Heterogeneous expectations and asset pricing models: The dynamical systems approach

UNIVERSITA' DI PISA

Department of Economics and Management

Master of Science in Economics

Heterogeneous Beliefs and Asset Pricing Models:

The Dynamical Systems Approach

SUPERVISOR:

CANDIDATE:

Prof. Mauro Sodini

Andrea Caravaggio

Contents

1 Introduction 7

2 Intruduction to Bifurcation and Chaos Theory 15

2.1 Non-convergence to a steady state point . . . 17

2.2 Bifurcations . . . 20

2.2.1 Period-doubling bifurcation (Flip Bifurcation) . . . . 21

2.2.2 Bifurcation diagram of the quadratic map . . . 22

2.2.3 Tangent bifurcation . . . 23

2.2.4 Creation of a period 3-cycle for xt+1 = xt(1 xt) . . . 24

2.2.5 Transcritical bifurcation . . . 25

2.2.6 Pitchfork bifurcation . . . 26

2.3 Extension to 2-D Dynamical Systems . . . 27

2.3.1 A important example: the Hénon Map . . . 27

2.3.2 De…nition of a strange attractor . . . 30

2.3.3 Hopf Bifurcation . . . 31

2.4 Chaos Theory . . . 36

2.4.1 Measures of Chaos . . . 38

2.4.2 A chaotic invariant Cantor Set . . . 39

2.4.3 Invariant Cantor set for > 4 . . . 41

2.4.4 Lyapunov Exponent . . . 42

2.5 Chaos and Autocorrelation . . . 44

2.6 A …nacial example of Chaos existence . . . 46

3 Adaptive Belief Systems in Asset Pricing Models 51 3.1 A …rst economic example: The simple Cobweb model. . . . 51 3.2 The Mean-Variance Optimization . . . 55 3.3 The Brock and Hommes model with di¤erent belief types 59 3.4 Examples with di¤erent belief types . . . 65 3.4.1 Fundamentalists versus trend chasers . . . 66 3.4.2 Fundamentalists versus opposite biases . . . 71 4 Technical Analysis Approach in Trading 77 4.1 Trading Principles and Dow Theory . . . 77 4.2 Technical Analysts Strategies . . . 81 5 Models With Sophisticated Price Predictions 85 5.1 A model with simple behavioural functions . . . 85 5.2 An extension with technical analysis principles . . . 96

Abstract

The main aim of this work is to analyse asset pricing models with the dynamical system approach. In particular, we assume behavioral rationality and heteroge-neous expectations. First, we describe Brock and Hommes paradigm characterized by adaptive beliefs systems in asset pricing models. They showed how, in a …nan-cial market with heterogeneous beliefs, instability and chaos may occur. Second, we develop two di¤erent models, analysing the evolution of dynamics when more sophisticated agents and expectations are considered. We show that if the hetero-geneity degree increases then chaotic dynamics becomes stronger.

Furthermore, in the …rst model we analyse the contrast between fundamental-ists and simple chartist believing that price follows a simple trend driven by the linear regression; instead, in the second one we emphasize the application of some technical analysis principles in the construction of chartists behavioural functions.

To my father, who sacri…ed his life to give me this chance, and to my mother, who every day helps me to dream a bright future

Acknowledgments

I would like to take the occasion to express my gratitude to my supervisor, Prof. Mauro Sodini who, during this experience, taught me not only all the required mathematical tools to face this work but also to be more "smart and brilliant" to face better my future. His scienti…c approach in solving each problem, sometimes by resorting sheet and pen rather than the pc, allowed me to understand that we have always to face problems and never skip them. Moreover, I thank the professor for his helpfulness and tolerance kept with a terrible student like me.

I am very thankful to Nicla, my girlfriend, who supports me in all my "battles" and spreads her irrepressible desire to "change our stars".

Thanks to my friend Francesco, who is like a brother for me; to all my close friends of a life, as Nico, Paolo, Davide, Luciano and Emiliano and especially thanks to the precious friends met along the way (Marco, Enrico, Giulio, Elena, Tommaso, Eleonora, Asya), partners of studies and games, able to inspire me in ways I have never believed possible.

Chapter 1

Introduction

The need of a nonlinear framework to describe economic phenomena, appeared between the 40s and 50s, when economists such as Goodwin or Kaldor developed non-linear and endogenous models in order to study business ‡uctuations. They demonstrated that linear models were not appropriate to describe these phenom-ena. Economy, being intrinsically unstable, required a system that could record ‡uctuations in variables. These results inspired the revolution on the rational expectations, (Muth, 1961) and besides, the famous "Lucas’critique" of 1976.

Discussing brie‡y the critique, very relevant for the recent history of economic modelling, Lucas argued as it is naive try to predict the e¤ects of a change in economic policy on the basis of relationships observed in historical aggregated data; in particular he suggested to investigate them by modelling the "deep pa-rameters", such as preferences, technologies and resource constraints, that were assumed to govern the individual behavior. The critique had a key role in the creation of "Dynamic Stochastic General Equilibrium" models, (see e.g. Clarida et al., 1999 and Woodford, 2003), which aim is to describe the economic trends by analysing the interaction of many microeconomic decisions. Moreover, these models are both dynamic and stochastic, taking into account the fact that the economy is a¤ected by random shocks as changes and ‡uctuations. However, it is important to underline that this kind of models represent a parallel radically dif-ferent way to analyse the economy as a complex system in respect to the approach

provided in this work; this dissertation will present an analysis that does not take into account possible external shocks and it will use di¤erently the numerical ap-proach to study equilibria and their stability. For example, in DSGE models the convergence (divergence) route is observed starting from initial values very close to the steady states. Further on, it will be discussed in details how di¤erently our approach works.

A great incentive in passing to analyze nonlinear models was given especially by the discovery of “deterministic chaos” (Lorenz, 1963). It can be de…ned as the sensitive dependence of the system by the initial conditions, that is, a small perturbation of the initial states generates a very di¤erent time path prediction in the long run. Moreover, Ruelle, Takens, et al. (1971) mathematically showed that a simple non-linear system of few di¤erential equations, without perturbations triggered from outside, can exhibit a complicated, irregular dynamical behavior. Hence, they introduce the notion of a strange attractor (brie‡y discussed in Chap-ter 2) to identify irregular behaviors in non-linear deChap-terministic dynamical systems, extremely useful in this work.

In the 1980s, many economists, inspired by the idea of "chaos", started looking for non-linear, deterministic models generating erratic time series. This search allowed theoriest to create new frameworks generating chaotic business ‡uctuations (Benhabib and Day, 1982).

By analyzing the problem of non-linearity from an economic point of view, it can be noticed how the agent’s expectations play a crucial role, that is, how today’s decisions of economic agents depend upon their expectations and beliefs about the future. A classical …nancial example of expectations power on the market is represented by the “Dutch tulip mania”, in the seventeenth century. In just one year, the beliefs of the investors concerning the high returns on their investments in which the beliefs of investors for excessive high returns on their investments in tulip bulbs accelerated the price explosion process and its crash back to the original price level, along the same year (Kindleberger, 1996). A more recent example is the 2008 …nancial crisis, in which the decline of …nancial markets of more than 50% was not completely driven by changes in economic fundamentals, but it was ampli…ed by pessimistic expectations and market psychology. Hence, it is clear that the expectation theory is central in the construction of a “law of

motion”governing the economic system, taking into account that “the economy is a highly nonlinear expectation feedback system” (Hommes, 2013). An important contribution of this theory was given by Muth with his introduction to the rational expectations (Muth, 1961).

Although a perfectly rational representative agent model nicely performs into a stable economy, this kind of assumption is not representative of a complex eco-nomic system and it does not allow us to approach the real economy (Kirman, 1992) & (Kirman, 2010). A …rst step toward complexity, as already suggested by Simon in the 1950s, is to construct models with boundedly rational agents, who adapt their behavior on the basis of their past experiences, leading to a highly non-linear dynamic system in which they do not know the law of motion and set up predictions upon observations along time; it is widely known as “adaptive learning approach”(Sargent, 1993). A crucial step in the literature is represented by the study about the stability of rational expectations equilibria under adaptive learning: if adaptive learning allows the convergence to a rational expectations equilibrium, the rational expectations hypothesis would be more fair as a descrip-tion of the economy. In this sense, in several examples the adaptive learning does not converge to rational expectations and displays endogenous, chaotic ‡uctuations and excess volatility (Hommes and Sorger, 1998).

Therefore, starting from the idea of bounded rationality and adaptive learning it is possible to approach also the idea of heterogeneous expectations, which rep-resents the most interesting …eld of analysis, the new frontier. Indeed, it enables to consider the economy as a complex evolving system composed of di¤erent, boundedly rational and interacting agents by means of initial di¤erent decision strategies (as the technical analysis in …nance), scienti…c methods (or heuristics) and forecasting rules. A …rst issue is that, in contrast with the rational expecta-tion hypothesis, the "wilderness of bounded raexpecta-tionality" leaves many degrees of freedom and seems far from clear. Hence, to avoid the problem, the idea was to discipline the class of expectations and decision rules, that is, to de…ne the types of agents, and, further, to set the forms of learning disciplining the class of deci-sion. In Chapter 3, it will be introduced and discussed extensively the approach that makes use of the heterogeneous strategy switching framework of Brock and Hommes (1997), Brock and Hommes (1998) of endogenous evolutionary selection

or reinforcement learning among heterogeneous decision or forecasting rules. Here, the core idea is that agents may switch to rules that have performed better, ac-cording to some economic performance measures, such as realized past pro…ts. A second learning form is given by some parameters changing over time following some adaptive learning process: for example, we can think about trend-following agents and the fact the trend extrapolated may change over time depending on market realizations (Kahneman, 2003).

Therefore, it is possible to understand how behavioral rationality and het-erogeneous expectations lead to highly non-linear dynamical systems, since the fractions attached to the di¤erent rules change over time; as a consequence, the evolutionary system may often be unstable and exhibit complicated, perpetual ‡uctuations. An emblematic example is the study of a Cobweb Model with het-erogeneous expectations and rational against naive producers resulting from the introduction of non-linearities in markets dynamics and the possibility of chaotic price ‡uctuations; in this model, agents can either buy rational expectation fore-casts on the basis of not freely available information or obtain a naive forecasting rule (Brock and Hommes, 1997). Here the information cost for rational expecta-tion represents a more sophisticated constraint for the predicexpecta-tion of future prices. In models like this, as shown by Brock and Hommes in "A Rational Route to Ran-domness", published in 1997, agents can either buy a rational expectation forecast at positive costs or freely obtain a simple, naive forecasting rule. It is important to remember that more sophisticated price predictions are more expensive than those based on simpler schemes. In the dynamics, the fractions of the two types change over time and agents are boundedly rational due to their tendence to switch strategies towards the one performing better in the recent past. In other words, the presence of heterogeneous expectations leads to a natural non-linearity and to an evolutionary dynamics exhibiting precisely a rational route to randomness, also due to the time-varying fractions of di¤erent agent types appear as multiplicative factors in the market equilibrium equation, and this means, probably, chaos.

The analysis provided above is useful to the study of the …nancial markets dynamics. This is con…rmed by the recent introduction of several structural agent models in …nance literature (LeBaron, 2006; Hommes, 2006; Anufriev, Bottazzi,

and Pancotto, 2006)„including nonlinear elements.

Typically, in these models, nonlinearity derives from agents’ trading rules of demand functions (Chiarella, 1992), from evolutionary switching between di¤erent strategies, based on …tness (or performance) measures (Brock and Hommes, 1998), and from phenomena of contagion between pessimistic and optimistic traders groups (Kirman, 1991). These nonlinearities originate some typical dynamics of complex models, such as long-run chaotic behavior in prices oscillation around an unstable steady state (the fundamental price), the existence of more than one equilibrium in the system, and the emergence of bubble and crashes.

Indeed, such models have the peculiarity of being able to capture an important determinant of price oscillations. Chartists use technical trading rules to forecast prices, believing in the persistence of bullish or bearish phases, and formulate their demands accordingly. In contrast, fundamentalists place their orders by assuming that prices will return towards their fundamental value. Hence, prices are set as a function of aggregate investors’ demand by assuming the market clearing condition, or the intermediation of market makers. Endogenous price dynamics are generated by the interaction between these di¤erent forces, operating in the market.

In general, by means of these models, we can achieve important results: …rst, we can know that price movements are at least partially driven by endogenous laws of motion; second, we may have the possibility to replicate a number of stylized facts of …nancial markets, such as excess volatility, bubbles and crashes, fat tails for the distribution of the returns and volatility clustering (Dieci and Westerho¤, 2009). Moreover, these models are useful for testing how regulatory measures recently introduced work. In this …eld, Wieland and Westerho¤ (2005) explore the usefulness of central bank interventions and Bauer et al. (2009) demonstrate that target zones may stabilize the …nancial market (as theoretically shown by Krugman in 1991), while He and Westerho¤ (2005) discuss how price caps may a¤ect the dynamics of some markets, as thosw dealing with commodities.

The literature we are talking about generally focuses on the dynamics of a single speculative market, driven by the interactions between di¤erent boundedly rational heterogeneous traders. In particular, the majority of studies provided in literature concerns the behavior of asset prices in the case of a market with a single

risky asset and a risk free one.

An interesting asset pricing model with heterogeneous beliefs was developed by Brock and Hommes in 1998, based on their own work published on Econometrika in 1997, "A Rational Route to Randomness". Here, agents can either buy a risky asset that pays an uncertain dividend, or invest in a risk-free asset that pays a …xed rate of return, and dividends follow an exogenous stochastic process, known by all the agents. Moreover, they can choose between a …nite set of future price’s predictors and change their beliefs in each period in a boundendly rational way, according to a "…tness measure" such as the pro…ts realized in the past. Hence, price deviations may be triggered by di¤erent choices in beliefs and ampli…ed by evolutionary dynamics between di¤erent schemes.

This means that di¤erences in trading rules play a central role in price ‡uc-tuations. Consequently, according to Hommes, "sophisticated traders, such as fundamentalists or rational traders usually act as stabilizing forces, pushing prices towards rational expectations fundamental value. Instead, technical traders, such as feedback traders, act as destabilizing forces, pushing prices away from the fun-damental".

The largest part of this work is based on this model, stylized and constructed including only one risky asset; but, in order to present here the state of literature, we have to display how, in the last years, this basic idea has been expanded to cases in which the dynamics concerns multiple risky assets and to more general analysis to explore the dynamics of interacting speculative markets. In this …eld, Böhm and Chiarella (2005) establish dynamic setups where prices and returns coevolve over time due to dynamic mean-variance portfolio diversi…cation and updating of heterogeneous beliefs. Following this line, Westerho¤ and Dieci (2006) model the interplay between di¤erent asset markets with fundamental and technical traders, where connections arise from traders switching between markets, depending upon relative pro…tability. Moreover, Corona et al. (2008) showed a model with in-teracting stock and foreign exchange markets studied via numerical simulations and calibrated to match some statistical properties of …nancial market dynamics. Finally, Brock et al. (2009), by extending the stylized model of 1998 through the inclusion of a market for derivative securities, demonstrate how the latter, by providing hedging opportunities, may destabilize …nancial markets.

In conclusion, we can generally see these last models as instruments to show how interactions may destabilize stable markets. Hence, these become a further source of nonlinearity and complex price dynamics, depending on the parameters which characterize agents’behavior.

This work is structured as follows.

Chapter 2 will introduce the instrument of analysis: the bifurcation and chaos theory; in particular will provide a series of examples concerning bifurcation in one and two dimensions. Moreover, there will be introduced measures of chaos with mathematical and …nancial examples.

Chapter 3 will focus on the presence of agent’s heterogeneous beliefs and the use of an adaptive beliefs system to write an asset pricing model. In this sense, the Brock and Hommes asset pricing model will be introduced and discussed as the crucial work in the literature and as the basis for the …nal extension.

Finally, Chapter 4 will provide an extension to the model provided, imple-mented on traders with di¤erent behavioral functions. Since the extension is based on the possible presence of trend-chaser investors, it will be necessary to brie‡y introduce the technical analysis grounded on the work conducted by Professor Fanelli. The …nal part of this chapter will be centered on the simulation results about the chaos in the dynamics and the analysis of the di¤erences with respect to the basic and linear models proposed in the early literature. It will be followed by a discussion about the utility of using a more complex analysis to describe the price dynamics in …nance considering bounded rationality and heterogeneous beliefs and advanced study possibilities.

Chapter 2

Intruduction to Bifurcation and

Chaos Theory

In the last decade, the Economic Research was concetrated on the construction of complex models about the Business Cycle or about the Financial Options pric-ing, using more complex instruments to analyse the stability and convergence to one or more steady states and the complex dynamics occurring when a convergence is not achievable. This is the reason why it is crucial to show some de…nitions and applications useful to the stability analysis in "non-linear" dynamical systems and introduce the "Chaos Theory" principles. This last one will be valuable to under-line the model’s sensitive dependence from the initial states and its consequences on the ‡uctuations.

First of all we have to consider a 1-D (one dimensional) non-linear dynamical system in its generical form:

Xt+1= f (Xt) (2.1)

where f represents a non-linear map on the real line depending on the para-meter .

Now, in order to study the model’s dynamics, it is essential to analyse the map’s convergence to a steady state point, that is, if there exists x s.t. f (x ) =

x (deriving from the Fixed Point theorem). In such case it would obtain that the orbit of the steady state is the point x .

Recalling the basic mathematical notions on convergence, there are di¤erent kinds of x stability:

1) x is asymptotically locally stable if 9I = (x "; x + "); " > 0; _{s.t 8x}0 I

the orbit of x0 converges to the steady state x : This implies that lim i!1f

i_(x 0) = x

2) x is globally stable if for all x0 belonging D(f ) and this implies that the

time path converges to x

3) x is unstable if 9I = (x "; x + "); " > 0 _{s.t 8x}0 I, x0 6= x , the orbit

of x0 leaves I and so 9i > 0 s.t. fi(x0) =2 I

Hence, the steady state is derivable, in a 1-D discrete system, computing the derivative of f at the point x :

1) if jf0(x )_{j < 1 we have that x is locally stable} 2) if jf0(x )_{j > 1 we have that x is locally unstable}

This formulation comes from the Taylor approximation at the …rst order, that is:

xt+1 = f (xt) x + f

0

(x )(xt x ) (2.2)

It is interesting to remark how the stability is dependent on the linear term of the approximation and it is not a¤ected by the higher order terms. It is evident only in the case jf0(x )_{j 6= 1; instead, for those cases in which jf}0(x )_{j = 1, higher} order derivatives have to be analysed.

However, recalling again the theory, there are di¤erent types of convergence (or divergence):

1) monotonic convergence ) 0 < f0(x ) < 1

2) unstable and monotonic divergence ) f0(x ) > 1

3) unstable and oscillatory convergence ) 1 < f0(x ) < 0 4) unstable and oscillatory divergence ) f0(x ) < 1

2.1

Non-convergence to a steady state point

Generally, it is absolutely not sure that an orbit converges to a steady state and, indeed, there is the possibility that it describes an asymptotically periodical ‡uc-tuation. First of all, it is crucial to state that x is a periodic point with period k (i.e. it is a point that after a period k comes back always to itself) if fk_{(x) = x}

and fi_(x)

6= x; 0 < i < k. Hence, a periodic point with period k is a …xed point of the k-th iteration fk _{of f , the orbit x; f (x); f}2_{(x); ::::::; f}k 1_{(x) is the periodic}

orbit (k-cycle), and this set has exactly k points.

In terms of 1-D dynamic systems, if the 1-D map is either increasing or de-creasing there are only three long run behavior possibilities:

1) xt converges to a steady state

2) xt converges to a period 2-cycle

3) xt is unbounded ) diverging oscillations in the interval (+1; 1)

Through this analysis, it is clean that, when f is increasing, the trajectory converges to a steady state point, or to (+1; 1):

If f is decreasing, then it may show a stable steady state x = 0 and a stable 2-cycle, in particular if that trajectory is increasing. Hence, drawing the graph of the II iterate f2, there will be f2 increasing for f decreasing, with 5 …xed points, that is, x = 0; x₁; x₂; y₁; y_{2, where the couple fx1}; x_{2g generates an unstable 2-cycle} and the other, fy1; y2g, generates a stable 2-cycle.

Depending on x0:

1) for x₁ < x0 < x2 the orbit converges to x = 0

2) for x0 < x1; x0 > x2 the orbit converges to the stable 2-cycle fy1; y2g

Generalizing, it can be concluded that, when the map function is decreasing, possible behaviors are:

- convergence to a steady state

- convergence to a 2-cycle (but here it must be considered the simplest case of monotonic f)

As an advanced step, a non monotonic f was introducted. The most common studied case is represented by the quadratic di¤erence equation, called "logistic map":

xt+1 = f (xt) = xt(1 xt) (2.3)

A map, constructed like this, represents an example of "unimodal map", that is, a 1-D map whose graph has one maximum or minimum. Calling x = c where 1-D map f has max or min (critical points of f ), for the map c = 1

2 where it

assumes its maximum and it has two steady state points: x = 0; x = 1 1_.

This kind of result depends on the fact that at the steady state xt+1 = f (xt) = xt) xt(1 + xt) = 0

) x = 0

) 1 + xt= 0 ) xt = 1 = 1 1

Here, the local stability is determined by _dxdf(x = x ) = 2 x. Hence, the steady states are stable if _dxdf(x = 0) and _dxdf(x = 1 1)are in the interval ( 1; 1). It leads to the following result:

1) for 0 1 the solution x = 0 is a unique steady state and also it’s globally stable;

2) for 1 there are two steady state x = 0; x = 1 1 but x = 0 is unstable. 3) for 1 < 3 x = 1 1 is stable and it attracts all time paths with x0

(0; 1)

In order to be very clear, it is better to do additional observations on the achieved result:

It can be stated surely that x = 0 is stable only for 1; for 1 < < 4 it will achieve that, substituting x = 1 1 into 2 x, the slope is 2 . Therefore, being less than a down-ward sloping _{4, it requires to have j2 j < 1. This means} that the process will converge to the …xed point only when < 3. Instead, for values of around 2, the convergence process is extremely fast (it is independent from the initial value); when is closer to 3, the convergence is extremely slow and several iterations are required.

series model, and using a time line of 100 and a starting point as x0 = 0:1, it can

be observed that:

1) for = 2:9 there is convergence to a stable steady state

2) for = 3:3the steady state is unstable and there is convergence to a stable period 2-orbit

3) for = 3:5the period 2-orbit is unstable and there is convergence to a stable 4-orbit period

4) for = 3:83the …rst 60 periods give an erratic path, called "transient chaos" (better explained later in this chapter), and eventually, after that, the time path settles down to a stable period 3-cycle;

5) for = 4, using as starting point x0 = 0:1 or x0 = 0:1001, a chaotic time

series is achieved, due to the "sensitive dependence on initial conditions".

(*) Fig. 1. Time Series of considered cases

About the last exposed case, it can be de…ned as the phenomenon in which the time paths of nearby initial states do not stay close to each other but diverge

expo-nentially so fast. It means that two trajectories, starting very close to each other, will eventually diverge. This lead to consequences for forecasting and simulating models. Therefore, reliable predictions depend on the accuracy of informations on the initial conditions. Then, it can be shown that, due to the sensitive dependence, long term prediction in a chaotic dynamics seems impossible, because in reality, initial conditions are known with …nite precision. Hence, a butter‡y e¤ect in the model will occurr and there will be signi…cant problems in making predictions.

In order to completely understand the theoretical and statistical result is nec-essary to provide a de…nition of "stable periodic orbit":

def : Let fx1; ::::::; xkg be a periodic orbit of period k, then each point xi is a

…xed point of the k-th iterate fk

if xi is a stable …xed point of fk, then fx1; ::::::xkg is a stable periodic

orbit; using the chain rule: (fk)0(xi) = (fk) 0 (x1) = f 0 (fk 1(x1))f 0 (fk 2(x1)):::::f 0 (f (x1))f 0 (x1) = k 1 Y i=0 f0(fi(x1)) (2.4)

for each 1 i kthe slope (fk₎0

(xi)of a periodic point of period k thus equals

to the product of the slopes f0(xi):

It is also provided a de…nition of aperiodic point. Indeed, a point x is called an aperiodic point if (1) the orbit of x is bounded; (2) the orbit of x is not periodic and (3) the orbit of x does not converge to a periodic orbit.

2.2

Bifurcations

A qualitative variation in the non-linear systems dynamics, as a model parameter change, is called bifurcation. For example, it occurs when the stability of the steady state changes and/or a periodic orbit is created or destroyed.

(*) Fig. 2. Stability graph of a map

2.2.1

Period-doubling bifurcation (Flip Bifurcation)

By recalling the previous analysis, _dxdf(x = 1 1) = 2 , and, as rises, the steady state becomes unstable (for example at = 3 because f0(x ) = 1). Moreover, there is another important change. Indeed, a new stable 2-cycle is created (draw graph for di¤erent levels of ).

Di¤erent cases, for values of closer to 3, will be discussed. Since (f2)0(x) = f0(f (x))f0(x), f2 _{has three critical points. Consequently, x = c =} 1

2 where f 2 _has

a local minimum and d1; d2, for which f (d1) = f (d2) = c, where f2 has a global

maximum.

As suggested by the graph, for = 2:9 3 there are two intersection points with y = x (the bisector), 0 and x . Hence, there are two …xed points of f . For

= 3 slope of f2 _{at x is:}

For > 3, it will be f0(x ) < 1 and (f2₎0_{(x ) > 1. Therefore, if > 3, the}

graph of f2 has at least four intersection with the bisector ( y = x ); 0; x and the other two set up a period 2-cycle

fy1; y2g where f (y1) = y2 and f (y2) = y1; this implies that f2(y1) = y1 and

f2(y2) = y2:

This analysis lead to the following property: at = 3 the quadratic map f exhibits a period-doubling bifurcation in which, for increasing , the x loses stability and a stable period 2-cycle is created. Moreover, at the period-doubling bifurcation the I order condition f0(x ) = 1 is satis…ed.

2.2.2

Bifurcation diagram of the quadratic map

(*) Fig. 3. Bifurcation Diagram of the Quadratic Map

For an increasing , the long run behavior of the system becomes more complex. A simple numerical method to investigate this complexity is represented by the construction of a bifurcation diagram.

The diagram shows that, for 3:45, a second period doubling bifurcation has occurred, in which the period 2-cycle becomes unstable and a new stable period 4-cycle is generated. The

bifurcation values nfrom a stable cycle of period 2n to a stable cycle of period

2n+1 _{converge to a limiting value}

For > ₁, sometimes stable cycles arise; but often the dynamical behavior does not converge to a periodic cycle and becomes "chaotic". For many other values, the time path does not settle down to a stable cycle with low period, and for this reason it is considered chaotic.

2.2.3

Tangent bifurcation

For the logistic map, period orbits with period 2n _{are created in subsequent}

period doubling bifurcation, when is increasing. However, the bifurcation, the period doubling one, may not

explain stable cycles creation. Indeed, the tangent bifurcation will be debated. Considering the model xt+1= x2t + c, the steady states are:

xt+1= fc(xt)) x2+ c = x) x2 x + c = 0) x1;2 =

1 p21 4c 2

1) for c > 1₄ there are not steady states 2) for c = 1

4 there is a unique steady state x and f

0

(x ) = 1

3) for c < 1₄ there are two steady state points, one stable and the other unstable. Being f_c0(x) = 2x, at the steady state we get f_c0(x ) = 1 p1 4c. For 0 < c < 1₄, one steady state will be stable and the other unstable. Therefore, fcexhibits a tangent bifurcation at c = 1₄. E¤ectively, at this point, the graph

of fc is tangent to y = x and two new steady state points are created. Tangent

bifurcation parameter value is described by the I order condition f_c0(x ) = 1where x is the steady state.

(*) Fig. 4. Tangent bifurcation at c = 1₄ for xt+1 = x2t + q

2.2.4

Creation of a period 3-cycle for

x

= x

(1

x

)

It is created in the same way, through a tangent bifurcation of the third iterate f3. Since (f3)0(x) = f0(f2(x))f0(f (x))f0(x), f3 has critical points at x = c = 1₂, at points d1; d2 for which

f (d1) = f (d2) = c and at an additional point ej (with 1 j 4), for which

f2_(e

j) = c. This implies that, for 3:8;, the map f3 has seven critical points,

and this case underlines the following property: for = 3:82 f has a tangent bifurcation in which two 3-cycles are created (one stable and the other unstable). At = equivalently the third f3 _{exhibits a tangent bifurcation in}

which simultaneosly six steady states are created (three stable, the other three unstable). In detail:

1) for < f has no 3-cycle

2) for = f _{has no 3-cycle fx}1; x2; x3g and (f3)0(xi) = 1 for 1 i 3

3) for > ( closed to ) f has two 3-cycles (one stable and the other unstable)

Hence, at the bifurcation value = , the graph of the third iterate f3 _is

2.2.5

Transcritical bifurcation

This kind of bifurcation occurs when two steady states collide and exchange sta-bility. It represents the standard mechanism for a situation in which a …xed point exists for all values of a parameter and it can never be destroyed but may change its stability. Generalizing, a model in which this bifurcation form may occur is

:

x = rx x2 _{= x(r x). It is very familiar, because it is the same form of the}

logistic map.

2.2.6

Pitchfork bifurcation

(2.1) (*) Fig. 6. Pitchfork Bifurcation Diagram

It is better to explain this kind of bifurcation with an example: Consider the 1-D map xt+1 = g (xt) = tanh( xt) = e

xt _e xt

e xt+e xt. The map g

is increasing S-shaped and its derivative is g0(x) = (1 tanh2( x)). It’s easy to verify that x = 0 is always a steady state and g_y0(0) = . Therefore, g satis…es the following properties:

1) for 0 1, x = 0 is the unique steady state and it is globally stable; 2) for = 1, x = 0 is the unique steady state with gy0(0) = 1and it is globally

stable. The graph of g is also tangent to the diagonal at the steady state x = 0 . 3) for > 1 we have three steady states: x = 0, that now is unstable, and 9x1 ; x2 both locally stable.

Finally, the conclusion is that, in the Pitchfork bifurcation case, the number of steady states changes from one to three.

In conclusion, tangent, transcritical and pitchfork bifurcation, on a steady state x , share the same I order condition:

2.3

Extension to 2-D Dynamical Systems

Consider a 2-D discrete dynamical system (xt+1; yt+1) = F (xt; yt) (2.6)

where F is a non-linear 2-D di¤erentiable map and a parameter. The orbit with initial state (x0; y0) is the set

f(x0; y0); (x1; y1); (x2; y2); :::g = f(x0; y0); F (x0; y0); F2(x0; y0); ::::g

Now, an orbit may converge to a stable steady state or to a stable k-cycle, but also to a much more complicated set.

2.3.1

A important example: the Hénon Map

Hénon, in 1976, introduced a simple 2-D quadratic map: xt+1= 1 ax2t + yt ,

yt+1 = bxt (2.7)

where a and b are parameters. We can observe that, in the case b = 0 , the map comes back to a 1-D quadratic map xt+1 = 1 axt, and this shows how the

Hénon map is a 2-D generalization of the 1-D quadratic map. Rewriting all as a di¤erence equation:

(xt+1; yt+1) = Ha;b(xt; yt) (2.8)

where Ha;b is the 2-D map Ha;b(x; y) = (1 ax2+ y; bx) and is called Hénon

map.

At this point, we have to investigate the stability of steady states of the map. The steady state must satisfy the equations

bx = y

by substituting the last one in the previous ax2 _{+ (1 b)x 1 = 0}

and the solutions are x1;2 =

b 1 p2 (1 b)2_+4a

2a . Therefore, steady states are:

(x1; y1) = ( b 1+p2 (1 b)2_+4a 2a ; bx1) (2.9) (x2; y2) = ( b 1 p2 (1 b)2_+4a 2a ; bx2) (2.10)

In order to investigate the stability of these points, we have to consider the Jacobian matrix and its eigenvalues at the steady states.

The Jacobian matrix of the Hénon map is given by: J Ha;b(x; y) =

2ax 1 b 0

!

, and the characteristic polynom is

2_{+ 2ax}

i b = 0,

so, the eigenvalues of the jacobian are

i = axi 2

p

(axi)2+ b (2.11)

Fixing a = 1:4 and b = 0:3 we can obtain as eigenvalues of J F (x1; y1)

1:92; 0:16 and as eigenvalues of J F (x2; y2) 3:26; 0:09. Therefore, for these

(*) Fig. 7. Phase Plot of a Hénon Map

This …gure shows the long term dynamical behavior of the Hénon map and we can call it the "attractor" of the Hénon map, starting with the same initial points. The attractor does not be properly a curve, but enlarging the graph, it can be denoted how the structure is composed by in…nitely many lines on a …ner scale. Indeed, the attractor seems to be the product of lines and a Cantor set (which will be later de…ned).

By providing di¤erent time series corresponding to the strange attractor, with di¤erent initial states, we can also show that the dynamical behavior of the at-tractor is chaotic, due to the its sensitive dependence on the initial values. Indeed, considering two di¤erent starting points, (x0; y0) = (0; 0)and (x0; y0) = (0:001; 0),

the two time series are di¤erent after some time.

2.3.2

De…nition of a strange attractor

We can de…ne an attractor as a set of points representing the long-term dynamical behavior of a system, but in a more mathematical manner:

An attractor of a K-dimensional system Xt+1 = F (Xt) is a compact (closed

and bounded) set A with the following three properties:

1. The set A is invariant. Hence, F (A) A;

2. There exists an open neighborhood U of A ( A U), such that all initial

states X0 2 U converge to the attractor. Hence, for all X0 2 U limn!1dist(Fn(X0); A) =

0

3. There exists an initial state X0 2 A for which the orbit is dense in A

The simplest example of an attractor is a stable steady state, in which the attractor is a single point, attracting all other orbits with initial states in some neighborhood. Moreover, another simple example is a stable k-periodic orbit, in which the attractor is composed by k points.

In some cases, more complicated attractors consisting of in…nitely many points may occur. These, called "strange attractors" arise as limiting sets on which long run chaotic dynamics arises. Hence, we can de…ne these attractors in the following manner:

An attractor A is called a strange attractor of the N-dimensional dynamic system xt+1 = F (xt), if the map F has sensitive dependence with respect to the

set of initial states converging to A

In 2-D dynamical systems, also the idea of bifurcation changes: Indeed, now, we will consider bifurcations as qualitative changes in the dynamical behavior of a one-parameter family F of 2-D maps, as the parameter varies. In 2-D systems we can consider again the types of bifurcation considered previously discussing 1-D ones, but it is also interesting to present and discuss, for its usefulness in the next chapters of this work, a new type of bifurcation that does not occur in the 1-D systems, the Hopf bifurcation.

2.3.3

Hopf Bifurcation

How should become clear shortly, the Hopf bifurcation is characterized by two complex eigenvalues crossing the unit circle as a parameter changes.

The best way to discuss this bifurcation type is through an example, the delayed logistic map. Indeed, by introducing an extra time delay in the quadratic logistic equation xt+1 = axt(1 xt), we obtain the second order delayed logistic di¤erence

equation

Nt+1= aNt(1 Nt 1) (2.12)

By translating the second order di¤erence equation into a 2-D …rst order dif-ference equation and substituting xt= Nt 1 and yt= Nt we get

xt+1= yt

yt+1 = ayt(1 xt)

This can be seen as a …rst order di¤erence equation (xt+1; yt+1) = Fa(xt; yt),

where Fa is the 2-D map

Fa(x; y) = (y; ay(1 x)) (2.13)

with a 0 as a parameter. In order to determine the steady states of the system, we can recall that they have to satisfy

y = x

ay(1 x) = y

Substituting the …rst in the second one, ax(1 x) = x, the solutions are x = 0 and x = a 1_a . Therefore, the steady states are (x1; y1) = (0; 0) and

(x2; y2) = (a 1_a ;a 1_a ). Then, as explained before, the stability of these steady states

is determined by the eigenvalues of the Jacobian matrix at the steady states.

J Fa(x; y) =

0 1

ay a(1 x) !

Hence, at steady states we have: J Fa(0; 0) = 0 1 0 a ! and J Fa(a 1_a ;a 1_a ) = 0 1 1 a 1 !

By computing the Characteristic Polynoms and their roots, we …nd that eigen-values of J Fa(0; 0) are 0 and a, so (0; 0) is a stable node for 0 a < 1 and

unstable for a > 1; the eigenvalues of J Fa(a 1_a ;a 1_a ) are 1 = 1₂ 1₂

p

5 4a and

2 = 1₂ +1₂

p

5 4a. 1 and 2 satisfy the properties:

1. 0 a < 1: real eigenvalues with 1 < 1 < 1 < 2, so (a 1_a ;a 1_a )is a saddle.

2. 1 < a < 5₄: real eigenvalues with 0 < 1 < 2 < 1, so (a 1_a ;a 1_a ) is a stable

node. 3. 5

4 < a < 2: complex eigenvalues with 1 2 = a 1 < 1, so ( a 1

a ; a 1

a ) is a

stable focus.

4. a > 2: complex eigenvalues with 1 2 = a 1 > 1, so (a 1_a ;a 1_a ) is an

unstable focus.

For the particular case, not mentioned before, a = 1, a transcritical bifurcation occurs; hence, (0; 0) and (a 1_a ;a 1_a ) coincide and there is an exchange of stability. From the properties (3) and (4) it derives the delayed logistic equation exhibits a Hopf bifurcation at a = 2. Indeed, as the parameter a increases and passes 2, the eigenvalues cross the unit circle from inside to outside and (a 1_a ;a 1_a ) loses stability.

By using a numerical analysis, it is interesting to see the dynamical behavior after the Hopf bifurcation, when the steady state is unstable.

(*) Fig. 9. Phase Plot after a Hopf Bifurcation

Through the provided …gure, we can see that, for values of a close to 2 (blue, red and green curves), the orbit converges to an attracting invariant circle, that is, a closed curve. Increasing a, as in the example, to 2.16, the invariant circle gets distorted. Finally, …xing a to 2.27 the attractor is no more an invariant circle, but a more complicated set. Hence, in this case, we have a strange attractor such as the Hénon map. This bifurcation route is called breaking of an invariant circle. Moreover, we can investigate the chaos triggered in the system by changes in the initial values, using a time series analysis.

(*) Fig. 10. Time Series after a Hopf Bifurcation

Considering time series from a = 2:01; 2:04 we can see how they are almost periodic but, considering also time series for a = 2:16,2:27 the time series become more complicated and chaos and sensitivity to initial states arise.

Finally, we can analyse properly the bifurcation diagram for the delayed logistic map with 1:95 < a < 2:27 and 2:22 < a < 2:27.

(*) Fig. 11. Bifurcation Diagrams

After a = 2, the Hopf bifurcation, periodic and quasi-periodic dynamics on an invariant circle arise. Quasi-periodic dynamics is interrupted with stable cycles; but for increasing values of a the dynamics becomes chaotic and a lot of period-doubling bifurcations occur.

In order to understand analysis and results of the next chapter, another de…n-ition has to be provided here: Homoclinic point.

First of all, we can introduce some new notions. Let p be a …xed point of the 2-D map F , the local stable manifold and local unstable manifold of p are de…ned as

W_locs (p) =_{fx 2 U j limn!1}Fn(x) = p_g (2.15) Wu

loc(p) =fx 2 U j limn! 1Fn(x) = pg (2.16)

where U is some small neighborhood of p. In general, for a non linear map F the local stable manifold Ws

loc and the local unstable manifold Wlocu are smooth curves

tangent to the stable and unstable eigenvectors of the Jacobian matrix J F (p): For linear maps, stable and unstable manifolds are simple and coincide with stable and unstable eigenvectors of a saddle point. For non linear ones, stable and unstable manifolds may have intersection point di¤erent from p and may be very complicated curves.

A key de…nition will be now the Homoclinic point:

A point q is called a homoclinic point if q 6= p and q is an intersection point of the stable and unstable manifolds of the saddle point p, that is, q 2 Ws_(p)

\Wu_(p):

This notion was introduced by Poincaré at the end on 19th _{century and he}

realized that the presence of a homoclinic point means very complicated behav-ior. Poincarè showed that the motion of sun, earth and moon not need to be periodic, but may become highly irregular and unpredictable. Hence, he proved that a homoclinic orbit implies chaotic motion and sensitive dependence on initial conditions.

To explain how the existence of a homoclinic point implies complicated dynam-ical behavior, a …rst observation is that if q is a homoclinic point, then there are in…nitely many homoclinic points, because each Fn(q); n_{2 Z, is also a homoclinic} point. Therefore, F (q); F2_{(q); F}3_{(q)and etc. are homoclinic and also F} 1_(q);

F 2_{(q); F} 3_{(q)etc. This means that wild oscillations of stable and unstable}

man-ifolds, with in…nitely many homoclinic intersections, are accumulating onto each other and hence, there is a sensitive dependence on initial conditions.

2.4

Chaos Theory

In applied mathematics, the "Chaos" is detected as the phenomenon in which simple non-linear models exhibit very complicated dynamics, and in particular this one is

called "Deterministic Chaos". This means that, in these models, chaotic solu-tions exhibit sensitivity to initial states, in the sense that small perturbasolu-tions on the initial conditions lead to a completely di¤erent time path.

First of all, some de…nitions, regarding the "chaos theory", will be provided: def (1): A di¤erence equation xt+1 = f (xt) dynamics is called "topologically

chaotic", if the following three properties are satis…ed:

(a) 9 an in…nite set P of unstable periodic points with di¤erent period;

(b) 9 an uncountable set S of aperiodic points, where they are mentioned as the ones with bounded orbits and non-converging to a periodic point;

(c) f has sensitive dependence on initial conditions in respect to a set = P _{[ S and there exists a positive distance c such that, for all x}0 2 and any

" neighborhood U of x0, 9y0 (initial state) with y0 2 \ U and a time T > 0

such that d(xT; yT) = d(fT(x0); fT(y0)) > c

Additional observations: the property (a) indicates that, in a chaotic system, the dynamics may follow many di¤erent, periodic patterns and the time paths do

not converge to a periodic orbit. Instead, the property (b) implies the idea of unpredictability in the long run.

In order to understand the concept of "Chaos", it is provided an example: cosider the logistic map, with = 4, xt+1 = 4xt(1 xt). This kind of model

satis…es all the three properties. It can be seen as f4(x) = 4x(1 x) and …nd its

maximum as f4(1₂) = 1. Hence, f4 increases from 0 to 1 in the interval 0;1₂ and

decreases from 1 to 0 in the interval 1₂; 1 . Considering also the II iterate f2 4, it

oscillates twice between 0; 1 on the interval [0; 1]. f2

4 has four intersection points

with y = x; two are steady state of f4 (x = 0; x = 1 1 = 3₄) and the remaining

form a period 2-cycle. Considering f3

4, there are four oscillations between 0; 1 in the interval [0; 1] :

They generate four maxima, all 1, and …ve minima, all 0, and this means that f3 4

has eight intersections with y = x where two of them are steady states of f2 4 and

the remaining create two unstable 3-cycles.

(*) Fig. 12. Di¤erence in the four iteration considered on the map for = 4 In general, for any n, the graph of fn

4 has the following properties:

1) fn

2) fn

4 oscillates 2n 1 times on the interval [0; 1];

3) the map f4n has 2n …xed points;

4) for any I of arbitrarily small length " 9N > 0 such that I contains x; y with f₄N(x) = 0; f₄N(y) = 1

For each positive integer n, f4 has a periodic point with period n and this

satis…es the I property of the Chaos. In addition, the sensitive dependence from the initial conditions follows from the point 4) since the points x; y (arbitrarily close) have distance 1 after N periods.

Therefore, the point becomes how to understand that a 1-D map exhibits a topological chaos. For this reason, it is quite useful to introduce the Li-Yorke theorem. It states:

Let xt+1 = f (xt) be 1-D di¤erence equation and f a continous map,

if 9x0 s:t:f3(x0) x0 < f (x0) f2(x0), then the dynamics is topological chaotic.

This theorem provides a su¢ cient but non-necessary condition for the topolog-ical chaos. For example, about the logistic map, it is right to de…ne the dynamics as topologically chaotic for all parameter values > ₁, i.e beyond the accumu-lation point of the period doubling bifurcation. Although for > ₁ the Li-Yorke condition need not be satis…ed for f , there exist some N 1 such that N-th iterate fN _{satis…es the Li-Yorke theorem. Obviously this is a general result, but}

there are two important conditions in the theorem: f has to be one-dimensional and continuous; indeed, it is quite simple to see that the implication "period 3 ) chaos" is not true for f discontinuous 1-D map.

2.4.1

Measures of Chaos

To sum up, a topological chaotic system has a set of initial states P with in…nitely many periodic points and a set S of aperiodic points, with sensitive dependence on the initial states in the set = P _{[ S. Furthermore, two other observations} are indispensable: the chaotic dynamics may be restricted to a set of initial states with "prob = 0", and may have a Lebesgue measure zero, where the Lebesgue measure represents the standard way to assign a measure to subsets of n-dimension euclidean space.

To begin this part of analysis, a precise de…nition of Lebesgue Measure must be provided:

def : given E subset of R s.t. E R; with the length of an (open, closed, semi-closed) interval I = [a; b] established by l(I) = b a. Then the Lebesgue Outer Measure is de…ned as:

(E) = inf ( ₁ X k=1 l(lk) : (Ik)k2N )

where it is a sequence of disjoint open

inter-vals with E

1

[

k=1

Ik (2.17)

The Lebesgue Measure of E is given by its Lebesgue Outer Measure (E) = (E) _{if 8A R} (A) = (A_{\ E) +} (A_{\ E}c_).

A crucial case is the presence of "Lebesgue Measure Zero". E¤ectively, a subset of Rn has Lebesgue measure zero if, for every " > 0 it can be covered with a numerable set generated by the product of n intervals whose volume not exceed ". This de…nition is useful in particular to show that a set is measurable by Lebesgue and this means that it is so important to …nd a Lebesgue measure zero set in order to …nd a positive or negative Lebesgue measure set generated by the union or intersection of open or closed sets.

Through the Lebesgue measure concept, it is possible to give a stronger de…n-ition of "chaos":

The dynamics of xt+1 = f (xt) is called "truly chaotic" if, 9 of positive

Lebesgue Measure s.t. f has sensitive dependence on initial conditions of , there exists positive distance c s.t. 8x0 2 and any " neighborhood U of x09y0 2 \U

and a time T > 0 s.t. the distance d(xT; yT) = d(fT(x0); fT(y0)) > c

Hence, by using this new de…nition of Chaos, it is interesting to study again the previous example on the logistic map with = 4, and, in this way, to de…ne it as a "truly chaotic system", without a stable periodic cycle.

In a higher level of complexity, it is necessary to state that, in non-linear dynamic systems, complex sets with a fractal structure frequently arise as invariant chaotic sets. Cantor, in 1883, introduced the concept of "fractal set", with a repeating pattern displayed at every scale and in which if the replication is exactly the same in every scale, the pattern is called "self-similar pattern".

In this analysis, if an orbit with an initial state in a chaotic Cantor set is considered , this never leaves the invariant set but it jumps irregularly over the fractal set. It will be provided an example of Cantor set, the "middle third Cantor set":

The construction begins with the unit interval C0 = [0; 1] and removing the

middle third interval (1₃;2₃) from C0

) C1 = 0;1₃ [ 2₃; 1 ) C2 = 0;1₉ [ 2₉;1₃ [ 2₃;7₉ [ 8₉; 1 )

::::::::::::_{) C}n

Cn consists of 2n intervals, each one with length (1₃)n. Letting n ! 1, the

Cantor set is obtained as the limiting set C₁. At this point, it is correct to think about how many points are contained in the Cantor set. It does not contain any arbitrarily small interval, because at the n-th stage of the construction, the length of each interval in Cn is (1₃)n. This means that the set has

in…nitely many points, but contains no intervals. In addition, it is possible to show (proved properly by Cantor) how a Cantor set is uncountable.

In order to demonstrate it, the existence of a one-to-one mapping from an interval, like [0; 1], to the Cantor set, must be shown. The trick, to complete the proof, consists in to exploit that any y 2 [0; 1] has a binary representation (y1;y2;y3);with yj = 0; yj = 1. This implies:

y = 1₂y1+ (1₂)2y2+ ::::::::::::: = 1 X j=1 (1₂)j_y j (2.18)

For each y 2 [0; 1], the binary representation is unique, with the exception of points of the form y = (1₂)k, k 1, having two binary representations.

Another property of the middle third Cantor set is that it has Lebesgue measure zero. Indeed, being Cn with length of each interval (1₃)n, its Lebesgue measure is

equal to the total length (2₃)n. Hence, the limiting Cantor set C₁ has Lebesgue measure equal to 0; and C1 is uncountable with Lebesgue measure zero.

2.4.3

Invariant Cantor set for

> 4

Since f (1₂) > 1, there is an interval A0 of initial states x0 for which x1 = f (x0) > 1

and their orbit escapes from the unit interval [0; 1] and at the end diverges to _1. Therefore, the next step is to understand which is the set of x0 whose orbit

remains into [0; 1] for all time (it is known how this set is non-empty because at least there are x = 0 and x = 1 1, called rispectively I0; I1). Hence, the invariant

set of points that never escape from [0; 1] is a Cantor set, and it has a chaotic dynamics .

Since f (I0) = f (I1) = [0; 1], there exist four subintervals I00; I01 I0 and

I10; I11 I1 s.t. f (I00) = f (I10) = I0; f (I11) = f (I10) = I1 and f2(I00) = f2(I01) =

f2_(I

11) = f2(I10) = [0; 1]

Each of these four subintervals IS0S1 contains two smaller intervals IS0S1S2 for

which f ( IS0S1S2) = IS1S2 and f

3_(I

S0S1S2) = [0; 1]with Si = 0; 1 and 0 i 2.

Let S0S1::::::Sn 1 be a sequence of 0 or 1; at the end there will be 2n intervals

with the properties:

1) fn _{is monotonic on I} S0S1:::::Sn 1 and f (IS0S1:::::Sn 1) = IS0S1::::::Sn; 2) fn(IS0S1::::::Sn 1) = [0; 1]; 3) 8x0 2 IS0S1:::::Sn 1 f k_(x 0)2 ISk for 0 k n 1

Let n set of points x 2 [0; 1] for which fk(x) 2 [0; 1] for all 0 k n; by

construction and the previous properties, for n ! 1 n has 2n intervals and

is a Cantor set. Moreover, it can be shown that the sum of the lengths of the 2n intervals n converges to 0 as n ! 1. Then, invariant set is a Cantor set with

Lebesgue Measure equal to 0.

The invariant Cantor set dynamics is described by a "symbolic dynamics". Let be the set of all one-sided symbolic sequences of 00_{1and 1}0_{1. Hence, a point}

2 is an in…nite

sequence = (S0S1:::::).

A further de…nition to provide is now the concept of Shiftmap (') on as '(S0S1::::) = (S1S2:::::). The dynamics of f (in the logistic map case) on the

Invariant Cantor set is equivalent to dynamics of ' on . This dynamics is topo-logically chaotic , since the properties hold:

1) ' has a periodic point of period n and each periodic sequence is a periodic point of ' ;

2) any aperiodic sequence is an aperiodic point of ' and this implies that ' has uncountably many aperiodic points;

3) ' has sensitive dependence on initial conditions in respect to .

To sum up, the dynamics of f , when > 4, on the Invariant Cantor set is also topologically chaotic.

2.4.4

Lyapunov Exponent

The Lyapunov exponent, in a one dimensional discrete chaotic system, measures the average exponential rate of divergence of nearby initial states. Consider the dynamic model:

xt+1 = f (xt) with two initial states x0; x0 + . After n periods the

separa-tion between two orbits is: jfn_(x

0+ ) fn(x0)j (fn)

0

(x0) . Introducing the

exponent (x0), the one measuring the average exponential rate of divergence,

it has to satisfy jfn(x0+ ) fn(x0)j (fn)

0

(x0) = en (x0)j j, or equivalently

en (x0) _{= (f}n₎0_(x

0) .

The Lyapunov exponent thus satis…es (x0) = _n1 ln (fn)

0

(x0); using the Chain

Rule for (fn₎0_(x

0) and, taking the limit t ! 1, we get:

(x0) = lim n!1 1 n n 1 X i=0 ln (f0(fi_(x 0))) (2.19)

Speci…cally, the Lyapunov exponent is the average of the absolute value loga-rithm of the derivative along the orbit. For x0 converging to a steady state, LE

represents the log of absolute value of the derivative at the steady state; for x0

converging to a periodic orbit with period k, LE is the log of absolute value of the derivative of fk_{, at the periodic point, divided by k; for x}

0 converging to a stable

steady state or a stable k-cycle, there will be LE < 0 (in the contrast a chaotic time path has LE > 0).

In general, LE and (x0) depend upon x. Hence, in the chaotic systems, such

as the logistic map, LE = (x0) is the same "number" for Lebesgue almost all

For example we can consider the piecewise linear tent map: T (X) = 2x 2(1 x) with T 0 (x) = 2 _{and x 6=} 1 2 (2.20)

(*) Fig. 13. Graph of Lyapunov Exponent for the logistic map

In this case LE, being the average of the log of absolute values of the derivatives along an orbit, must be (x0) = ln(2). But the question is: Hold this result for all

x0 2 [0; 1]? For sure, it does not hold for x0 = 1₂, that is, the critical point. Indeed,

since T is a linear map, it is not di¤erentiable at 1₂. This condition does not hold also for every point x0 mapped onto 1₂, that is, for points such that Tk(x0) = 1₂,

for some k 1. This exceptional set contains in…nitely points and it is "dense" in [0; 1]. It is also coutable and has Lebesgue Measure equal to 0. For this reason, LE is well de…ned and LE = ln(2). Using these last notions, it can be provided a numerical de…nition of chaos:

def : dynamics of a di¤erence equation xt+1 = f (xt) is called chaotic if there

exists a set of initial states of positive Lebesgue Measure such that LE = (x0) > 0

In the proposed case, LE > 0 and then, for many , the dynamics is chaotic. However, the chaotic region is interspensed with small parameter windows with negative Lyapunov exponents, when stable cycles occur. Therefore, it is correct to de…ne the Lyapunov exponent graph (as a function of ) as a fractal curve with in…nitely many downward spikes.

In other terms, the Lyapunov exponent is a measure of long term predictability (or unpredictability) in a chaotic model. Recalling that, for = 4, LE = (x0) =

ln(2), on average, the separation factor between initial states after ten time periods is e10 ln(2)_{= 2}10 _{= 1024.}

In a chaotic system, prediction of uncertainty after ten periods is more than a thousand times as big as initial measurement of uncertainty. Then, the conclusion is that the larger is the Lyapunov exponent, more di¢ cult the long run prediction in a chaotic system becomes.

2.5

Chaos and Autocorrelation

As a …nal step, it should be possible to study the proposed models as time series because this approach may be very useful in order to perform econometric analysis. By using time series generated by chaotic maps, it is possible to …nd that, in these cases, the autocorrelation is zero.

The problem is related to the di¢ culty in distinguish between chaos and a pure white noise. In this sense, the autocorrelation functions (ACF) of asymmetric tent maps exactly coincide with the ACF of linear stochastic AR(1) processes.

Consider the linear map T : [0; 1] ! [0; 1], de…ned as:

T (x) = 2x 1+ 2x 1 (2.21)

where the critical point of the function is x = +1₂ and 1 < < 1

T _{increases from 0 ! 1 on the I = [0;} +1₂ ] _{and decreases from 1 ! 0 on the} I = [ +1₂ ; 1]. Hence, the dynamics of the asymmetric tent map is "truly" chaotic.

This kind of map is called "Tent map" and it is represented by the function f := min_{fx; 1} x_{g and it has this name due to the tent-like shape of its graph.}

(2.2)

Since the tent map is expanding, with slopes ₁₊2 > 1and ₁2 < 1, T may have not stable steady states or stable cycles and LE = (x0) > 0

Therefore, the dynamics of the map xt+1= T (xt) follows the properties:

1) 8j 2 Z s.t. j 1 T has a periodic point of period j and all periodic orbits are unstable;

2) According to Lebesgue Measure Theory almost all initial states x0 2 [0; 1]

the time path fxtg10 is chaotic and dense on I = [0; 1];

3) According to Lebesgue Measure Theory x0 2 [0; 1], the sample average of

the chaotic time path is x = lim

T !1 1 T +1 T X t=0 xt= 1₂;

4) According to Lebesgue Measure Theory almost all initial states x0 2 [0; 1],

the sample autocorrelation coe¢ cient at lag j is j = j

In order to be clearer, the following additional observations are provided: - The property 1) follows by looking all the graph of Tj(that is, the T map composed with itself j times);

- According to the 2), the chaotic system time paths range through the interval [0; 1], getting close to every point in the interval, at some time;

- According to 3), the average of these chaotic series exists and it is 1₂;

- From the 4), the sample autocorrelation function of a tipical chaotic series of T is identical to the autocorrelation function of the linear stochastic AR(1) process:

xt= 1₂ + (xt 1 1₂) + "t where "t iid(0; 1)

- Then, from 4), the sample autocorrelation function of the chaotic time series by the asymmetric tent map depends upon

For example, by observing the graph of T for = 0, it is intuitively clear that the typical chaotic time series has zero sample autocorrelation at all leads and lags and for this reason it would be indistinguishable from a white noise.

(2.3)

Therefore, generalizing, the deterministic chaotic asymmetric tent map may be seen as a stochastic AR(1) process.

2.6

A …nacial example of Chaos existence

The purpose of this example is to show that even the simplest deterministic exchange rate system can display very complex dynamic behaviour. It will be assumed an exchange rate determined by the interaction of speculators and traders. The non-linearity enters the model through the speculators’demands for foreign currency and this non-linearity generates the chaotic dynamics for some parameter values. In particular, we assume:

St= (e

e

et 1); with 0 (2.22)

where et represents the home currency price of foreign currency and ee the

expected future exchange rate, with as a sensitive parameter. Clearly, if = 0 there are no speculative demands for foreign currency and if =_{1 any deviation} of the exchange rate will lead to in…nite net demands and that any such deviation will be eliminated instantaneously. Remaining on , if it is small, it follows that small undervaluations do not generate large demands. The trade balance, called Tt,

is a simple linear function of the current and last-period exchange rates, expressed as the deviation from the corresponding expected value:

Tt= (et ee) + (et 1 ee) with ; > 0 (2.23)

The expected exchange rate is the steady state value at which speculators wish neither to buy nor sell, and it is determinated by the so called "fundamental". It includes variables such as the interest rate di¤erential and relative money supplies:

e = ee_{(i; i}f_{; m; m}f_{; ::::)}

When the exchange rate is at its steady state value, et = et 1 = e , the trade

balance must clear. Indeed, for reasons of tractability, it is assumed to be e = 1. The time frame of the model is the short run, because the fundamental is not expected to change in the planning horizon. The foreign exchange market clears within each period, so that:

St= Tt (2.24)

This solves to give et 1e2t [( + )e et 1 e2t 1 e ]et e et 1 = 0.

There are two roots for et, one negative and one positive. For positive values

of , ; ; the positive root is choosen.

In order to better analyse the model, numerical simulations for di¤erent pa-rameter values may be provided. The result is that, for some parametrizations, the resulting time paths are chaotic. Through experiments with di¤erent parame-ter values, chaos arises in this model when the trade balance is su¢ ciently more sensitive to past rather than to current values of the exchange rate.

(*12) graph of the tangency between the function in et 1 and the bisector:

(2.4)

The …gure shows a single-peaked mapping, with a critical value (ec)that maps

to a value (em). Once, in the interval formed by these upper and lower bounds, the

exchange rate is captured and cannot escape unless one of the exogenous variables changes. As stated before, a system with this particular parametrization is chaotic, according also to the Li-Yorke theorem. Therefore, starting from a point e1, the

exchange rate increases for two periods to ec and then falls to em < e1, ful…lling

the requirements of the theorem.

E¤ectively, The exchange rate moves in an unpredictable manner, never re-peating itself. While the model, and resulting time path, is far from mimicking the movement of any existing exchange rate, it does display some features of actual exchange rates.

However, two of the characteristics of chaos (sensitivity to initial conditions and sensitivity to parameters) make all but short-run prediction impossible for any forecast if the correct model is chaotic, even if he could correctly specify the functional form of the relationship between present and past exchange rates. This could be demonstrated simulating the model for di¤erent starting values and parameter values.

About the analyse of the expected (steady state) exchange rate, it is a para-meter, and forecasting requires that this should be known with complete accuracy.

It is important also to take in account that any exogenous shocks, changing the expected rate, will put the actual change rate on a new path. Such changes may be substantial, but they do not have to be large to have signi…cant e¤ects on a chaotic system. Therefore, while "news" is su¢ cient to cause large divergences of the exchange rate from its predicted paths, it is not necessary.

In general, these cases demonstrate that, if the dynamics of the economy are non-linear, and possibly chaotic, it is not su¢ cient to know the model in order to make useful forecasts over anything but the very short run; the forecaster has to know also the past values of the exchange rate with approximately 100% accuracy, and this is almost impossible.

2.7

Conclusions

By this introductory mathematical chapter, notions of bifurcation and chaos are provided, in order to approach the "non-linear world", in which there are complex ‡uctuations (random), though models are based on deterministic laws. Then, they are useful to understand the long run behavior of the economic system (in speci…c sectors) and its reaction to changes. In particular, it was described how a system behavior may change with empirical casuality due to its sensitive dependence on the initial states and changes in them. It was introduced the concept of Chaos with di¤erent de…nitions (one of these based on the Li-Yorke theorem and the Lebesgue measure). A de…nition and an example of "fractal set" was also provided, in order to speak about the concept of Invariant Cantor set. As measures of chaos, the Lebesgue Measure, as a measure of the initial conditions set with 0 probability, and the Lyapunov exponent, as a measure of the average exponential rate of divergence on initial states, were stated. In conclusion, it was stressed the idea of "chaotic deterministic Time Series" and its unpredictability similar to stochastic processes, like the AR(1) process. This …nal discussion represents, properly, the focus of the "chaos theory", that is, the possibility to look, in an exponential sensibility