A TUTORIAL INTROOUCTION TO NONLlNEAR OYNAMICS ANO CHAOS, PART I: TOOLS ANO BENCHMARKS


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1 A TUTORIAL INTROOUCTION TO NONLlNEAR OYNAMICS ANO CHAOS, PART I: TOOLS ANO BENCHMARKS Luis Antonio Aguirre Centro de Pesquisa e Desenvolvimento em Engenharia Elétrica Universidade Federal de Minas Gerais Av. Antônio Carlos 6627 : Belo Horizonte, M.G.. Brazil Fax: , Phone: (0:31) :3 aguirrelgcpc!ee.ufmg.br ABSTRACT  The relevance of nonlinear dynamics and chaos in science and engineering cannot be overemphasized. Unfortunately, the enormous wealth of techniques for the analysis of linear systems which is currently available is totally inadequate for handling nonlinear systems in a systematic, consistent and global way. This paper presents a brief introduction to some of the main concepts and tools used in the analysis of nonlinear dynamical system and chaos which have been currently used in the literature. The main objective is to present a readable introduction to the subject and provide several references for further reading. A number of well known and well documented nonlinear models are also included. Such models can be used as benchmarks not only for testing some of the tools described in this paper, but also for developing and troubleshooting other algorithms in the comprehensive fields of identification, analysis and control of nonlinear dynamical systems. Some of these aspects will be addressed in a companion paper which follows. 1 INTRODUCTION An important step towards the analysis of real systems is to realize that virtually ali systems are in a sense dynamieal. This does not mean to say, of course, that the dynamics of every system are always necessarily relevant to the analysis. Thus although it is sometimes justifiable to regard certain systems as being sialie, in most cases it is worthwhile taking into account the dynamics inherent in the systems to be analyzed. oartigo submetido em 15/07/94; 1a Revisão em 01/08/95 2 a Revisão em 27/11/95 Aceito por recomendação do Ed. Consultor Prof.Dr. Liu Hsu Mathematica.l1y, dynamical systems are described by differential equations in continuoustime and by difference equations in discretetime. On the other hand, most static systems are, of course, described by algebraic equations. A second step in the analysis of real dynamical systems is to take into consideration the nonlinearities, which are often as important to the system as the dynamics. The use of linear models in science and engineering has always been common practice. A good linear model, however, describes the dynamics of the a system only in the neighborhood of the particular operating point for which such a model was derived. The need for a broader picture of the dynamics of real systems has prompted the development and use of dynamicalmodels which included the nonlinear interactions observed in practice. In this paper a few basic concepts related to nonlinear dynamical systems are brief!y reviewed. The objective is twofold, namely to provide a brief introduction to nonlinear dynamics and chaos and to indicate a few basic references which can be used as a starting point for a more detailed study on this subject. In particular, this paper will describe a few mathematieal tools sueh as Poinearé seetions, bifurcation diagrams, Lyapunov exponents and correlation dimensions. AIso, some of the most commonly used nonlinear modeis which display chaotic behavior will be presented thus providing the reader with both, a few basic tools and well established benchmarks for testing such tools. The ambit of the techniques developed for nonlinear systems with chaotie dynamics can be appreciated by considering the wide range of examples in which chaos has been found. Different types of mathematical equations ex SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril
2 hibit chaotic solutions, for instance ordinary differential equations, partial differential equations (Abhyankar et alii, 1993), continued fractions (Corless, 1992) anel delay equations (Farmer, 1982). Chaos is also quite common in many fields of control systems such as nonlinear feed back systems (Baillieul et alii, 1980; Genesio anel Tesi, 1991), adaptive control (Mareels anel Bitmead, 1986; Mareels anel Bitmeael, 1988; Golelen anel Yelstie, 1992) and eligital control systems (Ushio and Hirai, 1983; Ushio and Hsu, 1987) Chaos seems to be the rule rather than the exception in many nonlinear mechanical and electrical oscillators and penelula (Blackburn et alii, 1987; Hasler, 1987; TvIatsumoto, 1987; Ketema, 1991; Kleczka et alii, 1992). Chaos, fractais and nonlinear dynamics are common in some aspects of human physiology (Mackey and Glass, 1977; Glass et alii, 1987; Goldberger et alii, 1990), population dynamics (May, 1987; Hassell et alii, 1991), ecology and epidemiology (May, 1980; Schaffer, 1985), and the solar system (Wisdom, 1987; Kern, 1992; Sussman and Wisdom, 1992). Models of eleetrical systems have been found to exhibit chaotic dynamics. A few examples inelude DCDC converters (Hamill et alii, 1992), digital filters (Lin and CIma, 1991; Ogorzalek, 1992), power electronic regulators (Tse, 1994), microelectronics (Vau Buskirk and Jeffries, 1985), robotics (Varghese et alii, 1991) and power system modeis (Abed et alii, 1993). There seems to be some evidence of low dimensional chaos in time series recorded from electroencephalogram (Babloyantz et alii, 1985; Babloyantz, 1986; Layne et alii, 1986) although such results are so far inconelusive. Other areas where there has been much debate concerning the possibility of chaotic dynamics are economics (Boldrin, 1992;.Ja.ditz and Sayers, 1993) and the elimate (Lorenz, 1963; EIgar and Kadtke, 1993). Many other examples in which chaos has apparently been diagnosed inelude the models of a rotor blade lag (Flowers and Tongue, 1992), force impacting systems (Foale and Bishop, 1992), belt conveyors (Harrison, 1992), neural systems (Harth, 1983), biological networks (Lewis anel Glass, 1991), spacecraft attitude control systems (Piper and Kwatny, 1991) and friction force (Wojewoda et alii, 1992), to mention a few. An advantage of focusing on chaotic systems is that chaos is ubiquitous in nature, science and engineering. Thus simpie systems which exhibit chaos commend themselves as valuable paradigms and benchmarks for developing and testing new concepts and algorithms which in principie would apply to a much wider elass of problems. Therefore most of the tools and concepts reviewed in this papel' are also very relevant to systems which display regular dynamics. In the companion papel' the tools and systems described here will be used in the identification, analysis and control of nonlinear dynamics and chaos. 2 NONLlNEAR DYNAMICS: CONCEPTS AND TOOLS This sedion l)l'ovides some concepts and tools for the analysis of nonlinear dynamics. Some of the tools considered in this section currently constitute active fields of research pe1' se. Although no attempt has been made to give a thorough treatment on such issues, a considerahle number of references has been ineludeel for further reading. 2.1 Differential and difference equations An 11. thorder continuoustime system can be described by the differential equation dy. f )  = y =. (y,t dt where y(t) E IR" is the.state at time t and f : IR r ~ IR" is a smooth function calleel the vecio1' field. f is saiel to generate a fiow <Pt : IR" ;. IR", where <Pt(y, t) is a smooth function which satisfies the group properties <Pt,+1.o = <Pt, o <Pto' and <p(y, O) = y. Given an initial condition, Yo E IR" anel a time to, a t1'ajecto1'y, o1'bit or.solution of equation (1) passing through (01' based at) Yo at time to is denoteel as <pdyo, to). Because the time is explicit in equation (1), f is said to be nonautonomou.s. Conversely, systems in which the vector fielel does not contain time explicit1y are called autonomou.s. A system is said to be time pe1'iodic (1) with period T if f(y, t) = f(y, t + T), 't/y, t. An 11. thorder nonautonomous system with period T can be converted into an ( ) thorder autonomous system by adding an extra state (J = 2/T in which case the state space will be transformed from the Euelidean space IR,,+l to the cylind1'ical.space IR" x Sl, where Sl = IR/T is the cirele of length T = 27r/w. It is noted that ali the nonautonomous systems considered in this work will be time periodic in most situations. A Fi:red point of for equilib1'úlm, y, is elefined as f(y) = O for continuoustime systems and as y = f(fj) for discretetime systems. Df is the Jacobian mat.rix of the system, defined as the matrix of first partial derivatives. Evaluating the Jacobian at a particular point on a trajectory of the system, that is Df(yd gives a local approximation of the vedar field f in the neighborhood of Yi, sometimes Df(Yi) is referred to as a linea1'ization of f at Yi. If D f(y) has no zero ar purely imaginary eigenvalues, then the eigenvalues of this matrix characterize the stability of the fixed point y. An 11. thorder discretetime system can be described by a difference equation of the form 30 SBA Controle & Automação IVo!.7 no. 1/Jan., Fev., Mar. e Abril 1996
3 y(k + 1) = f(y(k), t). (2) A trajectory or orbit of a eliscrete system is a set of points {y( k + I)}k=Q. The elefinitions for discrete systems are analogous to the ones elescribeel for continuoustime systems anel therefore will be omitteel. For eletails see (Guckenheimel' anel Holmes, 1983; Parker anel CIma, 1989; Wiggins, 1990). 2.2 NlImerical simlllation of dynamical systems Generating time series for a system elescribeel by a elifference equation is quite straightforwarel since y(k), k = n y, n y + 1, n y + 2,... can be computeel by simply computing repeatedly an equation like (2) from a set of n y initial conelitions. If the system is elescribeel by an ordinary elifferential equation, simulation cannot be perfol'lneel as easily since an equation like (1) shoulel be integrateel. Fortunately, there are a number of well know algorithms available for performing this task such as Euler, trapezoielal, RungeKutta, AelamsBashforth, AelamsMoulton anel Gear's algorithm (Parker and CIma, 1989). The fourthoreler RungeKutta is uneloubtedly the most commonly useel algorithm for integrating ordinary elifferential equations. An important question when integrating elifferential equations on a eligital computei' is the choice of the integration interva.1. In the case of linear systems 01' nonlinear systems with relatively slow elynamics the choice of the integration interval is not usually criticai. For some nonlinear systems, however, if such an interval is not sufficiently short spurious chaotic regimes may be ineluced when integrating the system using, for instance, a fourthoreler RungeKutta algorithm, whilst seconeloreler RungeKutta algorithms may ineluce spurious elynamics even for fairly short integration intervals (Grantham anel Athalye, 1990). It has also been reported that in some cases the location of the bifurcation points depenei on the integration interval if it exceeels a criticai value (Aguirre anel Billings, 1994a). Irrespective of the type of the elynamical equations or the algorithm useel to solve such equations, an important question which shoulel be answereel is whether the simulateel results are representative of the 'real solution'. This is a nontrivial matter, anel to aelelress it woulel involve a detaileel look into the shadowing lemma (Guckenheimer anel Holmes, 1983). For the purposes of this tutorial, it suffices to mention that there is abunelant evielence that computei' simulations are generally reliable as numerical toois for the analysis of elynamical systems (Sauer anel Yorke, 1991). However, it shoulel also be borne in minei that pitfalls exist (Troparevsky, 1992), some of them as a consequence of the extreme sensitivity to initial conelitions exhibiteel by some systems. This characteristic is one of the most peculiar features of a chaotic system anel wiii be briefly illustrateel in section 2.9. Extreme sensitivity to initial conelitions eloes not invalidate numerical computations but certainly calls for caution in analyzing the results. 2.3 Spectral methods One of the first tools useel in diagnosing chaos was the power spectrum (Mees anel Sparrm'", 1981). The a.ppearance of a broael spectrum of frequencies of highly structureel humps near the loworeler resonances is usually creeliteel to chaos in loworeler systems (Blacher anel Perelang, 1981). However, broaelbanel noise anel the existence ofphase coherence can make it elifficult to eliscriminate experimentally between chaotic anel perioelic behavior by means of power spectrum (Farmer et alii, 1980). More recently the raw spectrnm (sum of the absolute values of the real anel imaginary components) anel the log spectrum (log of the raw spectrum) have been compareel with more dassica.l techniques in the context of chaotic time series analysis (Denton anel Diamonel,1991). Recently, the application of spectral techniques to the analysis of chaotic systems has concentrateel on the bispectrum anel trispectrum (Pezeshki et alú, 1990; Subba Rao, 1992; Chanelran et alii, 1993; EIgar anel Chanelran, 1993; EIgar anel Kenneely, 1993). See (Nikias anel Menelel, 1993; Nikias anel Petropulu, 1993) for an introeluction on higheroreler spectral analysis. Such techniques have been useel to eletect anel, to a certain extent, to quantify the energy transfer among frequency modes in chaotic systems. 2.4 Embedded trajectories One technique useel in the analysis of nonlinear elynamical systems is to plot a steaelystate trajectory of a system in the phasespace. Thus if y(t) is a trajectory of a given system this can be achieved by plotting ij(t) against y(t). For loworeler systems this proceelure can be useel to elistinguish between elifferent elynamical regimes. In many praetical situations, however, only one variable is measureel. In these cases an alternative proceelure is to plot y(tt p ) against y(t) where Tj) is a time lago These variables can be useel in the reconstruction of attractors (Packarel et alii, 1980; Takens, 1980) anel such variables also define the socalleel pseuelophase plane. This is motivateel by the fact that y(t  Tj)) is, in a way, reiateei to ij(t) anel consequently the embeeleleel trajectories representeel in the pseuelophase plane shoulel have properties similar to those of the original attractor representeel in the phase plane (Moon, 1987). A further aelvantage of this technique is that it enables the comparison of trajectories computeel from continuous systems where ij(t) is usually available, anel from eliscrete moelels where ij(t) is often not available anel would have to be estimateel. The choice of T p for graphical representation purposes is not criticai anel plotting a trajectory onto the pseuelophase plane for varying values of T p may give some insight regareling the information flow on the attractor (Fraser anel Swinney, 1986). Phase portraits anel plots of trajectory embeelelings can be useel not only as a means of elistinguishing elifferent elynam SBA Controle & Automação jvol.? no. 1jJan., Fev., Mar. e Abril
4 ical regimes, but also to demonstrate qualitative relationships between original and reconstructed attractors. 2.5 Dynamical attractors If a deterministic and stable system is simulateel for a sufliciently long time it reaches.steady.state. In state space this corresponds to the trajectories of the system falling on a particular 'object' which is called the attrac1.o7'. Asymptotically stable linear systems excited by constant inputs have point attractors which have dimension zero anel corresponel to a constant time series. Nonlinear systems, on the other hanel, usually elisplay a wealth ofpossible attractors. To which attractor the system will finally settle elepenels on the system itself anel also on the initi ai condi tions. An aelvantage of consielering attractors in state space as alternative representations of time series is that a number of geometrical anel topological results can be used. For the purposes of this tutorial, it will suflice to point out that the.shape anel dúnension of the attractors in state space are elirectly linked to the complexity of the dynamics of the respective time series. Thus simpie low dimensional attractors correspond to simpie time series dynamics. The most common attractors are the poin1. at1.ractor (dimension zero), limi1. cycles (dimension one) and 1.ori (dimension two). Another type of attractor which has recently attracted a great deal of attention are the socalled strange or chaoiic attractor.s which are fractal objects. The determination of the dimension of such attractors will be briefly addressed in section Bifurcation diagrams Another useful tool for assessing the characteristics of the steaelystate solutions of a system over a range of parameter values is the bifurcation diagram which reveals how the system bifurcates as a certain parameter, called the bifurcation parameter, is varied. Roughly, a system is saiel to undergo a bifurcation when there is a qualitative change in the trajectory of the system as the bifurcation parameter is varied. At the bifurcation point, the.j acobian of the system has at least one eigenvalue with the real part equal to zero for continuoustime systems or on the unit circle for eliscretetime systems. There are a number of known bifurcations. The most common codimension one bifurcations are the pi1.chfork, the saddlenode, the 1.ranscri1.ical, the Hopf bifurcation, anel the fiip or period doubling, which only occur in discrete maps or periodically driven systems. For an introduction to bifurcation and a description of the aforementioneel types see (Guckenheimer and Holmes, 1983; Mees, 1983; Thompson and Stewart, 1986). Approaches to calculate bifurcation diagrams include the bruie force, pa1.h following (Parker and CIma, 1989), the celliocell rnapping technique (Hsu, 1987) anel frequency domain methods (Moiola anel Chen, 1993). For reasons of simplicity, the brute force approach is described in what follows. This approach is simpie and robust but in general it is computationally intensive. Thus a point l' of a bifurcation diagram of a nonautonomous systems driven by A cos(w i) with A as the bifurcation parameter is elefined as l' = { (y, A) E IR x [ I y = y( t.;), A = Ao; i, =1.0 +[\55 X 2íT/w}, where [ is the interval [ =[Ai Ar] C IR, :S ia :S 2íT/w anel I\55 is a constant. This means that the point l' is chosen by simulating the system for a sufliciently long time [\55 X 27r/w with A = Ao to ensure that transients have died out before plotting y( [\55 x 27r/w) against Ao. In practice for each value of the parameter A, nb points are taken at the instants Clearly, the input frequency w can also be used as a bifurcation parameter. For autonomous systems a bifurcation eliagram can be obtained in an analogous way by choosillg i, = ia + (I\ss + i), A bifurcation eliagram will therefore reveal at which values of the parameter A E [ the solution of the system bifurcates anel how it bifurcates. ''''hen studying chaos such diagrams are also useful in detecting parameter ranges for which the system behavior is chaotic. (3) i = 0,1,...,nb1. (4) i=0,1,...,nb 1. As an example of a bifurcation diagram consider figure 1. This diagram and the respective system will be described in some detain in section 4.5. Throughout this tutorial, bifurcation parameters are denoted by A.. Thus, figure 1 shows some of the different types of attractors displayed by the system as A is varied. In particular, for A. = 4.5, 9 and 11 the system elisplays periodone, periodthree and chaotic dynamics, respectively. For clarity the respective attractors represented in the cylindrical statespace (see section 2.1) are also shown Poincaré sections A bifurcation diagram shows the different types of attractors to which the system settles to as the bifurcation parameter is varieel. However, a bifurcatioll diagram provide> very little information concerning the shape of the attractors in statespace. In order to gain further insight into the geometry of attractors one may use the socalled PoincarE maps. Such a map is a cross sectioll of the attractor and car be obtaineel by defining a plane which should be transversa to the flow in state space as showll in figure 2. (.5 ) 32 SBA Controle & Automação IVo!.7 no. 1/Jan., Fev., Mar. e Abril 1996
5 3.r 3[. . ~ "'.. ii ij + y3 = A cos(t) 2 1'C:~ V~.'. /""/~ 1.5~ ~/~', 2 ;~../.., a.5! i ;.. l~. lq> :~:'.".'.'...""... A=9 ld. I >,,:. v."' ~;> A=4.5 A=ll Figure 1  Bifurcation diagram for the DuffingUeda oscillator, see section 4.5. A, the amplitude of the input, is the bifurcation parameter. More precisely, consider a periodic orbit A( of some fiow <Pt in IR n arising from a nonlinear vector field. Let ~ C IR n be a hypersurface of dimension n1 which is tl'ansverse to the fiow <Pt. Thus the first return 01' Poincaré map P = ~ ~ ~ is defined for a point q E ~ by P(q) where TIO is the time taken for the orbit <pdq) basec1 at q to first return to ~. This map is very useful in the analysis of nonlinear systems since it takes place in a space whieh is of lower dimension than the actuai system. It is therefore easy to see that a fixed point of P col'l'esponds to a perioc1ic orbit of period 27r/w for the fiow. Similarly, a subharmonic of periocl I{ x 27r/w wiu appear as I\. fixed points of P. Quasiperiodie and chaotic regimes can also be readily recognized using Poincaré maps. For instanee, the firstreturn map of a ehaotic solution is formed by a welldefined and finelystructured set of points for noisefree dissipative systems. Such maps ca be used in the validation of identified modeis and reconstructed attractors (Aguil'l'e and Billings, 1994b). From the above c1efinition it is clear that if a system has n > 3, the Poincaré map would require more than two dimensions for a graphical presentation. In order to restrict the plots to twodimensionai figures, y(t  ~») is plotted against y(t) at a constant period. For periodicaily driven systems the input period is a natural choice and the resulting plot is caued a Poincaré section. (6) y(t) This procedure amounts to defining the Poincaré plane ~P in the pseudophasespace and then sampling the orbit represented in such a space. The choice of~) is not criticai but it should not be chosen to be too smau nor too large compared to the col'l'elation time of the trajectory. Otherwise the geometry and fine structure of the attractor would not be weu represented. The qualitative information conveyed by both Poincaré maps and sections are equivalent as demonstrated by the theory of embeddings (Takens, 1980; Sauer et alii, 1991). y(t) Although the Poincaré sections are usuauy obtained by means of numerical simulation, it is possible, although not always feasible, to determine Poincaré maps analytically (Guckenheimer and Holmes, 1983; Brown and CIma, 1993). 2.8 Routes to chaos Figure 2  A Poincaré section is obtained by defining a plane in state space which is transversal to the fiow. The image formed on such a plane is the Poincaré section of the attractor and wiu display fractal structure if such an attractor is chaotic. y(t) In the study of chaotic systems it is somewhat instructive to consider the c1ifferent routes to chaos in order to gain further insight about the dynamics of the system under investigation. As pointed out "the benefit in identifying a particular prechaos pattern of motion with one of these now classic models is that a body of mathematicai work on each exists which may offer better understanding of the chaotic phenomenon under study" (Moon, 1987,page 62). Because a thorough study of the routes to chaos is be SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril
6 yond the immediate scope of this work, some of the most weuknown patterns wiu be listed with some references for further reading. Some of the routes to chaos reported in the litel'ature indude period do'/lbling ca8cade (Feigenbaum, 1983; Wiesenfeld, 1989), q1ta81periodic ro'/lte to cha08 (Moon, 1987), iterm.ittency (ManneviUe and Pomeau, 1980; Kadanoff, 198:3), freq1tency locking (Swinney, 1983). For other routes to chaos see (Robinson, 1982) and referenees therein. 2.9 Sensitivity to initial conditions (o) ~ J"'',! J I / (b) " I o 0.1 C. U o.t.'''<i,'s (c) (d) :~ ~W~~I~~W{ oll nt G.4 OI (c) (f) I Probably the most fundamental property of chaotic systems is the sensitive dependenee on initial conditions. This feature arises due to the local divergence of trajectories in. state space in at least one 'direction'. This wiu be also addressed in the next section. In order to iuustrate sensitivity to initial conditions anel one of its main consequences, it will be helpful to consider the map y(k) = A [1  y(k  1)] y(ã~  1). (7) In order to iterate equation (7) on a digital computei', an initial conelition y(o) is required. Using this value, the right hanel side of equation (7) can be evaluated for any vaiue of A. This proeluces y( 1) which should be 'fedback' anel useel as the initiai conelition in the following iteration. This procedure can be then repeateel as many times as neeessary to generate a time series y(o), y(i), y(2),... A graphical way of seeing this is iuustrateel in figure 3. It should be noteel that the right hand siele of equation (7) is a parabola, as shown in figure 3a. Thus to evaluate equation (7) is equivalent to finei the value on the parabola which eorresponds to the initial conelition. This is representeel in figure 3a by the first vertical line. The feeding back of the new value is then representeel by projecting the value found on the parabola on the bisector. This completes one iteration. Choosing the initia! condition y(o) = 0.22 anel A = 2.6, figure 3a shows the iterative procedure and reveals that after a few iterations the equation settles to a point attraetor. The respective time series is shown in figure 3b. The same procedure was fouowed for the same initial condition and A = 3.9. The results are shown in figures 3ed. Clearly, the equation does not settle onto any fixed point anel not even onto a limit cycle. In fact, it is known that equation (7) displays chaos for A = 3.9. What happens if instead of a single initial eondition an interval of initial eonditions is iterated? This is shown in figures 3ef. For A = 2.6, the map wiu eventuauy settle to the same point attractor as before. This is a typica! result for regular stable systems and it iuustrates how au the trajectories based on the initiai conditions taken from the original interval converge to the same attractor. Figure 3  GraphieaI iteration of the logistic equation (I) (a) regular motion (A = 2.6) and (b) respective time series. (c) ehaotie motion (A = 3.9), anel (el) respective time series. In these figures the same initial eonditiol1 has been used namely y(o) = In figures (e) and (f) an interval oj initial eonditions has been iterated for the same values oj A as above. The intervais useel were y(o) E [ ' and y(o) E [ ], respectively. Note how sueh ai; interval is amplifieel when the system is chaotie, (f). Thi, is due to the sensitive elependence on initial conelitions. Considering a much narrower interval of initial conditiom anel proeeeeling as before yieldeel the results shown in figure 3f for whieh A = 3.9. Clearly, the interval of initial conelitions was wielened at each iteration. Sueh an intervai ean be interpreted as an errar in the original initial conelition. y(o) = In practice errors in initial conditions wiu be aiways present due to a number offactors such as noise, eligitalization effects, rouneloff errors, finite worellength precision, etc. It is this effeet of amplifying errors in initial eonelitions which is known as the sensitive depenelenee on initial conelitions anel an immeeliate consequence ofthis feature is the impossibility of making longtenn preelietion for chaotic systems. The next section describes indices which quantify the sensitivity to initial conditions lyapunovexponents Lyapunov exponents measure the average divergenee 01 nearby trajectories aiong certain 'elirections' in state space. A chaotic attracting set has at least one positive Lyapunov exponent anel no Lyapunov exponent of a nonchaotic attracting set can be positive. Consequently such exponent, have been useel as a criterion to eletermine if a given attracting set is 01' is not chaotic (Wolf, 1986). Recently the eoncept of local Lyapunov exponents has been investigateel (Abarbanel, 1992). The local exponents describe orbit instabilities a fixed numbel' of steps aheael rather than an infinite number. The (global) Lyapunov exponents of an attraeting set of length N can be defined as 1 1 Many authors use log2 in this definition 34 SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril 1996
7 1 ),i = lim ;c loge ji(n), NHXJ Iv i=1,2,...,n, (8) of an attracting set of length N can be elefineel as (see also equation (8)) where loge = In and the {ji(n)}i'=l are the ahsolute values of the eigenvalues of N ),1 == ~ lim "" log 11 Dk+1 II N N;= ~ e II Dk II ' (10) [Df(YN )][Df(YN1 )]... [Df(Y1)], where Df(Yi) E IR" x n is the J acobian matrix of the n dimensional differential equation (or discrete map) evaluated at Yi, and {ydt~l is a trajectory on the attractor. Note that n is the dynamical oreler of the system. In l11any situations the reconstructeel or identifieel modeis may have a dimension which is larger than that of the original systems and therefore such moelels have more Lyapunov exponents. These 'extra' exponents are calleel S]J'lIrious Lyap'llnov exponents. The estimation of Lyapunov exponents is known to be a nontrivial task. The simplest algorithms (Wolf et alii, 1985; Moon, 1987) can only reliably estimate the largest Lyapunov exponent (Vastano and Kostelich, 1986). Estimating the entire spectrum is a typically illconelitioned problem and requires more sophisticated algorithms (Parker and CIma, 1989). Further problems arise when it comes to decieling which of the estil11ated exponents are tnle anel which are sp'llrious (Stoop anel Parisi, 1991; Parlitz, 1992; Abarbanel, 1992). The estimation of Lyapunov exponents is currently an active field of research as can be verifieel from the following references (Sano and Sawaela, 198.5; Eckl11ann et a/ii, 1986; Bryant et alú, 1990; Brown et alii, 1991; Parlitz, 1992; Kadtke et alii, 199:3; Nicolis and Nicolis, 1993; Chialina et alii, 1994). For application of Lyapunov exponents in the quantification of real elata see (Branelstiíter et alii, 198:3; Wolf and Bessoir, 1991; Vastano anel Kostelich, 1986). In view of such elifliculties and the fact that the largest Lyapunov exponent, ),1, is in many cases the only positive exponent 2 anel that this gives an indication of how far into the future accurate preelictions can be made, it seems appropriate to use ),1 to characterize a chaotic attracting set (Rosenstein et alii, 1993). Ineleeel, the largest Lyapunov exponent has been useel in this way and to compare several identified models (Abarbanel et alii, 1989; Abarbanel et alli, 1990; Principe et alii, 1992). The algorithm suggested in (Moon, 1987) for estimating ),1 is described below. A similar algorithm which simultaneously estimates the correlation dimension to be elefineel in section 2.11 has been recently investigateel in (Rosenstein et alii, 1993). Consider a point Xo on the trajectory x(k) (for the moment it is assumed that such a trajectory is available a priori), say Xo =x(o), anel a nearby point Xo+Do. For simplicity it is assumeel that x(k) E IR, but in general higherdimensional systems will be the case. The largest Lyapunov exponent 2In this case /\1 ;::: (9) h, where h is the KolmogorovSinai or ITletric entf,0py. Note that for dissipative systeitls (chaotic and nonchaotic) Li=l '\; < O (Eclunann and Ruelle, 1985; Wolf, 1986). where Dk is the distance betweell two points on nearby trajectories at time k. The estimation of ), 1 is a simulationbased calculation (Moon, 1987: Parker anel CIma, 1989). The main idea is to be able to eletermine the ratio II Xl  (;C1 + DIl II II a:o  (xo + Do) II D= Df(x;) D, II Dl II II Do II (11) where.r1 is another point on the trajectory x(k), namely.r(êll), ;C1+D1 is a point obtaineel by following the evolution of the ranelomly chosen initial conelition ;co + Do over the interval êll where êll will be referred to as the Lyapunov interval. From the last equation it is clear that one onlv needs to follow the evolution of perturbations Di along th~ reference trajectory x( k). It is well known that the Jacobian matrix D f( Xi) elescribes the elynamics of the system for small perturbations in the neighborhood of Xi. Thlls the computation of the largest Lyapunov exponent, ),1, consists in solving the variational equations (12) where Df(Xi) is the J acobian matrix of f( ) evaluated at Xi, and also of simulating the system if the trajectory x(k) = {Xi}~O x = f(x) (13) ' is not available in advance. Equations (12) anel (13) are simulated and the ratio II Dk+1 II / I1 Dk II is ca.lculated once at each êll interval. Therefore estimating ),1 consists in successively preelicting the systems governed by Df O anel f O êll seconds into the future and assessing the expansion of the perturbations Di. Some of the ideas described above are illustrateel in figures 4ab. The former figure is the bifurcation diagram of the logistic equation (7). Figure 4b shows the largest Lyapunov exponent of such an equation for a range of values of A. The largest Lyapunov exponent was ca.iculated as elescribeel above. Note that ),1 = O at bifurcation points and that ),1 > Ofor chaotic regimes as predicted by the theory. These figures a.iso reveal the narrow windows of regular dynamics which are surrounded by chaos Correlation dimension Another quantitative measure of an attracting set is the fractal elimension. In theory, the fractal dimension of a SBA Controle & Automação jvol.? no. IjJan., Fev., Mar. e Abril
8 O''~~L' A chaotic (nonchaotic) attracting set is noninteger (integer). An exception to this rule are fai fractais which have integer fractal dimension which is consequently inadequate to describe the properties of such fractais (Farmer, 1986). Nonetheless, like the largest Lyapunov exponent, the fractal dimension can be, in principie, used not only to diagnose chaos but also to provide some further dynamical information in most cases (Grassberger ei alú. 1991). A deeper treatment can be found in (Russell ei ahi, 1980; Farmer ei alú. 198:3; Grassberger anti Procaccia, 198:3a; Atten ei alii. 1984; Caputo ei alii ) for raw data and in (Badii and Politi, 1986; Badii ei alii, 1988; Mitschke, 1990; Brown ei alá, H)92; Sauer and Yorke. 19~);3) for filtered time series. The fractal dimension is related to the amount of information required to characterize a certain trajectory. If the fractal dimension of an attracting set is D + li, D E ~+, where O< li < 1. then the smallest number of firstorder differential equations required to describe the data is D+1. There are several types of fractal dimension such as the poiniwise dimension, correlation dimension, informaiion dimension, Hausdor/J dimension, Lyapnnov dimension, for a comparison of some of these dimensions see (Farmer, 1982; Hentschel and Procaccia, 1983; Moon, 1987). For many strange attractors, however, such measures give roughly the same value (Moon, 1987; Parker and Cima, 1989). The correlation dimension: 3 (Grassberger and Procaccia, 1983b), however, is clearly the most widely used measure of fractal dimension employed in the literature. OI::::=="~ ;o>';,rrfHt' '+fjj A time series {Y;}~l can be embedded in the phase space where it is represented as a sequence of dedimensional points Yj = [Yj Yj1... Yjd e +1]' Suppose the distance between two such points is 4 S'ij =1 Yi  Yj I then a correlation function is defined as (Grassberger and Procaccia, 1983b) ' ' ' ~'' A Figure 4  (a) Bifurcation diagram of the logistic map, and (b) respective largest Lyapunov exponent,.\1. Note that.\1 = O at bifurcation points and that.\1 > O for chaotic regll11es. C(e) = lim ~ (number of pairs (i, j) with S'ij < é). N~co N (14) The correlation dimension is then defined as D I loge C(é) loge é c = 1n1. eco (15) For many attractors De will be (roughly) constant for values of é within a certain range. In theory, the choice of de does not influence the final value of De if de is greater than a certain value. In particular, it has been shown that provided there are sufficient noisefree data, de = Ceil(D e ), where Ceil( ) is the smallest integer greater than or equal to De (Ding ei alii, 1993) and that this result remains trne in the case the data have been filtered using finiie imp1tlse 3This measure can be seen as a genera./ised dimension and is considered to be the easiest to estimate reliably (Grassberger, 1986b) and thus remains the most popular procedure so faro 4 Several nonns can be used here such as Euclidean, 1'1, etc. 36 SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril 1996
9 o.!lr~~~~~~~"";'>..., Ui' C al Õ 'aí 60..Q log(eps) Figure 5  Logarithm of the correlation function C(t:) plotted against log( c) for embedding dimensions de = 2 to de = 10. The correct value, De ~ 2.0 is attained for de 2': 5. response (FIR) filters (Sauer and Yorke, 199:3). In practice, due to the lack of data aud to the presence of noise, de > Ceil( De), thus several estimates of the correlation dimension are obtained for increasing values of de. If the data were produced by a lowdimensional system, such estimates would eventually converge. Of course, these results depend largely on both the amount and quality of the data available. For a brief account of data requirements, see seetion 3.1 below. In order to illustrate the estima.tion of De a time series with N = data points was obtained by simulating Chua's circuit (see section 4.3) operating on the double scroll attractor. The correlation function C(t:) was then calculated for 2 ::; de ::; 10 and plotted in figure 5. For small embedding dimensions (de = 2) the correlation dimension is De ~ 1.8 but as de is increased the scaling region converges to the correct value De ~ 2.0 for de 2': 5. One of the properties ofsome fractais is selfsimilarity. This is illustrated in figure 6 which shows the well known Hénon attractor (see aiso section 4.2) and an amplification of a small section of one of its legs. It should be observed tha.t what appears to be a single 'line' in the attractor turns out to be two lines (see zoom in figure 6). However, if each of these lines were zoomed again it would become apparent that they are composed of other two lines each and this continues ad infinitum. This particular fractal structure is sometimes referred to as having a Cantor set structure. Probably the greatest application of the correlation dimension is to diagnose if the underlying dynamics of a time series have been produced by a loworder system (Grass Figure 6  Fractal structure of the Hénon attractor. berger, 1986a; Lorenz, 1991). Because this is an important problem, the estimation of correlation dimension has attracted much attention in the last years. Many papers have focused on determining the causes of bad estimates (Theiler, 1986), estimating errorbounds (Holzfuss and MayerKress, 1986; Judd and Mees, 1991) and suggesting improvements on the original algorithm described in (Grassberger and Procaccia, 1983b) Other invariants There are a number of less used invariants of st.range attractors report.ed in the literature such as the Kolmogorov 01' metric entropy, topologicai entropy, generalised entropies and dimensions, partial dimensions, mutuai information, etc. (Grassberger aud Procaccia, 1984; Eckmann and Ruelle, 1985; Fraser, 1986; Grassberger, 1986b). Vlith few exceptions (Hsu and Kim, 1985), statistics have received little attention as invariant. measures of strange attractors. Apparent.ly, the most useful such measure is the probability densiiy funetion (Packard ei alii, 1980; Moon, 1987; Vallée et alii, 1984; Kapitaniak, 1988) The estimation of unstable limit cycles has also been put forward as a way of charaeterizing strange attractors. The motivation behind this approach is that because a strange attractor can be viewed as a bundle of infinite unstable limit cycles, the number of the periodic orbits, the respective distribution and properties should be representative of the attractor dynamics. Indeed, from such information other invariants such as entropies and dimensions can be estimat.ed (Auerbach ei alli, 1987). For more information on this subjeet, see (Grebogi et alii, 1987; Cvitanovié, 1988; Lathorp and Kostelich, 1989; Lathorp and Kost.elich, 1992) 3 DIAGNOSING CHAOS In general, the problem of diagnosing chaos can be reduced to estimating invariants which would suggest that the data are chaotic. For instance, positive Lyapunov exponents, noninteger dimensions and fractal structures in Poincaré SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril
10 sections would suggest the presence of chaos. The main question is how to confidently estimate such properties from the data, especia.lly when the available records are relatively short and possibly noisy. The tedmiques that have been suggesteel in the literature can be elivided in two major groups. N onparaluetric luethods. These incluele the use of tools which take the data anel estimate dynamical invariants which, in turn, will give an indication of the presence of chaos. Such tools incluele power spectra, the largest Lyapunov exponent. the correlation dimension, reconstructed trajectories, Poincaré sections, reiative rotation rates etc. Detailed description and a.pplication of these techniques can be founel in the literature (Moon, 1987; Tufillaro et alii, 1990; Dellton anel Diamond, 1991). For a recent comment of the practical difficulties in using Lyapunov exponents anel dimensions for diagnosing chaos see (Mitschke anel Dammig, 1993). Two practical difficulties common to most of these approaches are the number of data POilltS available and the noise present in the data. These aspects are briefiy discussed in the following section. Poincaré sections are very popular for detecting chaos because for a chaotic system the Poincaré section reveals the fractal structure of the attractor. However, in order to be able to elistinguish between a fractal object and a fuzzy doud of points a certain amount of data is necessary. Moon (1987) has suggested that a Poincaré section shoulel consist of at least 4000 points before dedaring a system chaotic. For nonautonomous systems this means 4 x 10 3 forcing periods which could amount to 4 x 10 5 data points. Predietionbased techniques. Some methods try to diagnose chaos in a data set baseel upon preeliction errors (Sugihara anel May, 1990; Casdagli, 1991; EIsner, 1992; Kennel anel Isabelle, 1992). Thus preelictors are estimateel from, say, the first half of the elata records and used to predict over the last half. CImos can, in principie, be diagnosed based on how the prediction errors behave as the prediction time is increased (Sugihara and May, 1990), 01' based on how the prediction errors related to the true data compare to the prediction errors obtained from 'faked' data which are random but have the same length anel spectral magnitude as the original data (Kennel and Isabelle, 1992). A related approach has been termeel the method of s'urrogate data (Theiler et alii, 1992a; Theiler et alii, 1992aa). Regardless of which criterion is used to decide if the data are chaotic 01' not, predictions have to be made. Clearly, the viability of these approaches depends on how easily predictors can be estimated and on the convenience of making predictions. Once a predictor is estimated criteria anel statistics such as the ones presented in (Sugihara and May, 1990; Kennel and Isabelle, 1992) can be used to diagnose chaos. 3.1 Data requirements The length and quality of the data records are crucial in the problem of characterization of strange attractors. At present, there seems to be no general rule which eletermines the amount of data required to learn the dynamics, to estimate Lyapunov exponents and the correlation elimension of attractors. However it is known that "in general the detailed diagnosis of chaotic elynamical systems requires long time series of high quality" (Ruelle. 1987). Typical values of elata length for learning the dynamics are 2 x 10 4 (Farmer and Sidorowich, 1987; Abarbanel et alii, 1990) for systems of elimension 2 to 3, 1.2 X X 10 4 (Caselagli. 1991). It has been argued that to estimate the Lyapunov exponellts forcing periods shoulel be used (Denton and Diamonel. 1991). Other estimates are N > lod (quoteel in (Rosenstein et alii, 1993) anel N > :30 D where D is the dimension ofthe system (W'olf et alii, 1985) but in some cases at least 2 x 30 D was required (Abarbanel et alii, 1990). Typical examples in the literature use 4 x X 10 4 (Eckmann et alii, 1986) 1.6 x 10 4 (Wolf anel Bessoir, 1991) anel 2 x 10 4 elata points (Ellner et alii, 1991). Fairly long time series are also requireel for estimating the correlation dimensiono In fact, it has been pointed out that dimension calculations generally require larger data recorels (Wolf and Bessoir, 1991). For a strange attractor, if insufficient data is useel the results would indicate the dimension of certain parts of the attractor rather than the elimension of the entire attractor (Denton anel Diamond, 1991). However, results have been reported which suggest that consistent estimates of the correlation dimension can be obtained from data sequences with less than 1000 points (Abraham et alii, 1986). On the other hanel, there seems to be evielence that "spuriously small dimension estimates can be obtained from using too few, too finely sampleel anel too highly smootheel data" (Grassberger, 1986a). Moreover, the use of short and noisy data sets may cause the correct scaling regions to become increasingly shorter and may cause the estimate of the correlation elimension to converge to the correct result for relatively large values of the embedding dimension (Ding et alii, 1993). Thus typical examples use 1.5 x X 10 4 (Grassberger and Procaccia, 1983b) and 0.8 x X 10 4 data points (Atten et alii, 1984). Thus there seems to be no agreeel upon rule to determine the amount of data required to estimate dimensions with confidence but it appears that at least a few thousanel points for low dimensional attractors are neeeleel (Theiler, 1986; Havstad and Ehlers, 1989; Ruelle, 1990; Essex and Nerenberg, 1991). In particular, N > lo D d 2 has been quoteel in (Ding et alii, 1993). It should be realized that the difficulties in obtaining long time series goes beyonel problems such as storage anel computation time. Indeeel, it has been pointeel out that for some real systems, stationarity cannot always be guaranteeel even over relatively short periods of time. Examples of this induele biological systems (May, 1987; Denton anel Diamond, 1991), ecological and epielemiologica.l elata (Schaffer, 38 SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril 1996
11 1985; Sugihara anel May, 1990). A test for stationarity has been recently suggesteel in (Isliker anel Kurths, 1993). 4 SOME NONLlNEAR MODELS The objective of this section is to proviele the reaeler will a. collection of well elocumenteel simpie nonlinear 1l10elels which elisplay a wiele variety of elynamical regimes which incluele selfoscillations, limit cycles, perioeleloubling cascaeles anel chaos. 4.1 THE logistic EQUATION In many fielels of science the elynamics of a particular system are better representeel by eliscrete maps, also calleel elifference equations, of the form This equation inelicates that the value of the variable y at time k + T is a function of the same variable at time k. The basic equation (16) applies to a number of situations which incluele switching power circuits (Tse, 1994), population elynamics, genetics, elemography, economics anel social sciences, see (May, 1975; May, 1976) anel references therein. The most well known, anel certainly one of the simplest, examples of (16) is equation (7) known as the logistic equation which was originally suggesteel as a population elynamics moelel (May, 1976) anel elisplays a variety of elynamical regimes as the bifurcation parameter, A., is varieel in the interval 2.8 ~ A. ~ 3.9. The bifurcation eliagram of this map, which is shown in figure 4a, is a classical example of the perioddoubling rmt.te to chaos. Due to its simplicity anel also because some elynamical invariants can be deriveel analytically for this moelel, the 10 gistic map is frequently useel as a bench test in the stuely of elynamical systems. For a review on firstoreler maps see (May, 1980; May, 1987). 4.2 The Hénon map The map (Hénon, 1976) { y(á~ + T) = F[y(k)]. x(k) = 1 ax(k _1)2 + y(k: 1), y(k) = bx(k 1) (16) (17) was proposeel by the French astronomer Michel Hénon as an approximation of a Poincaré mapping of the Lorenz system (see section 4.6). In the literature the parameter values usua.lly useel are a =1.4 anel b=0.3. Thus these values leael to the socalleel Hénon attractor shown in figure 6. For a more eletaileel elescription of the elynamical properties of this map see (Moon, 1987; Peitgen et alii, 1992). tl L C, (b) m Figure 7  (a) The CIma circuit anel (b) voltagecurrent characteristic of the piecewise linear component, NR, known as Chua's elioele. As the logistic equation, the Hénon map is a popular benchmark useel to test a variety of algorithms concerning the analysis anel signal processing of nonlinear elynamical systems. 4.3 The Chlla circliit The normalizeel equations of Chua's circuit can be written as (CIma et alii, 1986; CIma, 1992; Maelan, 1993; CIma anel Hasler, 1993) ~ = n(y  h(x)) y=:ry+z { i = f3y m 1 x+(m O m 1 ) h(x) = mox { m1a.:  (mo  In1) x > 1 I x I~ 1 x <1 (18) where mo = 1/7 anel m1 = 2/7. Varying the parameters n anel f3 the circuit elisplays several regular anel chaotic regimes. This system is a particular case of the more general unfolded Chu,a's circu,it (CIma, 1993). The well known elouble scroll attractor, for instance, is obtaineel for n = 9 anel f3 = 100/7, see figure. 8. For this attractor À 1 = 0.23 (Matsumoto et alii, 1985; Chialina et afii, 1994). The estimateel value of the correlation elimension for this attractor was De =1.99 ± Chua's circuit is certainly one of the most well stuelieel nonlinear circuits anel a great number of papers ensure that the elynamics of this circuit are also well elocumenteel, see for instance (CIma anel Hasler, 199:3; Maelan, 1993; Matsumoto et afii, 1993). The only nonlinear element in Chua's circuit is a twoterminal piecewiselinear resistor, calleel Chua's elioele, which has also' been implemented as an integrated circuit (Cruz anel CIma, 1992), however the entire circuit can be implementeel with 'offtheshelf' components (Kennedy, 1992). 4.4 The DlIffingHolmes oscillator One of the classical bench tests in mechanics is the Duffing oscillator (Duffing, 1918). Two elifferent versions of this oscillator have been investigated in connection with chaos SBA Controle & Automação jvo!.? no. 1jJan., Fev., Mar. e Abril
12 N y(t) o o x(t) 2 o Figure 8  The double sero11 Chua's attractor. by Philip Holmes and Yoshishuke Ueda. The equations whieh deseribe the dynamies of sueh osei11ators are briefiy deseribed in this and the fo11owing seetion, respectively. The we11 known DuflingHolmes equation is eommonly used to model mechanieal osei11ations arising in twowe11 potential problems (Moon, 1987) beeause this system is eharaeteristie of many of the structural nonlinearities eneountered in praetiee (Hunter, 1992). The equation whieh models this system is (Holmes, 1979; Moon and Holmes, 1979) 1.5' ~ ".J A a ~, :ij + ó ij  f3 y + y3 = A cos (w t). (19) 0.5 The bifureation diagram for this system for w = 1 rad/s and 0.22:s A :S 0.35 is shown in figure 9a. For Ó = 0.15, f3 = 1, A=0.3 anel w=lrad/s, this system settles to a strange attraetor whieh is shown in figure 9b. The largest Lyapunov exponent of this attraetor is )'1 = 0.20 and the eorrelation elimension equals De = 2.40 ± Equation (19) has been used to model a wiele range of systems in seienee and engineering, a few examples include (MeCa11um anel Gilmore, 1993; Parlitz, 1993) O y(k) The DufFingUeda oscillator The DuflingUeda equation (Ueda, 1985) y+kij+y3=u(t) (20) b Figure 9  (a) bifureation eliagram, and (b) Poinearé seetion ofthe DuflingHohnes osei11ator, for A = 0.3 and 11, = 5. was originally proposed as a model for nonlinear osei11ators anel has become a beneh test for the study of non 40 SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril 1996
13 Figure 10  The DuffingUeda oscillator. linear dynamics. It has also been considered as a simple paradigm for chaotic dynamics in electrical science (Moon. 1987). Consequently this system has attracted some attention and has been used to investigate nonlinear elynamics in different situations (Feigenbaum, 198:3; Kapitaniak, 1988). One of the main reasons for this is that in spite of being simpie this moelel can produce a variety of dynamical regimes, from periodone motions to chaos (Ueda, 1980; Ueda, H)85; Kawakami, 1986). The Poincaré sections and bifurcation diagrams were obtained as indicated respectively in sections 2.7 and 2.6 for the input 1l.(i) = A cos(wi). A was used as the bifurcation parameter in the bifurcation diagrams. The bifurcation shown in figure 1 was obtained by taking k = 0.1, w = 1 rad/s and simulating equation (20) digitally using a fourthorder RungeKutta algarithm with an integration interval equal to 7r/3000. Figures lia and llb shows the Poincaré section of the attractors at A = 5.7 and at A = 11, respectively. The largest Lyapunov exponent of these attractors are respectively.\1 = and.\1 = 0.11 and the carrelation dimensions equal De = 2.10 ± and De = 2.19 ± The bifurcation diagram in figure 1 reveals a number of dynamical regimes displayeel by this oscillator in the range of values of A considered. In particular, at A::::::: 4.86 the system undergoes a period doubling (flip) bifurcation. This happens again at A::::::: 5.41 and characterizes the well known period doubling route to chaos (Feigenbaum, 1983). Another similar cascade begins at A ::::::: 9.67 preceding a different chaotic regime. Two chaotic windows can be distinguished at approximately 5.55 :S A :S 5.82 and 9.94 :S A :S At A ::::::: 6.61 the system undergoes a supercritical pitchfork bifurcation and at A::::::: 9.67 it undergoes a subcriticai pitchfork bifurcation. The bifurcation diagram begins and ends with period1 regimes and displays period3 dynamics for 5.82 :S A :S It should be noted that in the range 4.5 :S A:S 5.5 there are two coexistent. attractars undergoing a sequence of perioddoublings, t.his becomes apparent by using a different set of initial conditions and explains the broken lines ~3.2..!. >; ~2.2 >; ,~ r,r, y(t) a y(t) , , ~, The lorenz equations The well known Lorenz equat.ions are (Lorenz, 1963) b {.~ = rr (y  x) y = px y z=xy(3z. xz (21) Figure 11  Poincaré sections for t.he DuffingUeda oscillat.ar, for T p =200 x 7r/3000 anel (a) A=5.7, (b) A=l1. Choosing rr=10, (3=8/3 and p=28, t.he syst.em described SBA Controle & Automação jvol.? no. 1jJan., Fev., Mar. e Abril
14 L R N20 10 O 50 O y(t) O x(t) Figure 13  The modifiecl van der Pol oscillator. anel Akamatsu, 1981) :ii + P (y~  lhi +:1/ = A. cos(w 1). (22) This is known as the modified van der Pai obcil/ator (Moon, 1987). Alike the eliverse versions of the Duffing system, this equation also has a cubic term which is absent in the original van der Pol equation. On the other hanel, alike the van eler Pol oscillator, the circuit governeel by equation (22) also exhibits selfsustained oscillations, that is, oscillations for A = Owith w = 1.62 rael/s. Figure 12  The Lorenz attractor. by equations (21) settles to the well kllown Lorenz attractor, shown in figure 12. Since the Lorenz equations were first published in 1963 they have become a standard for studying complex dynamics. Fairly detailed descriptions of the dynamics of this system can be found in (Sparrow, 1982; Thompson anel Stewart, 1986; Peitgen et aiii, 1992). The popularity of the Lorenz equations is not only due to the fact that it was one of the first systems published in connection with chaos but also because it is a physically motivated modei. Further, although the Lorenz equations moelel some aspects offluid dynamics, it has been shown that the instabilities of such a model are identical with that of the single moele laser and applicahle to underdampeellaser spikes (Haken, 1975) anel that the openioop dynamics of smoothairgap brushless De motors can also be modeled by such equations (Hemati, 1994). The bifurcation diagram of this system is shown in figure 14a. This bifurcation eliagram presents a number of coexistent attractors anel dynamical regimes interwoven in a very complicated manner. It. is worth pointing out that for O< A:::; 3 the oscillator presents quasiperioelic motions which cannot be distinguisheel from the chaotic dynamics baseel upon bifurcation diagrams only. Taking p = 0.2, 1'1=17 and w = 4 rad/s, this system settles to the strange attractor shown in figure 14b. The largest Lyapunov exponent of this attractor is À 1 = 0.33 and the correlation elimension equals De =2.18 ± The Rõssler equations The Hénon map elescribed in section 4.2 was originally suggesteel as a model for the Poicaré map of the Lorenz equations. Similarly, Rossler proposed the following set of equations (Rossler, 1976) The largest Lyapunov exponent of the attractor shown in figure 12 is)'1 =0.90 (Peitgen et aiii, 1992) and the correlation dimension equals De =2.01 ± { X = (y + =) y=x+o'y :: = O' + = (x  p,), (23) 4.7 The modified van der Pol oscillator During the twenties van der Pol investigated an electrical circuit in which the nonlinearity was introduced by a vacuum valve. One of the principal characteristics of such a circuit was that it exhibiteel selfsustained oscillations aiso called relaxation oscillations, thus lending itself as a model for a number of real oscillating systems such as the heart (van der Pol and van der Mark, 1928). The normalized equations of a modified version of the van der Pol oscillator, which has negative resistance, are (Veda as a simplifieel version of the Lorenz system, 01' in Rossler's words a "modei of a model". The simplification attained by Rossler can be appreciated by noticing that the attractor exhibiteel by (23), with [1'=0.2 and p=5.7 see figure 15., is composed of a single spiral which resembles a lviobius band insteael of the two spirals which can be clearly elistinguisheel in the Lorenz attractor. The largest Lyapunov exponent of the attractor shown in figure 15 is À 1 = 0.074, the Lyapunov dimension and the correlation dimension of this attractor are DL = 2.01 and De = 1.91 ± 0.002, respectively. 42 SBA Controle 1St Automação jvol.7 no. 1jJan., Fev., Mar. e Abril 1996
15 ,r,~, o 12 o 20 y(t) o x(t) A Figure 15  The Rossler attractor. a The field of possible applications of equations of the type of (23) ranges from astrophysics, via chemistry and biology, to economics (Rossler, 1976). In fact, the Rossler and Lorenz attractors are regarded as paradigms for chaos thus the chaotic dynamics of some systems are sometimes classified as "Rosslerlike" or "Lorenzlike" chaos (Gilmore, 1993). 5 DISCUSSION ANO FURTHER REAOING o ' ' ' ' ' ' 1 O 2 3 y(t) b Figure 14  (a) bifurcation diagram, and (b) Poincaré section of the modified van der Pol oscillator, A=17, w=4rad/s and 1;) =16. The analysis and quantification of chaotic dynamics is a relatively recent area. Nevertheless there is an immense collection of scientific papers and books devoted to this subject and any attempt to produce a survey on nonlinear dynamics and clla.os, no matter how thorough, would be, in all certainty, just a rough sketch on this fascinating subject. The main objective of this paper has been to review in a very pragmatic way a few concepts which are believed to be basic. Since it would be inappropriate to produce an indepth review, a rather generous number of references has been cited for further reading. Needless to say, the reference list does not exhaust the wealth of papers and books currently availahle. The following references seem to be a good starting point. The books (Gleick, 1987) and (Stewart, 1989) are a good introduction for the average reader. A more formal coverage is given by (Thompson and Stewart, 1986) and (Moon, 1987). For a mathematical exposition on the subject see (Guckenheimer and Holmes, 198:3) and (\Viggins, 1990). Some practical aspects of bifurcation and chaos are discussed in (Matsumoto et alii, 1993) and a good account on computer algorithms for nonlinear systems applications can be found in (Parker and CIma, 1989). See also (Abra. SBA Controle & Automação jvol.7 no. 1jJan., Fev., Mar. e Abril
16 h3m and Shaw, 1992) for a beautifully illustrated introduction to nonlinear dynamics and bifurcations. The following papers are also good introductions to nonlinear dynamics and chaos (Mees and Sparrow, 1981; Shaw, 1981; Mees, 1983; Eckmann anel Ruelle, 1985; Crutchfielel et alii, 1986; Mees anel Sparrow, 1987; Parker anel CIma, 1987; Argyris et alú, 1991; Thompson anel Stewart, 1993). Gooel surveys on modeling anel analysis of chaotic signals can be founel in (Grassberger et a.lú, 1991; Abarbanel et alú, 199;3). See also (Hayashi, 1964; Atherton and Dorrah, 1980) for a rather 'classica.l' approach to the analysis of nonlinear oscillations. Finally, it is worth pointing out that some software packages are availahle for analysis of nonlinear anel chaotic systems. In particular, the program kaos (Guckenheimer anel Kim, 1990; Guckenheimer, 1991) which runs on Sun workstations and can be obtained by ftp on macomb.tn.comell.edu The programs MTR CHAOS anel MTRLYAP (Rosenstein, 199:3) can be sueel for analysing chaotic signals and estimating correlation elimension anel largest Lyapunov exponents. These programs are available under request to Some of the concepts anel mathematical tools eliscusseel in this papel' will be useel in a companion papel' which will aelelress some aspects ofthe ielentification, analysis anel control of nonlineal' elynamics anel chaos. Acknowledgements FinanciaI support from CNPq (Brazil), uneler grant /959, is gratefully acknowleelgeel. REFERENCE5 Abarbanel, H. (1992). Local anel global Lyapunov exponents on a strange attractor. In Casdagli, M. anel Eubank, S., eelitors, Nonlinear Modeling and Forecasting, pages Aelelison Wesley, New York. Abarbanel, H. D. 1., Brown, R., anel Kaeltke, J. B. (1989). 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