Pergamon Comput. & Graphics, Vol. 21, No. 2, pp. 253-262, 1997

Q 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain

PII: SOO97-8493(96)00088

0097-8493/97 $17.00 + 0.00

Chaos & Graphics

A TUTORIAL AND DIAGNOSTIC TOOL FOR CHAOTIC OSCILLATORS AND TIME SERIES

SATISH CHANDRA

Structures Division, National Aerospace Laboratories, Bangalore, India e-mail: satish@cmmacs.ernet.in

Abstract-A tutorial and diagnostic visual tool is used to compute a set of strange attractors for a double oscillator. The tool provides a visual environment to detect chaotic behaviour from a time series. Here, computation and visualization are coupled together in a single environment. Common techniques to detect chaos involve the visualization of phase portraits and Poincare maps, apart from the computation of fractal dimension, Lyapunov exponents and Fast Fourier Transforms (FFT). Results from these computations must be viewed graphically before a decision about the chaotic behaviour of a system can be made. In many cases, the use of a single technique may not always guarantee conclusive evidence that a system’s behaviour is chaotic. A few numerical integration schemes are built into the environment to generate a time series. The environment is demonstrated first with the use of the Duffing oscillator and the chaotic behaviour of a double oscillator is then presented, which provides an interesting set of strange attractors. 0 1997 Elsevier Science Ltd

1. INTRODUCI’ION

Chaotic behaviour of many physical systems has received a great deal of attention over the past decade and has interested engineers [l, 21. Chaotic behaviour is now a term assigned to that class of motions in deterministic physical and mathematical systems whose time history has a sensitive depen- dence on initial conditions. Often, the diagnosis of non-periodic oscillations by experimentalists was limited to the observance of the time history of the signal, phase portraits (also called phase plane maps), Poincare maps and the Fourier spectrum. In some cases, variation of system parameters was conducted to look for bifurcations and routes to chaos. However, it has been observed that a rigorous definition of chaotic motion is only limited to certain classes of mathematical problems. In other cases, it becomes important to use more than one test to obtain conclusive proof that the motion is chaotic. Even in cases where the time histories are not generated by experiments, but from numerical methods used, for example in transient analysis of structural systems, a few of these tests may have to be applied to classify the motions as chaotic. The tests are now established, well documented and probably available freely in software form. The aim of this paper is to show that an integrated tool can be useful for tutorial and diagnostic purposes. The use of experimental data may also be possible in the environment, though the quality of experimental data will presently have to be improved by pre- processing before being used by the software proposed here. A benchmark problem is selected to validate the software and a forced double oscillator is shown to produce an interesting set of strange attractors.

The visualization of time histories by themselves may not always provide definite evidence of chaotic behaviour or classify the nature of chaos (e.g. transient chaos). As a result qualitative methods to study the behaviour of the system further include the visualization of phase plane and Poincare maps. Phase portraits by themselves may provide a clue to chaos, but a modified technique called Poincare maps, which is basically plots of stroboscopic sampling of the phase plane will indicate if the structure of the plot is fractal. More details of fractals can be found in work by Barnsley [3], Farmer [4] and Mandelbrot [5]. A map which looks fractal is a strong indicator of chaos. Enlargements of the plot will indicate if a strange attractor exists. A strange attractor is basically a chaotic attractor, which visually looks as if complex stretching and folding have taken place. Parametric studies can indicate if more than one attractor exists, and plots of basins of attraction between the attractors can be obtained for a set of initial conditions [6, 71.

However, problems become apparent if the system has noisy input, or if there is large dissipation. In problems where the dimension is greater than 3, a visual examination of the Poincare map alone is not sufficient. It has been generally accepted that a positive Lyapunov exponent and a non-integer fractal dimension is a good indicator of chaos. Reliably obtaining the Lyapunov’s exponent or fractal dimen- sion with a limited amount of data needs careful attention and the generation of the fractal dimension in particular needs intensive computation. For many numerically produced time series, where noise is not much of a problem, the computation of the largest Lyapunov exponent can be definitive. There are, however, the difficult issues of choosing a proper time

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254 S. Chandra

delay and embedding dimension. These problems, however, are beyond the scope of this paper, as the aim here is to present a visual tool to detect chaotic behaviour, though the software provides some help regarding input and refers the user to the considerable literature available on the experience of computing the quantitative measures and the reader is referred later to the work done in computing Lyapunov exponents and fractal dimension.

In many structural dynamics problems, where transient dynamics data is obtained from finite element simulations using time integration schemes like the central difference and average acceleration scheme as shown in this paper, it is useful to have a visual tool that provides a number of tests, including phase portraits and FFTs for frequency computation that have been commonly used in the linear regime. Quality of data produced by numerical schemes used in finite element simulations is dependent on a number of factors related to time step selection and primarily the computational power that is available to handle large scale matrix manipulation. It is not uncommon to use the central difference scheme (explicit integration) for large scale crashworthy analysis of automobiles and aircraft. The incorpora- tion of these schemes into the visual tool enables simple models of physical systems to be built in the form of coupled non-linear oscillators. It may be equally useful to acquire data from established transient dynamics simulation systems like DY- NA3D used in impact problems and to process the data to check for chaotic behaviour. This paper makes no attempt to discuss in detail the techniques used to detect chaos, but attempts to integrate them in an environment that provides an option to use any of them. The tool can also be regarded as a beginner’s tutorial tool, when combined with the literature and software for the understanding of chaotic behaviour.

Graphic user interfaces like MOTIF (based on X- Windows implemented on UNIX) and MS-Windows on DOS, can be used to develop a visual environ- ment for the detection of chaotic behaviour from a time series. Graphic user interfaces are developments in human-computer interaction that have found application in a vast number of areas and have made substantial contributions to make interaction with a computer faster and more efficient. In this paper, an interface that drives the visual environment has been developed using MOTIF, an X-based system on SCO-UNIX implemented on an Intel 486 platform.

2. THE STRUCTURE OF THR VISUAL ENVIRONMENT

The framework for visual numerical environments have been discussed by Spiteleri [8] and an example of a multigrid technique as implemented is detailed. The computational and the visual tools are inte- grated in a single environment. There are consider- able advantages for use in tutorial or diagnostics of chaotic behaviour. Initial data can be checked quite

easily and the performance of the method itself can be monitored and the results can also be interpreted very efficiently. Spiteleri classifies three components of the environment, numerical, image and evalua- tion. The numerical environment is basically the use of mathematical models, the image environment provides the visual images and the cvaluzation environment is a supervisory process that contains both computational and visual tools to allow the user to make judgements. Use of graphic user interfaces makes this process more friendly and conducting parametric studies in the numerical environment is much easier.

A time series can typically be computed numerically or acquired after suitable preprocessing from an experiment or from transient dynamics simulations and can include displacements, velocities and accel- erations in the case of vibrational problems. Pre- processing of experimental data is, however, beyond the scope of this paper though the acquisition and preprocessing of data from experimental sources is considered formidable due to contamination with noise. Acquisition could be in the form of a single variable time series which can be used to generate phase portraits with a psuedo-phase space method that uses single variable measurements (also called the embedding space method) first described in Wolf et al. [91.

On the other hand, a time series can be generated by a number of different time integration schemes. Three schemes are built into the present environment suitable for vibration and impact problems, (1) the Runge-Kutta method [lo], (2) the average accelera- tion method 11 I] and a central difference scheme [ 121. The user can choose any of the schemes to generate the time series. The schemes are briefly described in the Appendix A.

Consider the behaviour of a specific equation

S+ ci+k,x+kzx- = Bcos r iI1

In engineering, such an equation is used to model the motion of a forced structure undergoing large elastic deflection and represents the Dulling’s oscillator. Using the methods described in the Appendix A, numerical integration of the equation is possible. For example, when c= 0.05, li, = 0, k2 = 1 .O, B= 1.5, x = 1, n = 3, the behaviour is known to be chaotic [1, 2, 131 and is now well established as the &da’s attractor. The demonstration of this case helps validate the software. The time step chosen for the simulation is 0.003817477. We generate the time series using the Runge-Kutta method here to illustrate the visual environment. We differentiate the techniques to detect chaos into qualitative techniques and quantitative techniques. In the next sections, these techniques are discussed briefly. All the techniques, however, require the visual environment to view the results.

The graphic interface has been designed using MOTIF. MOTIF is a higher level graphical user

A tutorial and diagnostic tool 255

interface under X [14], which allows widgets to be created. A widget is a re-usable, configurable piece of code that operates independently of the application except through prearranged interactions. The pur- pose of MOTIF, OSF and other higher level systems is to provide an object-oriented layer that supports the user-interface abstraction which is called a widget. The widget separates application code from user-interface code and provides ready to use inter- face components such as buttons and scroll bars. Widgets to control the input data acquisition, computation process, present results and provide help are available. The present development has been made on SCO-UNIX running on an Intel 486 platform, however, the application can be ported easily to other machines.

For example in Fig. 1, Push button widget x-t, when activated, produces plots of the time series itself. Push button widgets also activate data hand- ling and input, apart from processing the data from the time series in all cases. Zoom, titling and printing capabilities also add to the usefulness of the soft- ware. A configuration menu helps the user select color for portraits and plots and change input streams especially if data acquired from elsewhere have to be analysed. Help menus are available for all methods, with references to published literature in cases where selecting input parameters are not straight forward.

3. QUALITATIVE METHODS

Qualitative methods include visualization of the time series. A time series plot is shown in Fig. 1 for

the Equation (1). The ragged appearance, with some repeated patterns is evident, but it is hard to make a definitive statement that the behaviour is chaotic. Plots of i - t (Fig. 2) also show some repetitive patterns, though it too provides no conclusive evidence of chaotic behaviour.

The phase portrait plane for a dynamical system is defined as the set of points (x, k). When motion is not periodic, as in chaotic motion, orbits (Fig. 3) do not close or repeat. The trajectory of the orbits tend to fill up a section of the phase space.

3.1. Poincare maps Given the nature of chaotic systems and their

sensitivity to initial conditions, two different initial conditions can lead to uncorrelated versions of an underlying pattern which is known as a chaotic attractor. The Poincare map is particularly useful to find this underlying pattern. Sampling x and i stroboscopically at times that are integer multiples of the forcing period 2ni, a sequence of points is accumulated and plotted [Fig. 4(a)]. In periodic systems, if the final motion is periodic, a single point can be found, sub harmonics would appear as a sequence of n points, but in Fig. 4(a) we see a pattern which shows that there is a complex structure and it clearly looks fractal. Quantitive measurements can be made to confirm the fractal nature as discussed in the next section. The pattern can be regarded as stretching and folding of chaotic trajectories and goes by the name strange attractor. An enlargement of a particular area (zoom) shows the hidden

Fipra 1. x-t plot

Fig. I. x-t plot.

s. Chdndrd

Flwe 2. x’-t plot

Fig. 2. x’- t plot.

similarity and the embedding that is common in 4.1. Fast .fourier tran@rm (FFT) fractal structures [Fig. 4(b)]. A strong indicator of chaotic behaviour is also the

appearance of a broad spectrum of Frequencies in the 4. QUANTITATIVE METHODS output when the input is a single frequency harmonic

Quantitative methods for the detection of chaotic motion [2]. This is computed by processing the time behaviour include the computation of FFTs, Lyapu- series through a FFT [lo] and plotting the frequency nov exponents and fractal dimension. vs power. Figure 5 shows a plot between frequency

CHRDSY

Fig. 3. Phase portrait.

A tutorial and diagnostic tool

(4

(b)

Fig. 4. Poincare section.

and normalized power, clearly indicating the driving frequency and some broad band components which are regarded as a possible indicator of chaos. However, in many systems with an unknown number of degrees of freedom, a multi harmonic output need not actually be chaos but representative of the degrees of freedom. As a result, this test must by itself not be regarded as the final indicator of chaos especially when large degrees of freedom systems are considered. One can, however, observe changes in

the spectrum by varying the driving frequency or amplitude to note changes in the system behaviour.

4.2. Lyapunov exponents The computation of the largest Lyapunov’s

exponent is particularly useful in finding out if the system behaviour is chaotic or not. The computation and review of Lyapunov exponents is given in [9] and will not be repeated here. In deterministic systems, the existence of chaos indicates a sensitive depen-

258 S. Chandra

Fig. 5. FFT.

dence on initial conditions. The principle behind the computation of Lyapunov’s exponents is the diver- gence of trajectories which start close to one another in a phase space and will move exponentially away from each other on average. One begins with a reference trajectory or fiducary [9] and a point on a nearby trajectory and measure d(t)& where 4 is the measure of the initial distance between two starting points. When d(t) becomes too large, a new trajectory is chosen and a new h(t) is defined. A Lyapunov exponent is given by the expression

R= (2)

The system is termed to have chaotic behaviour if 1> 0 and regular behaviour if A > 0. We will consider the data to be basically a measurement of a single observable as input to the program. This is to facilitate the use of the program for data generated numerically by other transient dynamics programs or preprocessed data from experiments. However, in the present example, we use the measurements (time series generated) from the numerical example chosen above. The use of delay co-ordinates will help us reconstruct the phase portrait. The two most important inputs to compute the Lyapunov exponent are the delay time and the embedding dimension and the selection of this is regarded as quite formidable especially for noisy experimental data. Considerable literature exists on the optimum values for both [ 15- 211. It is, however, important to note that the Lyapunov’s exponent must be used with other methods presented here to confirm suspected chaos.

Figure 6 shows a Lyapunov’s exponent for 400 cy- cles for the oscillator described in Section 3. Clearly, the largest exponent here is positive and the system can be regarded as chaotic.

4.3. Fractal dimension We have seen from the Poincare map, a strange

attractor which looks fractal. However, we will use quantitative measures to define the fractal nature. Details regarding fractals can be found elsewhere [3, 51 and the measurement of fractal dimension itself is given in great detail by Grassberger and Proccacia [22] and Farmer [4]. It is generally accepted that a non-integer fractal dimension is a good indicator of chaos [4, 61. A useful measurement is the correlation dimension, which is defined in the following way.

If one has a Poincare map (discrete set of points) and calculates the distances between pairs say so = 1x( - Xj]. A correlation function is then defined as

C(r) = lim, .-+ M --&$c H(r - [Xi - -ril,l t3) I i

whereH(s)=l ifs>OandH(s)=OifscOwhererisa chosen radius or length of a sphere or cube. With various values of r, a correlation or fraetal dimension is defined using the middle slope of log Cvs log r curve. For the Poincare map shown in -Fig. 4, the correlation function log C vs log r is plotted in Fig. 7. The middle slope works out to be 1.58. Although, sampling of 10000 to 40000 data points is de&a& [4} to obtain the fractal ,dimension, the intensive computations involved and the speed of the computer used have limited the number of data points used here to 45QO.

A tutorial and diagnostic tool 259

Fig. 6. Lyapunov exponent.

5. CHAOTIC BFXAVIOUR OF A LMMJBLE OSCILLATOR

[ml&f) + [cl-lq + [hl{x~ + hl{xY = if)

A double oscillator is shown in Fig. 8(a) and where ml = 1 .O, m2= 1.0, cl = 0.832, c2 = 0.832, represented by the equation of motion in the form kn=4.0. kn=4.0, k,,l=3.0, k,n=8.0 and fi=F sin

/ -4.53

I

-5, ,2

~ -650

-73

I FLyse 7. Fractal dirmtm

Fig. 7. Fractal dimension.

260 S. Chandra

(a)

(b)

Fig. 8. (a) Double oscillator. (b) Strange attractors of the double oscillator.

w t, where F= 5.0338, a = 2.2136 with initial condi- tions x= 0.4 and P = 0.4. Using the central difference scheme as shown in the Appendix A, we can obtain time series representations consisting of displace- ments, velocities and accelerations of the two oscilla- tors. We can see chaotic behaviour of both the oscillators when the Poincare maps are constructed [Fig. S(b)]. The fractal dimension of the strange attractors are 1.72 (blue) and 1.79 (red), respectively.

the now well known Ueda’s attractor that can be generated under certain conditions of behaviour of a Duffing oscillator. Data acquired from transient dynamics simulations using explicit or implicit numerical integration schemes can also be processed using the tool. It may be possible to use the tool for data acquired from experiments if the data can be pre-processed to eliminate noise. Additional numer- ical techniques, evaluation tools and techniques to separate noise from experimental data are being incorporated in the environment.

6. DISCUSSION

A set of strange attractors and their fractal dimension for a double oscillator have been pre- sented using a visual environment that includes numerical, visual and evaluation tools. The valida- tion of the visual tool developed for the purpose of detecting chaotic behaviour has been described using England, 1986.

REFERENCES

1. Thompson, J. M. T. and Stewart, H. B., Nonlinrur Dymzmics und Chuos. John Wiley and Sons, Chichester.

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I.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

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Moon, F. C., Chaotic und Fructul Dynamics: w Introduction for Applied Scientists and Engineers. John Wiley, New York, 1992. Barnsley, M., Fructals Everywhere. Academic Press, New York, 1988. Farmer, J. D., Ott, E. and Yorke, J. A. The dimension of chaotic attractors. Physicu D, 1983, 7, 153-170. Mandelbrot, B. B., Fractuls, Form, Chance, and Dimension. W. H. Freeman, San Francisco, 1977. Moon, F.C. and Li, G.-X. The fractal dimension of the two well potential strange attractor. Physicu D, 1985, 17, 99-108. Grebogi, C., Ott, E. and Yorke, J. A. Metamorphoses of basin boundaries in nonlinear dynamical systems. Physics Review Letters, 1986, 56, 101 l-1014. Spitaleri, R. M. Visual numerical environment refer- ence models, methods and tools. Computer Aided Design, 1994, 26, 907-916. Wolf, A., Swift, J. B., Swinney, H. L. and Vasano, J. A. Determining Lyapunov exponents from a time series. Physicu D, 1985, 16, 285-317. Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T., Numericul Recipes: the Art of Scientz>c Computing. Cambridge University Press, Cambridge, 1986. Timoshenko, S., Young, D. H. and Weaver, W. Jr., Vihrution Prohlems in Engineering. John Wiley, New York, 1974. Hughes, T. J. R., Transient algorithm and stability, In Computational Methou% for Transient Analysis, eds T. Belytscko and T. J. R. Hughes, North Holland, 1981. Ueda, Y. Randomly transitional phenomena in system governed by Duffing’s equation. Journul of Statistical Physics, 1979, 20, 181-196. O’Reilly, T. et al., The X Window System. O’Reilly and Associates, CA, 1990. Bryant, P., Brown, R. and Aberbanel, H. D. I. Lyapunov exponents from observed time series. Physics Review Letters, 1990, 65, 1523-1526. Casdagli, M., Eubank, S., Farmer, J. D. and Gibson, J. State space reconstruction in the presence of noise. Physica D, 1991, 51, 52-98. Gibson, J. F., Farmer, J. D., Casdagli, M. and Eubank, S. An analytical approach to practical state space reconstruction. Physicu D, 1992, 57, I-30. Par&z, U. Identification of true and spurious Lyapu- nov exponents from time series. Internutionul Journul of Bifurcation Chuos, 1992, 2, 3403-3411. Kennel, M. B., Brown, R. and Aberbanel, H. D. I. Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phy- sics Review Letters, 1992, A45, 185-187. Eckmann, J. P. and Ruelle, D. Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems. Physic0 D, 1992, 56, 33293232. Zeng, X., Eykholt, R. and Pielke, R. A. Estimating the Lyapunov exponent spectrum from short time series of low precision. Physics Review Letters, 1991, 66, 349- 352. Grassberger, P. and Proccacia, I. Characterization of strange attractors. Physics Review Letters, 1983, 50.

APPENDIX A

The equation of motion used to compute chaotic behaviour in this paper is given by Equation (1) and is represent in the form f = f(x, 1, t).

The standard Rung+Kutta fourth order integration process is to convert the equation of motion into a set of ordinary differential equations (ODES) in the form x =.f(x,f,l, t).

Az-1 = xi-1 + xi-iAti/2

Bz-t = Xi-1 +li_iAti/2 Lety=i j = (f‘sin wr - ci - kx)/m The iterative type of solution requires some criterion for

The Runge-Kutta method computes four functions to evaluate new positions at each time step given in the form

kl =f(x",ht)

kz =f (xn + h/‘& , tn + h/2)

ks =f (x,, + h/S, tn + h/2)

k4 =f (x,, + hh, t,, + h)

x,,+t = x,, + h/6(k, + 2kz + 2kj + k4)

The explicit integration (EI) scheme, a variant of the central difference scheme, is given in the following way where Equation (1) can represent multi-degrees of freedom systems in the conventional fashion as represented by the finite element method. Accelerations are computed recur- sively from Equation (1)

ii+, =f’- cii - (kzxz + k2Y;)

Displacements and velocities are in the form

X,+1 = Xi + At/P + (At2/2) 1 - 2p)j;i + 2pfi+i

ii+! = Xi + At( 1 - 7)x; + yxz+t

where y = 0.5 and /I = 0.0 for the explicit integration scheme. It must be noted that with y =0.5 and /?=0.25, the scheme reduces to the implicit integration scheme or the average acceleration method.

The implicit integration scheme is described in [1 1] and is also known as the average acceleration method or trapezoidal rule or the Newmark l/4 rule and is given in the following form

2 =.f(t,.i,x)

We evaluate the initial acceleration (at time t = 0) which yields

i” =.f’(O,i”,i”)

and the velocity li at any time ti is approximated as

X, = ii-1 + (ii-l + jti)Ati/2

where ii-1 is the velocity at the preceding time ti- i. Similarly the displacement x, is approximated by the trapezoidal rule as the expression

Xi = Xi-1 + (it-1 + ii)Ati/2

Since the value of xi is not known in advance, it is iterative within each step; and the following recurrence expressions represent jth iteration of the ith step.

(if), = Az-z + (Xi)j-iAti/2

(Xi), = Bz-1 + (Pi)jAti/Z

262 S. Chandra

stopping and can be the convergence of lower order A time step i, =0.009817477, tolerance of ~=0.00001, derivatives like xi given as maximum number of iterations= 10,OW was chosen to

compute the Ueda attractor using the average acceleration I(xi)j - Cxi)j-l < cXl(xi),l scheme.