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) = aN(t)(1" b N(t)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

!

MR =ddx(px) = p + x

dpdx

= A " 2Bx + 3Cx 2 " 4Dx 3

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 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

!

qt+1 = qt +"#t $#t$1

qt $ qt$1

where

!

"t ="(xt ) = (MRt #MCt )d$0

xt

% is the profit at time t, qt is the quantity of goods

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

!

xt+1 = yt

yt+1 = yt +"#(yt ) $#(xt )

yt $ xt

Replacing the coefficients above, we obtain

!

xt+1 = ytyt+1 = yt +"(3.6 # 2.4(xt + yt ) + 0.6(xt

2 + xt yt + yt2) # 0.05(xt

3 + xt2yt + xt yt

2 + yt3)

(2.1.1)

(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

!

xt+1 = yt+1 = yt = xt in the above system of equations (2.1.1):

!

(3.6 " 4.8x t+1.8xt2 " 0.2xt

3) = 0 The roots of this equation are

!

xa = 3 and

!

xb,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 In order to calculate the stability of the fixed points, we can rewrite the system of equations at the fixed point as

!

xt+1 = xt +"(3.6 # 2.4(xt + xt ) + 0.6(xt2 + xt xt + xt

2) # 0.05(xt3 + xt

2xt + xt xt2 + xt

3) 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(xa ) = 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(xb ) = J(xc ) = 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"K2

>2(1+"K)

2>1 for K>0

On the other hand,

!

!

(1+"K) ± (1+"K)2 + 4"K2

<1

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

!

xb,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

!

x =(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).

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

!

U(x,y) =x

x + y" ax

V (x,y) =y

x + y" by

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

!

x =ya" y

y =xb" x

which can be discretized as

!

xt+1 =yta" yt

yt+1 =xtb" xt

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.,

!

xt+1 =yta" yt yt #

1a

xt +$ else

% & '

( '

yt+1 =xtb" xt xt #

1b

yt +$ else

% & '

( '

Solving for the singular points, we obtain

!

x0 =b

(a + b)2

y0 =a

(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 or

!

3 " 2 2 # b /a # 3+ 2 2 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

parameter

!

0.5 " ab" 6.25 , for initial conditions minimally perturbed with respect to the

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

!

xt+1 = µ(1" xt )xt 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

!

xt+1 = xt , i.e., for the map above

!

xt = µ(1" xt )xt # µ(1" xt ) =1# xt =µ "1

µ and

!

xt = 0

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

!

xt+1 = f (xt ) 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

!

f (x) " f (a) + f ' (a)(x # a), from which

!

xn+1 " axn " a

=f (xn ) " axn " a

#f ' (a)(xn " a)

xn " a= f ' (a). The

derivative measures the rate at which successive iterates approach or diverge from a. Tool 1: Box plot tool Graphically, an appropriate tool to study the conditions of the theorem is the box plot of 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.

!

"(x0) = limn#$

1n

ln f ' (xt%1)t=1

n

&

!

"(x0) > 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,

!

" = limn#$

1n

ln Jt % vtt=1

n

&

where

!

vt =Jt"1# vt"1Jt"1# vt"1

is the direction vector (an implementation of this in Matlab can be

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:

!

xt =1" axt"12 + yt"1

yt = bxt"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 (xày). 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

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.

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