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

Luis Antonio AguirreCentro de Pesquisa e Desenvolvimento em Engenharia Elétrica

Universidade Federal de Minas GeraisAv. Antônio Carlos 6627

:31270-901 Belo Horizonte, M.G .. BrazilFax: .55 31 499-5480, Phone: (0:31) 499-.548:3

E-mail: aguirrelgcpc!ee.ufmg.br

ABSTRACT - The relevance of nonlinear dynamics andchaos in science and engineering cannot be overemphasized.Unfortunately, the enormous wealth of techniques for theanalysis 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 briefintroduction to some of the main concepts and tools used inthe analysis of nonlinear dynamical system and chaos whichhave been currently used in the literature. The main objec­tive is to present a readable introduction to the subject andprovide several references for further reading. A number ofwell known and well documented nonlinear models are alsoincluded. Such models can be used as benchmarks not onlyfor testing some of the tools described in this paper, butalso for developing and troubleshooting other algorithms inthe comprehensive fields of identification, analysis and con­trol of nonlinear dynamical systems. Some of these aspectswill be addressed in a companion paper which follows.

1 INTRODUCTION

An important step towards the analysis of real systems is torealize that virtually ali systems are in a sense dynamieal.This does not mean to say, of course, that the dynamics ofevery system are always necessarily relevant to the analysis.Thus although it is sometimes justifiable to regard certainsystems as being sialie, in most cases it is worthwhile takinginto account the dynamics inherent in the systems to beanalyzed.

oArtigo submetido em 15/07/94;1a Revisão em 01/08/95 2 a Revisão em 27/11/95Aceito por recomendação do Ed. Consultor Prof.Dr. Liu Hsu

Mathematica.l1y, dynamical systems are described by differ­ential equations in continuous-time and by difference equa­tions in discrete-time. 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 totake into consideration the nonlinearities, which are oftenas important to the system as the dynamics. The use oflinear models in science and engineering has always beencommon practice. A good linear model, however, describesthe dynamics of the a system only in the neighborhood ofthe particular operating point for which such a model wasderived. The need for a broader picture of the dynamicsof real systems has prompted the development and use ofdynamicalmodels which included the nonlinear interactionsobserved in practice.

In this paper a few basic concepts related to nonlineardynamical systems are brief!y reviewed. The objective istwofold, namely to provide a brief introduction to nonlineardynamics and chaos and to indicate a few basic referenceswhich can be used as a starting point for a more detailedstudy on this subject. In particular, this paper will de­scribe a few mathematieal tools sueh as Poinearé seetions,bifurcation diagrams, Lyapunov exponents and correlationdimensions. AIso, some of the most commonly used nonlin­ear modeis which display chaotic behavior will be presentedthus providing the reader with both, a few basic tools andwell 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 hasbeen found. Different types of mathematical equations ex-

SBA Controle & Automação jVol.7 no. 1jJan., Fev., Mar. e Abril 1996 29

hibit chaotic solutions, for instance ordinary differentialequations, 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 (Mareelsanel Bitmead, 1986; Mareels anel Bitmeael, 1988; Golelenanel Yelstie, 1992) and eligital control systems (Ushio andHirai, 1983; Ushio and Hsu, 1987)

Chaos seems to be the rule rather than the exception inmany nonlinear mechanical and electrical oscillators andpenelula (Blackburn et alii, 1987; Hasler, 1987; TvIatsumoto,1987; Ketema, 1991; Kleczka et alii, 1992).

of nonlinear dynamics and chaos.

2 NONLlNEAR DYNAMICS: CONCEPTSAND TOOLS

This sedion l)l'ovides some concepts and tools for the anal­ysis of nonlinear dynamics. Some of the tools consideredin this section currently constitute active fields of researchpe1' se. Although no attempt has been made to give a thor­ough treatment on such issues, a considerahle number ofreferences has been ineludeel for further reading.

2.1 Differential and difference equations

An 11. th-order continuous-time system can be described bythe differential equation

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).

where y(t) E IR" is the .state at time t and f : IRr~ IR" is a

smooth function calleel the vecio1' field. f is saiel to generatea fiow <Pt : IR" -;. IR", where <Pt(y, t) is a smooth functionwhich satisfies the group properties <Pt,+1.o = <Pt, o <Pto' and<p(y, O) = y.

Chaos, fractaIs and nonlinear dynamics are common insome aspects of human physiology (Mackey and Glass,1977; Glass et alii, 1987; Goldberger et alii, 1990), pop­ulation dynamics (May, 1987; Hassell et alii, 1991), ecologyand epidemiology (May, 1980; Schaffer, 1985), and the solarsystem (Wisdom, 1987; Kern, 1992; Sussman and Wisdom,1992).

Models of eleetrical systems have been found to exhibitchaotic dynamics. A few examples inelude DC-DC 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).

dy . f )- = y =. (y,tdt

(1)

There seems to be some evidence of low dimensional chaosin time series recorded from electroencephalogram (Babloy­antz et alii, 1985; Babloyantz, 1986; Layne et alii, 1986)although such results are so far inconelusive. Other areaswhere 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 beendiagnosed inelude the models of a rotor blade lag (Flow­ers and Tongue, 1992), force impacting systems (Foaleand Bishop, 1992), belt conveyors (Harrison, 1992), neu­ral systems (Harth, 1983), biological networks (Lewis anelGlass, 1991), spacecraft attitude control systems (Piper andKwatny, 1991) and friction force (Wojewoda et alii, 1992),to mention a few.

An advantage of focusing on chaotic systems is that chaos isubiquitous in nature, science and engineering. Thus simpIesystems which exhibit chaos commend themselves as valu­able paradigms and benchmarks for developing and testingnew concepts and algorithms which in principIe would ap­ply to a much wider elass of problems. Therefore most ofthe tools and concepts reviewed in this papel' are also veryrelevant to systems which display regular dynamics.

In the companion papel' the tools and systems describedhere will be used in the identification, analysis and control

Because the time is explicit in equation (1), f is said to benon-autonomou.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 with period T iff(y, t) = f(y, t + T), 't/y, t. An 11. th-order non-autonomoussystem with period T can be converted into an (11. + 1) th­order autonomous system by adding an extra state (J = 2/Tin which case the state space will be transformed from theEuelidean space IR,,+l to the cylind1'ical .space IR" x Sl,where Sl = IR/T is the cirele of length T = 27r/w. It isnoted that alI the non-autonomous systems considered inthis work will be time periodic in most situations.

A Fi:red point of for equilib1'úlm, y, is elefined as f(y) =Ofor continuous-time 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. Evaluatingthe Jacobian at a particular point on a trajectory of thesystem, that is Df(yd gives a local approximation of thevedar 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 nozero ar purely imaginary eigenvalues, then the eigenvaluesof this matrix characterize the stability of the fixed pointy.

An 11. th-order discrete-time system can be described by adifference equation of the form

30 SBA Controle & Automação IVo!.7 no. 1/Jan., Fev., Mar. e Abril 1996

y(k + 1) = f(y(k), t) . (2)

2.3 Spectral methods

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 continuous-time systemsanel 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 initialconelitions.

If the system is elescribeel by an ordinary elifferential equa­tion, simulation cannot be perfol'lneel as easily since anequation like (1) shoulel be integrateel. Fortunately, thereare a number of well know algorithms available for per­forming this task such as Euler, trapezoielal, Runge-Kutta,Aelams-Bashforth, Aelams-Moulton anel Gear's algorithm(Parker and CIma, 1989). The fourth-oreler Runge-Kuttais 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 integrationinterva.1. In the case of linear systems 01' nonlinear systemswith relatively slow elynamics the choice of the integrationinterval 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 thesystem using, for instance, a fourth-oreler Runge-Kutta al­gorithm, whilst seconel-oreler Runge-Kutta algorithms mayineluce spurious elynamics even for fairly short integrationintervals (Grantham anel Athalye, 1990). It has also beenreported that in some cases the location of the bifurcationpoints depenei on the integration interval if it exceeels acriticaI value (Aguirre anel Billings, 1994a).

Irrespective of the type of the elynamical equations or thealgorithm useel to solve such equations, an important ques­tion which shoulel be answereel is whether the simulateelresults are representative of the 'real solution'. This is anontrivial matter, anel to aelelress it woulel involve a de­taileel look into the shadowing lemma (Guckenheimer anelHolmes, 1983). For the purposes of this tutorial, it sufficesto mention that there is abunelant evielence that computeI'simulations are generally reliable as numerical tooIs for theanalysis 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 theextreme sensitivity to initial conelitions exhibiteel by somesystems. This characteristic is one of the most peculiar fea­tures of a chaotic system anel wiII be briefly illustrateel insection 2.9. Extreme sensitivity to initial conelitions eloesnot invalidate numerical computations but certainly callsfor caution in analyzing the results.

One of the first tools useel in diagnosing chaos was the powerspectrum (Mees anel Sparrm'", 1981). The a.ppearance of abroael spectrum of frequencies of highly structureel humpsnear the low-oreler resonances is usually creeliteel to chaosin low-oreler systems (Blacher anel Perelang, 1981). How­ever, broael-banel noise anel the existence ofphase coherencecan make it elifficult to eliscriminate experimentally betweenchaotic 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 thecontext 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 bispectrumanel trispectrum (Pezeshki et alú, 1990; Subba Rao, 1992;Chanelran et alii, 1993; EIgar anel Chanelran, 1993; EIgaranel Kenneely, 1993). See (Nikias anel Menelel, 1993; Nikiasanel Petropulu, 1993) for an introeluction on higher-orelerspectral analysis. Such techniques have been useel to ele­tect anel, to a certain extent, to quantify the energy transferamong frequency modes in chaotic systems.

2.4 Embedded trajectories

One technique useel in the analysis of nonlinear elynamicalsystems is to plot a steaely-state trajectory of a system inthe phase-space. Thus if y(t) is a trajectory of a given sys­tem this can be achieved by plotting iJ(t) against y(t). Forlow-oreler systems this proceelure can be useel to elistinguishbetween elifferent elynamical regimes.

In many praetical situations, however, only one variable ismeasureel. In these cases an alternative proceelure is to ploty(t-Tp ) against y(t) where Tj) is a time lago These variablescan be useel in the reconstruction of attractors (Packarel etalii, 1980; Takens, 1980) anel such variables also define theso-calleel pseuelo-phase plane. This is motivateel by the factthat y(t - Tj)) is, in a way, reIateei to iJ(t) anel consequentlythe embeeleleel trajectories representeel in the pseuelo-phaseplane shoulel have properties similar to those of the originalattractor representeel in the phase plane (Moon, 1987).

A further aelvantage of this technique is that it enablesthe comparison of trajectories computeel from continuoussystems where iJ(t) is usually available, anel from eliscretemoelels where iJ(t) is often not available anel would have tobe estimateel.

The choice of Tp for graphical representation purposes isnot criticaI anel plotting a trajectory onto the pseuelo-phaseplane for varying values of Tp may give some insight re­gareling the information flow on the attractor (Fraser anelSwinney, 1986).

Phase portraits anel plots of trajectory embeelelings can beuseel not only as a means of elistinguishing elifferent elynam-

SBA Controle & Automação jVol.? no. 1jJan., Fev., Mar. e Abril 1996 31

ical regimes, but also to demonstrate qualitative relation­ships between original and reconstructed attractors.

2.5 Dynamical attractors

domain methods (Moiola anel Chen, 1993). For reasons ofsimplicity, the brute force approach is described in whatfollows. This approach is simpIe and robust but in generalit is computationally intensive.

Thus a point l' of a bifurcation diagram of a nonautonomoussystems driven by A cos(w i) wit-h A as the bifurcation pa­rameter is elefined as

If a deterministic and stable system is simulateel for a sufli­ciently long time it reaches .steady-.state. In state space thiscorresponds to the trajectories of the system falling on aparticular 'object' which is called the attrac1.o7'. Asymptot­ically stable linear systems excited by constant inputs havepoint attractors which have dimension zero anel corresponelto a constant time series.

l' = { (y, A) E IR x [ I y = y( t.;), A = Ao;i, =1.0 +[\55 X 2íT/w} ,

(3)

Clearly, the input frequency w can also be used as a bifur­cation parameter. For autonomous systems a bifurcationeliagram can be obtained in an analogous way by choosillg

A bifurcation eliagram will therefore reveal at which valuesof the parameter A E [ the solution of the system bifurcatesanel how it bifurcates. ''''hen studying chaos such diagramsare also useful in detecting parameter ranges for which thesystem behavior is chaotic.

where [ is the interval [ =[Ai Ar] C IR, °:S ia :S 2íT /w anelI\55 is a constant. This means that the point l' is chosen bysimulating the system for a sufliciently long time [\55 X 27r /wwith A = Ao to ensure that transients have died out beforeplotting y( [\55 x 27r /w) against Ao. In practice for each valueof the parameter A, nb points are taken at the instants

(.5 )

·i = 0,1, ... ,nb-1 . (4)

i=0,1, ... ,nb- 1 .i, = ia + (I\ss + i),

2.6 Bifurcation diagrams

Nonlinear systems, on the other hanel, usually elisplay awealth ofpossible attractors. To which attractor the systemwill finally settle elepenels on the system itself anel also onthe initi aI condi tions.

An aelvantage of consielering attractors in state space asalternative representations of time series is that a numberof geometrical anel topological results can be used. For thepurposes of this tutorial, it will suflice to point out thatthe .shape anel dúnension of the attractors in state spaceare elirectly linked to the complexity of the dynamics ofthe respective time series. Thus simpIe low dimensionalattractors 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 recentlyattracted a great deal of attention are the so-called strangeor chaoiic attractor.s which are fractal objects. The deter­mination of the dimension of such attractors will be brieflyaddressed in section 2.11.

Another useful tool for assessing the characteristics of thesteaely-state solutions of a system over a range of parame­ter values is the bifurcation diagram which reveals how thesystem bifurcates as a certain parameter, called the bifur­cation parameter, is varied. Roughly, a system is saiel toundergo a bifurcation when there is a qualitative change inthe trajectory of the system as the bifurcation parameteris varied. At the bifurcation point, the .J acobian of the sys­tem has at least one eigenvalue with the real part equal tozero for continuous-time systems or on the unit circle foreliscrete-time systems.

As an example of a bifurcation diagram consider figure 1.This diagram and the respective system will be describedin some detain in section 4.5. Throughout this tutorial,bifurcation parameters are denoted by A.. Thus, figure 1shows some of the different types of attractors displayed bythe system as A is varied. In particular, for A. = 4.5, 9 and11 the system elisplays period-one, period-three and chaoticdynamics, respectively. For clarity the respective attractorsrepresented in the cylindrical state-space (see section 2.1)are also shown. .

There are a number of known bifurcations. The most com­mon co-dimension one bifurcations are the pi1.chfork, thesaddle-node, the 1.ranscri1.ical, the Hopf bifurcation, anel thefiip or period doubling, which only occur in discrete mapsor periodically driven systems. For an introduction to bi­furcation and a description of the aforementioneel types see(Guckenheimer and Holmes, 1983; Mees, 1983; Thompsonand Stewart, 1986) .

Approaches to calculate bifurcation diagrams include thebruie force, pa1.h following (Parker and CIma, 1989), thecell-io-cell rnapping technique (Hsu, 1987) anel frequency

2.7 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 state-space. In order to gain further insight into thEgeometry of attractors one may use the so-called PoincarEmaps. Such a map is a cross sectioll of the attractor and carbe obtaineel by defining a plane which should be transversato the flow in state space as showll in figure 2.

32 SBA Controle & Automação IVo!.7 no. 1/Jan., Fev., Mar. e Abril 1996

ii + 0.1 iJ + y3 = A cos(t)

More precisely, consider a periodic orbit A( of some fiow <Ptin IRn arising from a nonlinear vector field. Let ~ C IRn bea hypersurface of dimension n-1 which is tl'ansverse to thefiow <Pt. Thus the first return 01' Poincaré map P = ~~ ~

is defined for a point q E ~ by

i·····;··..·l~·.lQ>

~"'.. :~:'.".'.' A=9 ld·.....""... I····>,,:

. v."' ~;>

where TIO is the time taken for the orbit <pdq) basec1 at q tofirst return to ~.

(6)P(q)

This map is very useful in the analysis of nonlinear systemssince it takes place in a space whieh is of lower dimensionthan the actuaI system. It is therefore easy to see thata fixed point of P cOl'l'esponds to a perioc1ic orbit of pe­riod 27r /w for the fiow. Similarly, a subharmonic of perioclI{ 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 first-return map ofa ehaotic solution is formed by a well-defined and finely­structured set of points for noise-free dissipative systems.Such maps ca be used in the validation of identified modeIsand reconstructed attractors (Aguil'l'e and Billings, 1994b ).

A=llA=4.5

3.r---·3[ . -.

2·1-'C:~·V~.'./""/~2· ;~ . ./.. ,

1.5~ --~/~',

a.5!

Figure 1 - Bifurcation diagram for the Duffing-Ueda oscil­lator, see section 4.5. A, the amplitude of the input, is thebifurcation parameter.

From the above c1efinition it is clear that if a system hasn > 3, the Poincaré map would require more than two di­mensions for a graphical presentation. In order to restrictthe plots to two-dimensionaI figures, y(t - ~») is plottedagainst y(t) at a constant period. For periodicaIly drivensystems the input period is a natural choice and the result­ing plot is caUed a Poincaré section.

y(t)

y(t)

This procedure amounts to defining the Poincaré plane ~P

in the pseudo-phase-space and then sampling the orbit rep­resented in such a space. The choice of~) is not criticaIbut it should not be chosen to be too smaU nor too largecompared to the cOl'l'elation time of the trajectory. Other­wise the geometry and fine structure of the attractor wouldnot be weU represented. The qualitative information con­veyed by both Poincaré maps and sections are equivalent asdemonstrated by the theory of embeddings (Takens, 1980;Sauer et alii, 1991).

Although the Poincaré sections are usuaUy obtained bymeans of numerical simulation, it is possible, although notalways feasible, to determine Poincaré maps analytically(Guckenheimer and Holmes, 1983; Brown and CIma, 1993).

2.8 Routes to chaos

y(t)

Figure 2 - A Poincaré section is obtained by defining aplane in state space which is transversal to the fiow. Theimage formed on such a plane is the Poincaré section ofthe attractor and wiU display fractal structure if such anattractor is chaotic.

In the study of chaotic systems it is somewhat instructiveto consider the c1ifferent routes to chaos in order to gainfurther insight about the dynamics of the system underinvestigation. As pointed out "the benefit in identifyinga particular prechaos pattern of motion with one of thesenow classic models is that a body of mathematicaI workon each exists which may offer better understanding of thechaotic 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 1996 33

yond the immediate scope of this work, some of the mostweU-known patterns wiU be listed with some references forfurther reading. Some of the routes to chaos reported inthe litel'ature indude period dO'/lbling ca8cade (Feigenbaum,1983; Wiesenfeld, 1989), q1ta81-periodic rO'/lte to cha08(Moon, 1987), iterm.ittency (ManneviUe and Pomeau, 1980;Kadanoff, 198:3), freq1tency locking (Swinney, 1983). Forother routes to chaos see (Robinson, 1982) and refereneestherein.

2.9 Sensitivity to initial conditions

(o) (b)

~J"'-----' " I

I/I

, ! Jo 0.1 C. U o.t • .'''<i ,'s

(c) (d)

:~ ~W~~I~~W{oll

nt G.4 OI 0.1 1 ..(c) (f)

Probably the most fundamental property of chaotic sys­tems is the sensitive dependenee on initial conditions. Thisfeature arises due to the local divergence of trajectories in

. state space in at least one 'direction'. This wiU be alsoaddressed in the next section.

In order to iUustrate sensitivity to initial conditions anelone of its main consequences, it will be helpful to considerthe map

In order to iterate equation (7) on a digital computeI', aninitial conelition y(O) is required. Using this value, the righthanel side of equation (7) can be evaluated for any vaIueof A. This proeluces y( 1) which should be 'fedback' aneluseel as the initiaI conelition in the following iteration. Thisprocedure can be then repeateel as many times as neeessaryto generate a time series y(O), y(I), y(2), ....

A graphical way of seeing this is iUustrateel in figure 3. Itshould be noteel that the right hand siele of equation (7) is aparabola, as shown in figure 3a. Thus to evaluate equation(7) is equivalent to finei the value on the parabola whicheorresponds to the initial conelition. This is representeelin figure 3a by the first vertical line. The feeding back ofthe new value is then representeel by projecting the valuefound on the parabola on the bisector. This completes oneiteration.

y(k) = A [1 - y(k - 1)] y(Ã~ - 1) . (7)

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 usednamely y(O) = 0.22. In figures (e) and (f) an interval oJinitial eonditions has been iterated for the same values oJA as above. The intervaIs useel were y(O) E [0.22 0.24'and y(O) E [0.220 0.221], 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 conditiomanel proeeeeling as before yieldeel the results shown in figurE3f for whieh A = 3.9. Clearly, the interval of initial coneli­tions was wielened at each iteration. Sueh an intervaI eanbe interpreted as an errar in the original initial conelition.y(O) = 0.22. In practice errors in initial conditions wiU beaIways present due to a number offactors such as noise, elig­italization effects, rounel-off errors, finite worellength preci­sion, etc. It is this effeet of amplifying errors in initialeonelitions which is known as the sensitive depenelenee oninitial conelitions anel an immeeliate consequence of this fea­ture is the impossibility of making long-tenn preelietion forchaotic systems. The next section describes indices whichquantify the sensitivity to initial conditions.

Choosing the initia! condition y(O) = 0.22 anel A = 2.6, fig­ure 3a shows the iterative procedure and reveals that aftera few iterations the equation settles to a point attraetor.The respective time series is shown in figure 3b. The sameprocedure was foUowed for the same initial condition andA = 3.9. The results are shown in figures 3e-d. Clearly, theequation does not settle onto any fixed point anel not evenonto 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 aninterval of initial eonditions is iterated? This is shown infigures 3e-f. For A = 2.6, the map wiU eventuaUy settleto the same point attractor as before. This is a typica!result for regular stable systems and it iUustrates how aUthe trajectories based on the initiaI conditions taken fromthe original interval converge to the same attractor.

2.10 lyapunovexponents

Lyapunov exponents measure the average divergenee 01nearby trajectories aIong certain 'elirections' in state space.A chaotic attracting set has at least one positive Lyapunovexponent anel no Lyapunov exponent of a non-chaotic 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 theeoncept of local Lyapunov exponents has been investigateel(Abarbanel, 1992). The local exponents describe orbit in­stabilities a fixed numbel' of steps aheael rather than aninfinite number. The (global) Lyapunov exponents of anattraeting 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

1),i = lim ---;c loge ji(N),

N--HXJ Ivi=1,2, ... ,n, (8)

of an attracting set of length N can be elefineel as (see alsoequation (8))

where loge = In and the {ji(N)}i'=l are the ahsolute valuesof the eigenvalues of

N

),1 == ~ lim "" log 11 Dk+1 IIN N-;=~ e II Dk II ' (10)

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 modeIsmay have a dimension which is larger than that of theoriginal systems and therefore such moelels have more Lya­punov exponents. These 'extra' exponents are calleel S]J'lI­

rious Lyap'llnov exponents. The estimation of Lyapunovexponents is known to be a nontrivial task. The simplestalgorithms (Wolf et alii, 1985; Moon, 1987) can only re­liably estimate the largest Lyapunov exponent (Vastanoand Kostelich, 1986). Estimating the entire spectrum isa typically ill-conelitioned problem and requires more so­phisticated algorithms (Parker and CIma, 1989). Furtherproblems arise when it comes to decieling which of the es­til11ated exponents are tnle anel which are sp'llrious (Stoopanel Parisi, 1991; Parlitz, 1992; Abarbanel, 1992). The es­timation of Lyapunov exponents is currently an active fieldof research as can be verifieel from the following references(Sano and Sawaela, 198.5; Eckl11ann et a/ii, 1986; Bryant etalú, 1990; Brown et alii, 1991; Parlitz, 1992; Kadtke et alii,199:3; Nicolis and Nicolis, 1993; Chialina et alii, 1994). Forapplication of Lyapunov exponents in the quantification ofreal elata see (Branelstiíter et alii, 198:3; Wolf and Bessoir,1991; Vastano anel Kostelich, 1986).

where Df(Xi) is the J acobian matrix of f(·) evaluated atXi, and also of simulating the system

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

(11)

(12)

II Dl IIII Do II '

D= Df(x;) D ,

II Xl - (;C1 + DIl IIII a:o - (xo + Do) II

where .r1 is another point on the trajectory x(k), namely.r(êlL), ;C1+D1 is a point obtaineel by following the evolutionof the ranelomly chosen initial conelition ;co + Do over theinterval êlL where êlL will be referred to as the Lyapunovinterval.

From the last equation it is clear that one onlv needs tofollow the evolution of perturbations Di along th~ referencetrajectory x( k). It is well known that the Jacobian ma­trix D f( Xi) elescribes the elynamics of the system for smallperturbations in the neighborhood of Xi. Thlls the com­putation of the largest Lyapunov exponent, ),1, consists insolving the variational equations

(9)[Df(YN )][Df(YN-1 )] ... [Df(Y1)] ,

if the trajectory x( k) ={Xi}~O 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. Thereforeestimating ),1 consists in successively preelicting the systemsgoverned by Df O anel f O êlL seconds into the futureand assessing the expansion of the perturbations Di.

In view of such elifliculties and the fact that the largestLyapunov exponent, ),1, is in many cases the only positiveexponent2 anel that this gives an indication of how far intothe 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 Lyapunovexponent has been useel in this way and to compare sev­eral identified models (Abarbanel et alii, 1989; Abarbanelet alli, 1990; Principe et alii, 1992).

x = f(x) (13)

The algorithm suggested in (Moon, 1987) for estimating ),1

is described below. A similar algorithm which simultane­ously estimates the correlation dimension to be elefineel insection 2.11 has been recently investigateel in (Rosensteinet alii, 1993).

Consider a point Xo on the trajectory x(k) (for the momentit is assumed that such a trajectory is available a priori),say Xo =x(O), anel a nearby point Xo+Do. For simplicity it isassumeel that x(k) E IR, but in general higher-dimensionalsystems will be the case. The largest Lyapunov exponent

Some of the ideas described above are illustrateel in figures4a-b. The former figure is the bifurcation diagram of thelogistic equation (7). Figure 4b shows the largest Lyapunovexponent of such an equation for a range of values of A.The largest Lyapunov exponent was ca.Iculated as elescribeelabove. Note that ),1 = O at bifurcation points and that),1 > Ofor chaotic regimes as predicted by the theory. Thesefigures a.Iso reveal the narrow windows of regular dynamicswhich are surrounded by chaos.

2.11 Correlation dimension

2In this case /\1 ;::: h, where h is the Kolmogorov-Sinai or ITletricentf,0PY. Note that for dissipative systeITls (chaotic and non-chaotic)Li=l '\; < O (Eclunann and Ruelle, 1985; Wolf, 1986).

Another quantitative measure of an attracting set is thefractal elimension. In theory, the fractal dimension of a

SBA Controle & Automação jVol.? no. IjJan., Fev., Mar. e Abril 1996 35

0.8

0.6

0.4

0.2

O'-----'-----~----~-----L------'

2.9 3.2 3.5 3.8A

OI---::::=="~----------;o>';,---rrfH-t-'--+-----'-+f--jj

chaotic (non-chaotic) attracting set is non-integer (integer).An exception to this rule are fai fractais which have in­teger fractal dimension which is consequently inadequateto 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 diagnosechaos but also to provide some further dynamical informa­tion in most cases (Grassberger ei alú. 1991). A deepertreatment can be found in (Russell ei ahi, 1980; Farmer eialú. 198:3; Grassberger anti Procaccia, 198:3a; Atten ei alii.1984; Caputo ei alii .. 1986) for raw data and in (Badii andPoliti, 1986; Badii ei alii, 1988; Mitschke, 1990; Brown eialá, 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 thefractal dimension of an attracting set is D + li, D E ~+,

where O< li < 1. then the smallest number of first-orderdifferential equations required to describe the data is D+ 1.

There are several types of fractal dimension such as thepoiniwise dimension, correlat-ion dimension, informaiiondimension, Hausdor/J dimension, Lyapnnov dimension, fora comparison of some of these dimensions see (Farmer,1982; Hentschel and Procaccia, 1983; Moon, 1987). Formany strange attractors, however, such measures giveroughly the same value (Moon, 1987; Parker and Cima,1989). The correlation dimension:3 (Grassberger and Pro­caccia, 1983b), however, is clearly the most widely usedmeasure of fractal dimension employed in the literature.

A time series {Y;}~l can be embedded in the phase spacewhere it is represented as a sequence of de-dimensionalpoints Yj = [Yj Yj-1 ... Yj-d e +1]' Suppose the distancebetween two such points is4 S'ij =1 Yi - Yj I then a cor­relation function is defined as (Grassberger and Procaccia,1983b)

-1

-2C(e) = lim ~ (number of pairs (i, j) with S'ij < é) .

N~co N(14)

-3

-4

-5'---------'--------'--------~-----'---'

2.9 3.2 3.5 3.8A

The correlation dimension is then defined as

D I· loge C(é)

c = 1n1 .e-co loge é

(15)

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 chaoticregll11es.

For many attractors De will be (roughly) constant for val­ues of é within a certain range. In theory, the choice ofde does not influence the final value of De if de is greaterthan a certain value. In particular, it has been shown thatprovided there are sufficient noise-free data, de = Ceil(De ),

where Ceil(·) is the smallest integer greater than or equalto De (Ding ei alii, 1993) and that this result remains trnein 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) andthus 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

-20

-30

-40

Ui'-50

C-al

-60Õ'aí..Q -70

-80

-90

-10070 80 100 110

log(eps)

Figure 5 - Logarithm of the correlation function C(t:) plot­ted against log( c) for embedding dimensions de = 2 tode = 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 datawere produced by a low-dimensional system, such estimateswould eventually converge. Of course, these results dependlargely on both the amount and quality of the data avail­able. For a brief account of data requirements, see seetion3.1 below.

In order to illustrate the estima.tion of De a time series withN = 15000 data points was obtained by simulating Chua'scircuit (see section 4.3) operating on the double scroll at­tractor. The correlation function C( t:) was then calculatedfor 2 ::; de ::; 10 and plotted in figure 5. For small em­bedding dimensions (de = 2) the correlation dimension isDe ~ 1.8 but as de is increased the scaling region convergesto the correct value De ~ 2.0 for de 2': 5.

One of the properties of some fractaIs is self-similarity. Thisis illustrated in figure 6 which shows the well known Hénonattractor (see aIso section 4.2) and an amplification of asmall section of one of its legs. It should be observed tha.twhat appears to be a single 'line' in the attractor turns outto be two lines (see zoom in figure 6). However, if each ofthese lines were zoomed again it would become apparentthat they are composed of other two lines each and thiscontinues ad infinitum. This particular fractal structure issometimes 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 timeseries have been produced by a low-order system (Grass-

-o.!lr-~~~~~~~"";'>...,

Figure 6 - Fractal structure of the Hénon attractor.

berger, 1986a; Lorenz, 1991). Because this is an impor­tant problem, the estimation of correlation dimension hasattracted much attention in the last years. Many pa­pers have focused on determining the causes of bad esti­mates (Theiler, 1986), estimating error-bounds (Holzfussand Mayer-Kress, 1986; Judd and Mees, 1991) and sug­gesting improvements on the original algorithm describedin (Grassberger and Procaccia, 1983b).

2.12 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 entropiesand 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 havereceived little attention as invariant. measures of strangeattractors. Apparent.ly, the most useful such measure is theprobability densiiy funetion (Packard ei alii, 1980; Moon,1987; Vallée et alii, 1984; Kapitaniak, 1988)

The estimation of unstable limit cycles has also been putforward as a way of charaeterizing strange attractors. Themotivation behind this approach is that because a strangeattractor can be viewed as a bundle of infinite unstablelimit cycles, the number of the periodic orbits, the respec­tive distribution and properties should be representativeof the attractor dynamics. Indeed, from such informationother invariants such as entropies and dimensions can beestimat.ed (Auerbach ei alli, 1987). For more informationon 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 reducedto estimating invariants which would suggest that the dataare chaotic. For instance, positive Lyapunov exponents,non-integer dimensions and fractal structures in Poincaré

SBA Controle & Automação jVol.7 no. 1jJan., Fev., Mar. e Abril 1996 37

sections would suggest the presence of chaos. The mainquestion is how to confidently estimate such properties fromthe data, especia.lly when the available records are relativelyshort and possibly noisy. The tedmiques that have beensuggesteel in the literature can be elivided in two majorgroups.

N on-paraluetric luethods. These incluele the use oftools which take the data anel estimate dynamical invari­ants which, in turn, will give an indication of the presenceof chaos. Such tools incluele power spectra, the largest Lya­punov exponent. the correlation dimension, reconstructedtrajectories, Poincaré sections, reIative rotation rates etc.Detailed description and a.pplication of these techniquescan be founel in the literature (Moon, 1987; Tufillaro etalii, 1990; Dellton anel Diamond, 1991). For a recent com­ment of the practical difficulties in using Lyapunov expo­nents anel dimensions for diagnosing chaos see (Mitschkeanel Dammig, 1993).

Two practical difficulties common to most of these ap­proaches are the number of data POilltS available and thenoise 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 thefractal structure of the attractor. However, in order to beable to elistinguish between a fractal object and a fuzzydoud of points a certain amount of data is necessary. Moon(1987) has suggested that a Poincaré section shoulel consistof at least 4000 points before dedaring a system chaotic.For non-autonomous systems this means 4 x 103 forcingperiods which could amount to 4 x 105 data points.

Predietion-based 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 estimateelfrom, say, the first half of the elata records and used topredict over the last half. CImos can, in principIe, be di­agnosed based on how the prediction errors behave as theprediction time is increased (Sugihara and May, 1990), 01'

based on how the prediction errors related to the true datacompare to the prediction errors obtained from 'faked' datawhich are random but have the same length anel spectralmagnitude 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 dataare 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 anelstatistics such as the ones presented in (Sugihara and May,1990; Kennel and Isabelle, 1992) can be used to diagnosechaos.

3.1 Data requirements

The length and quality of the data records are crucial inthe problem of characterization of strange attractors. Atpresent, there seems to be no general rule which eleterminesthe amount of data required to learn the dynamics, to es­timate Lyapunov exponents and the correlation elimensionof attractors. However it is known that "in general the de­tailed diagnosis of chaotic elynamical systems requires longtime series of high quality" (Ruelle. 1987).

Typical values of elata length for learning the dynamics are2 x 104 (Farmer and Sidorowich, 1987; Abarbanel et alii,1990) for systems of elimension 2 to 3, 1.2 X 104 - 4 X 104

(Caselagli. 1991).

It has been argued that to estimate the Lyapunov expo­nellts 103 -104 forcing periods shoulel be used (Denton andDiamonel. 1991). Other estimates are N > lOD (quoteel in(Rosenstein et alii, 1993) anel N > :30D where D is the di­mension ofthe system (W'olf et alii, 1985) but in some casesat least 2 x 30D was required (Abarbanel et alii, 1990).Typical examples in the literature use 4 x 104 - 6.4 X 104

(Eckmann et alii, 1986) 1.6 x 104 (Wolf anel Bessoir, 1991)anel 2 x 104 elata points (Ellner et alii, 1991).

Fairly long time series are also requireel for estimating thecorrelation dimensiono In fact, it has been pointed outthat dimension calculations generally require larger datarecorels (Wolf and Bessoir, 1991). For a strange attractor,if insufficient data is useel the results would indicate thedimension of certain parts of the attractor rather than theelimension of the entire attractor (Denton anel Diamond,1991). However, results have been reported which suggestthat consistent estimates of the correlation dimension canbe obtained from data sequences with less than 1000 points(Abraham et alii, 1986). On the other hanel, there seemsto be evielence that "spuriously small dimension estimatescan be obtained from using too few, too finely sampleel aneltoo highly smootheel data" (Grassberger, 1986a). More­over, the use of short and noisy data sets may cause thecorrect scaling regions to become increasingly shorter andmay cause the estimate of the correlation elimension to con­verge to the correct result for relatively large values of theembedding dimension (Ding et alii, 1993). Thus typicalexamples use 1.5 x 104 - 2.5 X 104 (Grassberger and Procac­cia, 1983b) and 0.8 x 104

- 30 X 104 data points (Atten etalii, 1984). Thus there seems to be no agreeel upon ruleto determine the amount of data required to estimate di­mensions with confidence but it appears that at least a fewthousanel points for low dimensional attractors are neeeleel(Theiler, 1986; Havstad and Ehlers, 1989; Ruelle, 1990; Es­sex and Nerenberg, 1991). In particular, N > lOD d 2 hasbeen quoteel in (Ding et alii, 1993).

It should be realized that the difficulties in obtaining longtime series goes beyonel problems such as storage anel com­putation time. Indeeel, it has been pointeel out that forsome real systems, stationarity cannot always be guaran­teeel even over relatively short periods of time. Examples ofthis 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

Figure 7 - (a) The CIma circuit anel (b) voltage-currentcharacteristic 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 theanalysis anel signal processing of nonlinear elynamical sys­tems.

1985; Sugihara anel May, 1990). A test for stationarity hasbeen recently suggesteel in (Isliker anel Kurths, 1993).

4 SOME NONLlNEAR MODELS

The objective of this section is to proviele the reaeler willa. collection of well elocumenteel simpIe nonlinear 1l10elelswhich elisplay a wiele variety of elynamical regimes whichincluele self-oscillations, limit cycles, perioel-eloubling 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 calleelelifference equations, of the form

tL

L C,

(b)

m

4.3 The Chlla circlIit

4.2 The Hénon map

Due to its simplicity anel also because some elynamical in­variants can be deriveel analytically for this moelel, the 10­gistic map is frequently useel as a bench test in the stuelyof elynamical systems. For a review on first-oreler maps see(May, 1980; May, 1987) .

This equation inelicates that the value of the variable y attime k + T is a function of the same variable at time k.The basic equation (16) applies to a number of situationswhich incluele switching power circuits (Tse, 1994), popula­tion elynamics, genetics, elemography, economics anel socialsciences, 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 elynamicalregimes as the bifurcation parameter, A., is varieel in theinterval 2.8 ~ A. ~ 3.9. The bifurcation eliagram of this map,which is shown in figure 4a, is a classical example of theperiod-doubling rmt.te to chaos.

(18)x > 1I x I~ 1x <-1

{

~ = n(y - h(x))y=:r-y+zi = -f3y

{

m 1x+(mO -m1 )

h(x) = moxm1a.: - (mo - In1)

Chua's circuit is certainly one of the most well stuelieel non­linear circuits anel a great number of papers ensure that theelynamics of this circuit are also well elocumenteel, see forinstance (CIma anel Hasler, 199:3; Maelan, 1993; Matsumotoet afii, 1993).

where mo = -1/7 anel m1 = 2/7. Varying the parametersn anel f3 the circuit elisplays several regular anel chaoticregimes. This system is a particular case of the more gen­eral unfolded Chu,a's circu,it (CIma, 1993). The well knownelouble scroll attractor, for instance, is obtaineel for n =9anel f3 = 100/7, see figure. 8. For this attractor À1 = 0.23(Matsumoto et alii, 1985; Chialina et afii, 1994). The esti­mateel value of the correlation elimension for this attractorwas De =1.99 ± 0.023.

The normalizeel equations of Chua's circuit can be writtenas (CIma et alii, 1986; CIma, 1992; Maelan, 1993; CIma anelHasler, 1993)

(16)y(Á~ + T) = F[y(k)] .

The map (Hénon, 1976)

{x(k) = 1- ax(k _1)2 + y(k: -1),y(k) = bx(k -1)

(17)

The only nonlinear element in Chua's circuit is a two­terminal piecewise-linear resistor, calleel Chua's elioele,which has also' been implemented as an integrated cir­cuit (Cruz anel CIma, 1992), however the entire circuit canbe implementeel with 'off-the-shelf' components (Kennedy,1992).

was proposeel by the French astronomer Michel Hénon as anapproximation of a Poincaré mapping of the Lorenz system(see section 4.6). In the literature the parameter valuesusua.lly useel are a =1.4 anel b=0.3. Thus these values leaelto the so-calleel Hénon attractor shown in figure 6. Fora more eletaileel elescription of the elynamical properties ofthis map see (Moon, 1987; Peitgen et alii, 1992) .

4.4 The DlIffing-Holmes oscillator

One of the classical bench tests in mechanics is the Duffingoscillator (Duffing, 1918). Two elifferent versions of thisoscillator have been investigated in connection with chaos

SBA Controle & Automação jVo!.? no. 1jJan., Fev., Mar. e Abril 1996 39

4

2

20N

-2

-40.5

oy(t)

-0.5 -2 ox(t)

2

1.5

0.5

o

-0.5

-1

0.350.310.26-1.5'----- ~ __"____ _.J

0.22A

Figure 8 - The double sero11 Chua's attractor.

by Philip Holmes and Yoshishuke Ueda. The equationswhieh deseribe the dynamies of sueh osei11ators are briefiydeseribed in this and the fo11owing seetion, respectively. a

The we11 known Dufling-Holmes equation is eommonly usedto model mechanieal osei11ations arising in two-we11 poten­tial problems (Moon, 1987) beeause this system is eharae­teristie of many of the structural nonlinearities eneounteredin praetiee (Hunter, 1992). The equation whieh models thissystem is (Holmes, 1979; Moon and Holmes, 1979)

:ij + ó iJ - f3 y + y3 = A cos (w t) . (19)

The bifureation diagram for this system for w = 1 rad/s and0.22: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 strangeattraetor whieh is shown in figure 9b. The largest Lyapunovexponent of this attraetor is )'1 = 0.20 and the eorrelationelimension equals De = 2.40 ± 0.019.

1.5.------.-----.----.--------.--..---~--,

0.5

-1

-1.5

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).

-1.5 -1 -0.5 Oy(k)

0.5 1.5

4.5 The DufFing-Ueda oscillator

The Dufling-Ueda equation (Ueda, 1985)b

y+kiJ+y3=U(t) (20)

Figure 9 - (a) bifureation eliagram, and (b) Poinearé see­tion ofthe Dufling-Hohnes 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

Figure 10 - The Duffing-Ueda oscillator.

43.53y(t)

a

2.52

2.8

2.6

2.4

1.5

4,------~-------r----,------r--____,

3.8

3.6

3.4

~3.2..!.>; 3

linear dynamics. It has also been considered as a simpleparadigm for chaotic dynamics in electrical science (Moon.1987). Consequently this system has attracted some atten­tion and has been used to investigate nonlinear elynamics indifferent 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 period-one 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 forthe input 1l.(i) = A cos(wi). A was used as the bifurca­tion parameter in the bifurcation diagrams. The bifurcationshown in figure 1 was obtained by taking k = 0.1, w = 1 rad/sand simulating equation (20) digitally using a fourth-orderRunge-Kutta algarithm with an integration interval equalto 7r/3000. Figures lIa and llb shows the Poincaré sectionof the attractors at A = 5.7 and at A = 11, respectively.The largest Lyapunov exponent of these attractors are re­spectively .\1 = 0.099 and .\1 = 0.11 and the carrelationdimensions equal De = 2.10 ± 0.050 and De = 2.19 ± 0.020.

32.5y(t)

1.5

1.8

1.6

1.4

3,--------,---------~------,

2.8

2.6

2.4

~2.2>; 2

The bifurcation diagram in figure 1 reveals a number of dy­namical regimes displayeel by this oscillator in the range ofvalues of A considered. In particular, at A::::::: 4.86 the systemundergoes a period doubling (flip) bifurcation. This hap­pens again at A::::::: 5.41 and characterizes the well known pe­riod doubling route to chaos (Feigenbaum, 1983). Anothersimilar cascade begins at A ::::::: 9.67 preceding a differentchaotic regime. Two chaotic windows can be distinguishedat approximately 5.55 :S A :S 5.82 and 9.94 :S A :S 11.64.At A ::::::: 6.61 the system undergoes a supercritical pitch­fork bifurcation and at A::::::: 9.67 it undergoes a sub criticaIpitchfork bifurcation. The bifurcation diagram begins andends with period-1 regimes and displays period-3 dynamicsfor 5.82 :S A :S 9.67. It should be noted that in the range4.5 :S A:S 5.5 there are two co-existent. attractars undergo­ing a sequence of period-doublings, t.his becomes apparentby using a different set of initial conditions and explainsthe broken lines.

4.6 The lorenz equations

The well known Lorenz equat.ions are (Lorenz, 1963) b

{.~ = rr (y - x)y = px y­z=xy-(3z.

xz (21)

Figure 11 - Poincaré sections for t.he Duffing-Ueda oscilla­t.ar, for Tp =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 1996 41

L R

50

40

30

-N20

10

O50

O

y(t) -50 -20 -10 Ox(t)

10 20

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 theoriginal van der Pol equation. On the other hanel, alike thevan eler Pol oscillator, the circuit governeel by equation (22)also exhibits self-sustained oscillations, that is, oscillationsfor 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 theyhave become a standard for studying complex dynamics.Fairly detailed descriptions of the dynamics of this systemcan be found in (Sparrow, 1982; Thompson anel Stewart,1986; Peitgen et aiii, 1992). The popularity of the Lorenzequations is not only due to the fact that it was one ofthe first systems published in connection with chaos butalso because it is a physically motivated modeI. Further,although the Lorenz equations moelel some aspects of fluiddynamics, it has been shown that the instabilities of such amodel are identical with that of the single moele laser andapplicahle to underdampeellaser spikes (Haken, 1975) anelthat the open-Ioop dynamics of smooth-air-gap brushlessDe motors can also be modeled by such equations (Hemati,1994).

The bifurcation diagram of this system is shown in figure14a. This bifurcation eliagram presents a number of co­existent attractors anel dynamical regimes interwoven in avery complicated manner. It. is worth pointing out thatfor O< A:::; 3 the oscillator presents quasi-perioelic motionswhich cannot be distinguisheel from the chaotic dynamicsbaseel upon bifurcation diagrams only.

Taking p = 0.2, 1'1=17 and w = 4 rad/s, this system settlesto the strange attractor shown in figure 14b. The largestLyapunov exponent of this attractor is À1 = 0.33 and thecorrelation elimension equals De =2.18 ± 0.028.

4.8 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 infigure 12 is)'1 =0.90 (Peitgen et aiii, 1992) and the corre­lation dimension equals De =2.01 ± 0.017. {

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 electricalcircuit in which the nonlinearity was introduced by a vac­uum valve. One of the principal characteristics of such acircuit was that it exhibiteel self-sustained oscillations aisocalled relaxation oscillations, thus lending itself as a modelfor 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 vander Pol oscillator, which has negative resistance, are (Veda

as a simplifieel version of the Lorenz system, 01' in Rossler 'swords a "modei of a model". The simplification attained byRossler can be appreciated by noticing that the attractorexhibiteel by (23), with [1'=0.2 and p=5.7 see figure 15., iscomposed of a single spiral which resembles a lVIobius bandinsteael of the two spirals which can be clearly elistinguisheelin the Lorenz attractor.

The largest Lyapunov exponent of the attractor shown infigure 15 is À1 = 0.074 , the Lyapunov dimension and thecorrelation dimension of this attractor are DL = 2.01 andDe = 1.91 ± 0.002, respectively.

42 SBA Controle 1St Automação jVol.7 no. 1jJan., Fev., Mar. e Abril 1996

30

3.------,------r------,----~---,

2

o

-1

-2

-3

-45 10 15 20

A

20

10

o20

y(t) -20 -10o

x(t)

20

a

o

-2

-3

-4'--------'------------'--------'--------'--------'

-1 O 2 3y(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.

Figure 15 - The Rossler attractor.

The field of possible applications of equations of the type of(23) ranges from astrophysics, via chemistry and biology, toeconomics (Rossler, 1976). In fact, the Rossler and Lorenzattractors are regarded as paradigms for chaos thus thechaotic dynamics of some systems are sometimes classifiedas "Rossler-like" or "Lorenz-like" chaos (Gilmore, 1993).

5 DISCUSSION ANO FURTHER REAOING

The analysis and quantification of chaotic dynamics is arelatively recent area. Nevertheless there is an immensecollection of scientific papers and books devoted to thissubject and any attempt to produce a survey on nonlin­ear dynamics and clla.os, no matter how thorough, wouldbe, in all certainty, just a rough sketch on this fascinatingsubject.

The main objective of this paper has been to review in avery pragmatic way a few concepts which are believed tobe basic. Since it would be inappropriate to produce anin-depth review, a rather generous number of referenceshas been cited for further reading. Needless to say, thereference list does not exhaust the wealth of papers andbooks currently availahle.

The following references seem to be a good starting point.The books (Gleick, 1987) and (Stewart, 1989) are a goodintroduction 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 accounton computer algorithms for nonlinear systems applicationscan be found in (Parker and CIma, 1989). See also (Abra.-

SBA Controle & Automação jVol.7 no. 1jJan., Fev., Mar. e Abril 1996 43

h3m and Shaw, 1992) for a beautifully illustrated introduc­tion to nonlinear dynamics and bifurcations. The followingpapers are also good introductions to nonlinear dynamicsand 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; Argyriset alú, 1991; Thompson anel Stewart, 1993). Gooel sur­veys on modeling anel analysis of chaotic signals can befounel in (Grassberger et a.lú, 1991; Abarbanel et alú, 199;3).See also (Hayashi, 1964; Atherton and Dorrah, 1980) for arather '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 anelchaotic systems. In particular, the program kaos (Guck­enheimer anel Kim, 1990; Guckenheimer, 1991) which runson Sun workstations and can be obtained by ftp on ma­comb.tn.comell.edu 128.84.237.12. The programs MTR­CHAOS anel MTRLYAP (Rosenstein, 199:3) can be sueelfor analysing chaotic signals and estimating correlation eli­mension anel largest Lyapunov exponents. These programsare available under request to MTRla@aol.com.

Some of the concepts anel mathematical tools eliscusseel inthis papel' will be useel in a companion papel' which will ael­elress some aspects ofthe ielentification, analysis anel controlof nonlineal' elynamics anel chaos.

Acknowledgements

FinanciaI support from CNPq (Brazil), uneler grant522538/95-9, is gratefully acknowleelgeel.

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