curX = reshape(XK' int2str(i) ', elements ,1);']);
vecsin=sin(theta);
veccos=cos(theta);
absval=((vecsin-2*a*curX.*veccos).^2+(b*veccos).^2).^0.5;
jacvtrow1=(vecsin-2*a*curX.*veccos)./absval;
jacvtrow2=(b*veccos)./absval;
theta=angle(complex(jacvtrow1,jacvtrow2));
partialexp(i,:)=log(absval');
%pause(1);
end
Figure A.4 – Relevant section of the MATLAB code to generate the figures above
1.4 Attractor Identification – Method of the Critical Lines
The method of critical lines permits to give a representation of the boundary of the
attractor in a chaotic system (what is technically called an absorbing area). While
the basin of attraction produces a sketch of the initial conditions that determine attraction
to chaos, this method actually sketches the geometry of the attractor itself to some extent.
Notice that not even this method can precisely identify the attractor, as the attractor has
typically a fractal shape, i.e., it is formed by a set of infinite points that may be separated
by an infinitesimal distance. The method is most useful with noninvertible maps.
Notably, the method of the critical lines also permits to understand the bifurcations
of a chaotic attractor to a chaotic repellor. In fact, when the absorbing area shoots
lines outside the attraction basin, it is easy to foresee that sooner or later any system in
the attractor will reach points outside the basin, and explode to infinity.
Technical description. Generally, a fold of the n-th iteration (necessarily present in
noninvertible maps) is critical in the sense that it separates points with two and zero
preimages, i.e., at the (n-1)-th iteration. Consider the first iteration, the preimage of the
fold identifies the points in the initial space (i.e., where we chose the initial values) that
map into the plane of the variables of the next iteration; notice that the fold line and its
preimage intersect. By repeating the iteration, and identifying a new folding line at each
iteration, and representing all folds in a common x-y plane (in the end, we are
considering an iterative system over the same R2 plane) we obtain a curve with increasing
curvature; this is because the folds of iteration i and i+1 are necessarily tangent due to the
folding. With a sufficient number of iterations (or a preimage of the first fold sufficiently
long), a closed curve is obtained that includes points that have a preimage, i.e., points
inside the curve map to points inside the closed curve. The closed curve is the absorbing
area and includes the attractor.