Discrete Dynamics in Nature and Society, Vol. 6, pp. 121-128Reprints available directly from the publisherPhotocopying permitted by license only

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Detection of the Onset of Numerical ChaoticInstabilities by Lyapunov Exponents

ALICIA SERFATY DE MARKUS

Centro de Estudios Avanzados en Optica and Centro de Astrofisica Tebrica, Facultad de Ciencias,La Heehieera, Universidad de Los Andes, Mkrida 5051, Venezuela

(Received 2 July 2000)

It is commonly found in the fixed-step numerical integration of nonlinear differentialequations that the size of the integration step is opposite related to the numericalstability of the scheme and to the speed of computation. We present a procedure thatestablishes a criterion to select the largest possible step size before the onset of chaoticnumerical instabilities, based upon the observation that computational chaos does notoccur in a smooth, continuous way, but rather abruptly, as detected by examining thelargest Lyapunov exponent as a function of the step size. For completeness,examination of the bifurcation diagrams with the step reveals the complexity imposedby the algorithmic discretization, showing the robustness of a scheme to numericalinstabilities, illustrated here for explicit and implicit Euler schemes. An example ofnumerical suppression of chaos is also provided.

Keywords: Numerical instabilities; Difference equations; Lyapunov exponents; Fixed-stepschemes; Chaos

I. INTRODUCTION

The numerical solutions of a nonlinear set ofdifferential equations are discrete approximationsof closed form solutions, which are usuallydifficult, impractical or impossible to obtain. Thereliability of such approximations will dependlargely on the integration algorithm and, accord-ingly, the first effort is devoted to the search of anaccurate scheme. The stability of the algorithmemployed may become the next important issue,followed, finally, by considerations regarding the

121

computational efficiency in terms of computa-tional speed. The unbalance of these three aspectsof a numerical scheme is mostly due in that theinterest in a system with a complex or even chaoticdynamics is more of an object of study than amatter of practical importance. Notoriously, con-siderations on speed have been traditionallyrelegated, as long as the scheme is accurate andstable. These requirements are inconsistent withthe dynamical nature of the systems being solved,and many problems may demand a fast as possiblecomputation of trajectories, because they are

122 A. SERFATY DE MARKUS

occurring in real time and/or because a simulationis being modeled by a large number of equations, asituation often found in engineering. A direct wayto explore the balance between accuracy, stabilityand speed is by means of numerical integrationwith conventional fixed-step schemes, in terms,specifically, of the size of the integration step.Typically, very "small" step-sizes usually guaran-tee the fidelity of the solution (even though toosmall steps may have the opposite effect [1]), but tothe expense of longer calculations.Widely used fixed-step schemes are the second

and fourth order Runge-Kutta, and the explicitand implicit Euler schemes. In the implementationof these schemes a continuous system of differ-ential equations is typically mapped into a discreterepresentation of difference equations, whichhopefully will share most of the properties of itscontinuous counterpart. Therefore, the diver-gences produced by this algorithmic discretizationwill be regarded as numerical instabilities, and thealgorithm is said to be numerical unstable.

This work presents a procedure to estimate thelargest possible integration step size of elementarystandard fixed-step algorithms, before the onset ofcertain numerically instabilities arising in non-linear systems of differential equations. Theinstabilities considered here are the numericallyinduction or suppression of chaos, because thenonlinear nature of the system being integratedallows the possibility of masking a chaotic or

periodic regime. Therefore, the method reported isbased upon a standard evaluation of the largestLyapunov exponent [2,3], on account of theobservation that this number changes distinctivelywith the step size.To this end we worked some numerical exam-

ples of physical importance known to have a limitcycle, and/or a chaotic attractor, depending on theparameter settings. Algorithmic discretization is

performed employing a standard forward Eulerscheme. This elementary scheme is importantbecause despite its simplicity, most sophisticatedmethods are based on simple Euler routines andused to solve very complex problems, such as

parabolic and hyperbolic partial differential equa-tions [4]. Moreover, in many other numerous casescombination with/from Euler schemes (like severalforms of predictor-corrector schemes) are com-monly used to solve systems of ordinary differ-ential equations. For comparison purposes, anonstandard backward Euler scheme is employed.In Section II are presented the examples with theEuler discretization. Section III presents resultsillustrating the procedure and a discussion on therelation of certain numerical effects to computa-tional errors. Finally, in Section IV are given someconclusion remarks.

II. NUMERICAL EXAMPLES

We employ three well-known examples of non-linear oscillators, namely, the Lewis oscillator, theVan der Pol oscillator and the two-well Dullingoscillator. The two first systems undergo a Hopfbifurcation, where a fixed point is losing stabilitygiving birth to a limit cycle; this type of bifurcationis analytical found by examination of the eigen-values when varying a system parameter. Likewise,the forced Duffing oscillator is able to display a

periodic cyclic behavior in addition to a welldefined chaotic dynamics. The equations for thesesystems are as follows:

(i) The Lewis oscillator is given by,

dty (la)

-x + e(1 Ix[)Y (lb)dt

with e being a parameter.(ii) The Van der Pol oscillator is given by,

d-- y (2a)- -x + ey- ex2y (2b)

where e is the parameter of the system.

NUMERICAL CHAOTIC INSTABILITIES 123

(iii) The driven two-well Duffing oscillator is givenby,

d2xdt2

ax + bx + C--z- F cos cotdt

(3)

In system form, the Duffing system isrepresented by,

dtdzdt

m=y

ax bx + ey + F cos (z) (4)

where x is position, y is velocity and z and Fare the phase and amplitude respectively ofthe periodic driving force. Parameters valueswere taken as a 1, b 1, c-0.5, co= andinitial conditions x0 Y0 z0 0.01.

We employ the Duffing oscillator to illustratethe algorithmic discretization. According to astandard forward Euler scheme (SFE), the firstorder differential equations (4) are mapped intodifference equations given by,

x+ x / hy (5a)

y/+l y/(1 hc)+ h(axl bx3 + Fcos (z/))(5b)

z+ z + hco (5c)

where the derivative terms were modeled followingthe basic calculus definitions xk+ xk/h y, withthe step size h--tg+l-t being the integrationtime interval; x, yk and z are the discretizedvariables. The discretization yields an iterativestructure (5) for the variables x+ 1, y+ and z+ l,

which is implemented straightforwardly.Analogously, it was constructed a backward

Euler discrete representation for the Duffingoscillator (4). Backward schemes are, in general,based on the implicit representations Xk+a= Xg+hf(Xk+ 1, tk+ 1), in contrast to the explicit

constructions as in Eqs. (5). In addition, anonstandard Euler construction is simply madeby replacing nonlinear terms by nonlocal repre-sentations, such as yZ--+y+ly, or y3__+y2(Yk+ +Yk)/2, etc. The derivative denominator,given as h, can be replaced by a more elaboratedfunction ff(h) of the integration step; in this workit was simply kept as (h)=h. For details onnonstandard techniques, see Ref. [5].An example of a nonstandard backward con-

struction is given by the Van der Pol equations (2);here, it was only considered the replacementY --+ Yg- instead of y - Yk in the nonlinear termof Eq. (2b), yielding the iterative equation,

Yk+I Yk (1 + he heXk+) hxk+ (6)

It will become manifest in the following discussionof results that these apparently "innocuous"changes provided great flexibility to the numericalintegration, by reducing considerably many of thenumerical instabilities founded in the standardcounterparts, including the usually more robustimplicit versions.

III. RESULTS AND DISCUSSION

Numerical instabilities are regarded as thosedissimilarities between the dynamical behavior ofthe continuous system and the numerical solutionproduced by a given scheme. In fixed-step schemesthere are a variety of numerical instabilities: themost common, by far, is the threshold instability,where beyond a critical step size, numericalsolutions began to differ [5]; creation instabilities,because of the appearance of spurious additionalpoints and typically found in higher orderintegrator [6]; chaotic instabilities [7-11, 14], werechaotic output is generated by the numericaldiscretization; numerical overflow, highly undesir-able because of the dramatic outbreak of thecomputation when a critical value of the step sizeis reached [9]; and computational alterations ofHopf bifurcations [10].

124 A. SERFATY DE MARKUS

In the present work we examine the onset ofunwanted chaotic/periodic behavior, which will bemonitored by the evaluation of the Lyapunovcharacteristic exponents (LCE). These Lyapunovexponents provide a powerful dynamical diagnos-tic of the chaotic status of a system, as they arerelated to the exponentially divergence or conver-gence of nearby orbits in phase space [2, 3].At this point is convenient to mention that in

one or higher dimensional maps is enough to haveone positive Lyapunov exponent for chaos, whilefor continuous dissipative system chaos is presentfor one or more positive Lyapunov exponents,provided we have no less than a 3-dimensionalsystem [1- 3]. Therefore, the presence of a chaoticattractor, that is, one LCE > 0 in the 2-dimen-sional Lewis (1) or Van der Pol (2) oscillators, willbe considered a numerical breaking resulting froma discretization effect.

III.1. Detection of Numerical Chaos

The 2-D Lewis oscillator provides a typicalexample of computational chaos. The real partof the eigenvalues A [1,3] of the continuous sys-tem (1) is given by Re[A] =e/2. The system has afixed point at the origin which is stable for e < 0and a stable limit cycle for e > 0. That meansthat the computed largest LCE for this systemshould be negative for e < 0 and equal to zero for>0.

Figure shows the largest Lyapunov exponentfor the Lewis oscillator (1) as a function of the stepsize when integrated according to a standardforward Euler algorithm ( =0.1 and initial con-ditions x0=Y0=0.01). This figure shows unequi-vocally that up to a step size h*0.45 thecomputed LCE remains approximately zero, asexpected for the limit cycle. Beyond that valuebegins an increasingly intricate dynamics, due toan assortment of numerically induced instabilities,that includes numerous n-periodicity windows,chaotic regions and finally a breakdown ath 1.178 due to numerical overflow. This complexdynamics is easily visualized with the help of the

corresponding bifurcation diagram, which com-

plements the information provided by the LCE vs.h plot, see Figure 1.

Figure shows the start of instabilities, clearlyindicated by evident changes in the LCEs forh > h*. Furthermore, is important to notice thatthis step size h* is more than a mark signal to limitcomputations; these limiting steps values werefound to be usually larger than the "typical"setting h 0.0001 0.01. This means that if thestep size could be safely increased by one or moreorders of magnitude, there will be a substantialreduction on the CPU time, which may be acrucial factor of practical importance.The Duffing oscillator offers an interesting

example of deterministic chaos appearing insystems with a nonlinear potential. In the presentexample, the amplitude F of the driving force willdetermine a behavior ranging from periodic, atF <_ 0.1, to fully chaotic, as in F--0.4. In Figure 2are plotted the largest Lyapunov characteristicexponent as a function of the step size for variousvalues of F. For F-0.1 the computed LCEs areapproximately zero, as expected for the limit cycle.At a step size h* 0.16 the LCEs becomes positive,explicitly indicating the onset of computationalchaos. Again, a complex dynamics sets in andfinally, at h-0.391, numerical overflow stopsfurther calculation.A chaotic dynamics is expected for F--0.4 and,

in agreement, the computed LCEs are positive.However, for h > 0.03, the shape of the computedattractor in phase space becomes different, and amore broaden and/or "scattered" chaotic attractortakes place as the step size approaches the valuethat overflows calculations. In fact, the scatteredappearance of the attractor appears to be a

frequent feature of numerically induced chaos, asobserved in other systems in our work [9-11]. Thisincreased degree of chaoticity (reflected by anincrease of the LCEs) in which a chaotic attractoris changing its shape, may be caused by theoccurrence of an internal crisis [12] forced byround-off and/or truncation errors [1, 4, 13, 14].For the same reason, a (internal) crisis may

NUMERICAL CHAOTIC INSTABILITIES 125

,I

28

X

0.30.20.1

0-0.1-0.2-0.3-0.4

0 0.2 0.4 0.6 0.8 1 1.2h

FIGURE Largest Lyapunov characteristic exponent and bifurcation plots as a function of the step size of the standard forwardEuler scheme used to integrate the Lewis oscillator (1) for e --0.1. The onset of numerical instabilities is marked distinctively abouth* 0.45.

0 0.1 0.2 0.3h

FIGURE 2 Largest Lyapunov characteristic exponent as a function of the step size of the standard forward Euler scheme used tointegrate the two-well Duffing oscillator (3) for various values of the amplitude of the driving force. For F=0.1 (where a limit cycle isexpected) is clearly observed the numerical induction of chaos at h* 0.16. In contrast, for F=0.7 (where a chaotic regime isexpected) it is shown a numerically suppression of chaos at h* 0.05, followed at h** 0.14 by a numerical transition periodicity/chaos similar to that for F--0.1.

produce other type of attractors, like the verypronounced quasiperiodic regime shown inFigure 2 for the valley in F= 0.4.

All these dynamical behavior could be conven-tionally understood as derived from the new

system created by the algorithmic discretization,

126 A. SERFATY DE MARKUS

with the "parameter" h as our control parameter.But more interesting is the connection of thisdynamics with the ideas of shadowing [15, 16].Formally, the critical step h* that signals the onsetof the new dynamics, could be related to themaximum step value defined in the shadowingtheories of Refs. [15-17]. According to theseideas, a true or suitable solution fi of a dynamicalsystem du/dt- F(u) follows closely or "shadows" acomputed solution, or more specifically, a pseudo-orbit Pn, which is the computed solution togetherwith the computational errors (round-off and localerrors). In single-step and multi-step discretiza-tion, the numerical method will reproduce, withina tolerance, the true orbit for h _< [17], such thatIt(nh)-pn] <_ ChI’:, where C and are positiveconstants and K is the order of the numericalmethod. In the context of the present work, clearly

h*. For h > h*, the computed solutions are nolonger reliable and a complete new dynamics takesplace, as shown distinctively in Figures and 2.Therefore, this h* represents a limit between (ap-proximately) continuous and discrete dynamics.

III.2. The Numerical Suppression of Chaos

This procedure of examining the LCEs as afunction of the step size also helps in the detectionof the numerical suppression of a chaotic dy-namics. And like the numerical induction of chaoson the previous examples, the actual suppressionoccurs in a well-located range of step-sizes. Thissituation is illustrated in Figure 2 for the thickcurve F 0.7. Here is possible to see that for astep size lower than h* 0.05, the computed LCEsare positive, as expected, but thereafter theysharply become negative for an extended rangeof steps sizes, where a numerically inducedperiodicity takes place [13, 18]. And again, towardsthe values of numerical overflow (h 0.168 forF= 0.7), a numerical crisis occurs, and thecharacteristic scattered numerical chaotic attractortakes place. In Figure 3 is shown the bifurcationdiagrams for the Duffing oscillator for F E [0.1, 1](eliminating 5000 first points), for h 0.001 and

X

-2

h=0.001

-2

h=0.147

0.1 F 1

FIGURE 3 Bifurcation plots as a function of the amplitude ofthe driving force for the Dutting oscillator (3), integrated with astandard forward Euler scheme with step size h 0.001 andh 0.147. Arrows indicate the suppression of chaos by theadvance of a numerically induced periodic front.

h 0.147. Notice the initial region for F < 0.3 atthe left of the bifurcation plots, where the expectedperiodic behavior is sustained. From then on, therough borders in Figure 3 indicate chaotic regionsand the subsequent smooth borders correspondsto periodic regions. The effect of the computa-tional suppression of chaos in the Euler schemeis strikingly clear in these plots: in the upper plot,at F 0.7 the system is well within a chaoticregion, which becomes artificially periodic with hincreasing, because a periodic front beginning atF 0.82 literally swallows the chaotic region upto F 0.490 (so then F 0.4 still remains chaotic)as indicated by the arrows pointing to the left.From Figure 3, is observed that at F the

Duffing system will be unaffected by the advanceof this numerical periodicity and will remain in theperiodic region for the range of step-sizes con-

sidered, including for the value h 0.147, veryclose to the step that overflows F= 1. Inconsequence, the LCEs for F 1, are approxi-mately zero, see the gray curve in Figure 2.

NUMERICAL CHAOTIC INSTABILITIES 127

111.3. Nonstandard Scheme

In Figure 4 is shown both the LCEs vs. h and thebifurcations plots for the Van de Pol system (2);integration is done according to the nonstandardimplicit scheme (6) described in Section II, fore 0.1 and initial conditions x0 y0 0.01. Thesame integration carried out with a forward Eulerstandard algorithm produced a much more com-plex bifurcation diagram (similar to Fig. 1) thanthat shown in Figure 4 for the nonstandardbackward Euler scheme. Compared to the bifurca-tion plots obtained for an explicit SFE, Figure 4shows the improvements of the nonstandardscheme, because the limit cycle is sustained forlarger values of the step, and the chaotic andbifurcations regions are much more reduced,preserving a more uniform dynamics. In theintegration of the Van der Pol system withthe nonstandard backward Euler scheme (6), thedeviation from the limit cycle was observed atabout h* 1.20 (omitting a small region ath 0.9) compared to h* 0.3 for the standard

tu -0,2

-0.4

-0.6

0 0.26 0.6 O.TS 1.26 1.6 1.7h

FIGURE 4 Largest Lyapunov characteristic exponent andbifurcation plots as a function of the step size of thenonstandard backward Euler scheme used to integrate theVan der Pol oscillator (2) for 0.1. Expected limit cycle and amostly instability-free dynamics is sustained for an extendedrange of steps, about h* 1.20.

explicit version. And the nonstandard implicitscheme overflows the calculations at a steph 1.71, which is comparatively much larger thanh 0.73, of the SFE integrator. Although is wellknown that implicit schemes are usually morerobust [1], superior results were obtained with thesimple nonlocal replacement already mentioned.In our work, the effect of nonstandard replace-ments was almost always to enhance stability andto allow larger steps before the onset of instabil-ities, which usually were greatly reduced or even

eliminated, see Refs. [5, 7, 9, 19]. Accordingly, thisexample confirms such properties by means of theLCEs vs. (h) and the bifurcation plots technique,see Figure 4.

III.4. Effect of the Computational Errors

Although there is an increasing evidence of theincidence of numerical chaotic instabilities due tocomputational errors [10, 13, 14, 18, 19], the exactmechanisms are still poorly understood. In part,because it is not well defined the build-up of sucherrors, and how the interaction model/schemeaffects the numerical output [9]. The relation ofcomputational errors (mainly truncation andround-off) to the step size has been regardedmainly as deterministic, like 0(hK), where K is aninteger, usually the order of the scheme [1,4];however, some evidence points out to a chaoticbuildup of errors with the step size [10], probably amore consistent approach when dealing withnonlinear dynamics.The algorithmic discretization of a continuous

nonlinear system introduces the step size as anextra parameter, affecting the dynamic of the"equivalent" difference system, which has now thedynamical properties of a nonlinear map, see Eqs.(5) and (6). For certain values of the h parameter,the spurious apparent chaos is presumably trig-gered by round-off errors in a mechanism similarto sensitivity to initial conditions. From this per-spective, it may be possible to explain the manifestcrisis events, that is, collisions with saddle-typeobjects being artificially generated and producing

128 A. SERFATY DE MARKUS

the many bifurcations shown in the bifurcationplots of Figures and 4.

In the other hand, the forced periodicity ob-served in Figure 2 for the Duffing system with F0.7, as well as the numerous periodic windowsshown in the bifurcation plots of Figures and 4,could be explained in terms of the finite arithmeticprecision of the computation, which forces chaotictrajectories to became periodic [18]. This could bebecause the truncation and round-off errors ex-cludes the possibility of an aperiodic (infinitedigits) dynamical evolution and after some timeof computing, the numerical trajectories may beginto repeat themselves.

In short, the combined effects of computationalerrors, enhanced by larger values of our parameterh in the discrete system, could affect the computedtrajectories by washing out the correlation (shad-owing) with a "true" trajectory after some time.

IV. CONCLUDING REMARKS

The manifest changes experienced by the LCEswith the step size provide explicit informationabout the numerical effects over periodic andchaotic dynamics. Construction of the LCEs vs. hplots allows establishing a range of workablevalues for step-sizes before instabilities, and showsa characteristic way to undergo numerical chaos.To complement the LCEs plots, the examina-

tion of the bifurcation diagrams facilitate thevisualization of numerically induced instabilities asa function of the step size and thus help to assessthe robustness and/or quality of the integrationscheme.

In summary, the procedure outlined in this workprovides a simple and direct criterion for theselection of much-larger-than-usual step-sizes ofcommonly used fixed step algorithms, under thepremises of minimum instabilities for the shortestcomputation time possible.

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