Modeling chaotic motions of a string from experimental data Kevin Judd and Alistair Mees Department of Mathematics The University of Western Australia March 3, 995 Abstract Experimental measurements of nonlinear vibrations of a string are analyzed using new techniques of nonlinear modeling. Previous theoretical and numerical work suggested that the motions of a string can be chaotic and a Shil'nikov mechanism is responsible. We show that the experimental data is consistent with a Shil'nikov mechanism. We also reveal a period doubling cascade with a period three window which is not immediately observable because there is sucient noise, probably of a dynamical origin, to mask the period-doubling bifurcation and the period three window. Introduction Bajaj and Johnson [2] have conducted an analysis of weakly nonlinear partial dierential equations describing the forced vibrations of stretched uniform strings. The equations take into account motions transverse to the plan of forcing, which are induced by a coupling with longitudinal displacements, and changes in tension that occur in large amplitude motions. The averaged equations of a resonant system can be reduced by successive approximation to a four-dimensional system of ordinary dierential equations. These equations have a complex bifurcation structure exhibiting Hopf, saddle-node, perioddoubling and homoclinic (Shil'nikov) bifurcations. The bifurcation parameters of interest are the damping and the detuning from resonance. Bajaj and Johnson discovered, for a xed damping, a Hopf bifurcation with subsequent period-doubling bifurcations as detuning is increased. This branch of orbits did not lead to chaos as the orbits undouble and the Hopf orbit nally collapses. However, an isolated branch of orbits, generated in a saddle-node bifurcation for the same damping, does have period-doubling bifurcations leading to chaos. Further increase of the detuning unfolds the chaos, undoes the doublings and, eventually, the saddle-node bifurcation too. On the other hand, decreasing the damping, with xed detuning, generated a cascade of isolated branches, which period-double and merge, leading ultimately to a homoclinic orbit. All of the complex bifurcations are consistent with the bifurcation structures that surround a Shil'nikov mechanism and Bajaj and Johnson do nd that the homoclinic orbit is attached to a suitable xed point. Molteno and Tullaro have performed experiments with guitar strings and have recorded periodic and apparently chaotic motions []. They generously provided us with their data. We describe here the construction and analysis of a model of the ow from a single time-series, from which we had hoped to infer that a Sil'nikov mechanism produces chaotic motions in the experimental system. We do not quite achieve this aim: although we come very close to
doing so some room for doubt remains, perhaps because the experimental system was not close enough to having an homoclinic orbit. Despite this slight uncertainty about identifying the global bifurcations we are able to identify local bifurcation cascades. A model of the Poincare section is constructed and we nd good evidence for a period-doubling route to chaos, which presumably occurs along one of the isolated branches. This identication is important since the recording of the system contains sucient noise, most likely of a dynamical origin, to mask the bifurcation from direct observation. We have used radial basis models generated under minimum description length constraints [7]. These methods appear to be very successful in avoiding problems of over-tting that often plague nonlinear modeling methods. Minimum description length methods appear to be robust, extracting only salient dynamical features from minimal data. Robustness and resistance to over-tting mean the models built from data can be analyzed in much the same way as one analyzes maps and ordinary dierential equations arrived at by theoretical considerations. 2 Shil'nikov bifurcations First we give a brief description of the Shil'nikov mechanism; for details the reader should refer to Glendinning and Sparrow [6, 5], and Mees and Sparrow [9]. Suppose a system _x = f(x; ), x 2 R 3, 2 R, and f analytic in x and, has a xed point with one real eigenvalue > and a complex conjugate pair i!, such that this xed point has a homoclinic orbit when = but not necessarily otherwise. Dene = j j=. The theorem of Shil'nikov describes the behaviour of this system in the neighbourhood of the homoclinic orbit for given. The naive interpretation of this theorem is, if the system is close to having a homoclinic orbit, that is, is close to zero, then one might expect to see a stable periodic orbit for > and chaotic motions for <. The theorem does not guarantee these observations, but they would be consistent with the invariant structures the theorem implies. See the above references for an accurate statement of the theorem and discussion of the global dynamics. The Shil'nikov mechanism is not a bifurcation in the usual sense, rather a sequence or family of bifurcations. As approaches zero from either direction, there is generated, in saddle-node bifurcations, a sequence of periodic orbits of longer and longer period; see gure. Each of these orbits gives rise to a period-doubling sequence so that near = the bifurcation diagram becomes a thick foam. Bajaj and Johnson do not describe for their string equations a bifurcation exactly as just described. They discovered an isolated branch of periodic doubling bifurcations that originating from a periodic orbit that appeared in a saddle-node bifurcation. They observed chaotic motions forming what they refer to as a Rossler-type attractor. They nd numerical evidence of a homoclinic orbit to a suitable xed point, and calculate that >, which would not be in accord with chaotic motions. They go on to describe what they call a Lorenz-type attractor, which is really a symmetric version of the above theorem where both halves of the unstable manifold of the xed point become homoclinic (see Glendinning and Sparrow) a true Lorenz-type attractor is an heteroclinic pair of xed points. In this symmetric situation one can observe two types of periodic and chaotic motions: () motions that pass through, and are symmetric about, a plane through the xed point, or (2) identical pairs of motions mirrored across a plane through the xed point, or inverted through the xed point. Note that asymptotic motions of the second type are usually asymmetric. Bajaj and Johnson's 2
T x µ µ Figure : A schematic of the bifurcation structure near a Shil'nikov-type homoclinic bifurcation. Solid lines represent stable orbits and dashed lines unstable orbits. The axes are the bifurcation parameter, the period of an orbit T and a state variable x. As approaches there is a sequence of saddlenode bifurcations generating periodic orbits of longer and longer periods. At the same time, as approaches the stable orbits begin period-doubling cascades. numerical simulations display examples of periodic and chaotic motions of both symmetric and asymmetric forms and a single symmetric Shil'nikov mechanism is implicated as the source of all these motions and accounts for all the bifurcations described by Bajaj and Johnson, although Bajaj and Johnson do not explicitly state this. 3 Experiments Molteno and Tullaro conducted a forced string experiment [], collecting copious data of which we have taken a single experiment's recording. During this experiment there was a monotone drift in the parameters, due to heating of the apparatus. Figure 2 shows Takens embedding plots of the beginning and end of a long time series that exhibits a transition from apparently chaotic to periodic motion. The mechanism of this transition was thought, by us and the experimenters, to be generated by a Shil'nikov mechanism as described above. If this supposition is true, then it is clear that the motions are of the asymmetric form. We investigated whether Molteno and Tullaro's data imply enough about the global dynamics to establish, or refute, there being a Sil'nikov mechanism at work. We would conrm a Shil'nikov mechanism by building a model of ow in phase space, then establishing several things of this ow: the ow has a xed point of the correct type whose eigenvalues satisfy Shil'nikov's theorem, the ow is close to having a homoclinic orbit, and the unstable manifold of the xed point of the model lies in the neighbourhood of the periodic and the apparently chaotic motions. Given these facts we might infer that the system does indeed have a Shil'nikov mechanism and that it is this mechanism that is responsible for the apparently 3
.5.5.5.5 x3.5 2 x3.5 2.5 2 x 2 2 x2.5 2 x 2 2 x2 Figure 2: The complete time series consisted of an approximately 2 minute run sampling at.346 khz giving over 28 thousand samples. Shown here are segments from the beginning and end of the times series each of 5 samples embedded in 3 dimensions with a lag of 28. The data shows a transition from apparently chaotic motion to periodic motion. chaotic motions. We may also infer, modulo intrinsic dynamic noise, that the observed complex motions are truly chaotic. We do not model the phase-space ow of the system, although this can be done [3], but rather model the system as a map induced by the ow over xed, short time-steps T. A model of the vector eld of the system would be more natural, but we had available sophisticated techniques [7] for constructing maps and everything that we need to establish can be got from a short-time-step map. Given a scalar time series Y t, t = ; : : : ; N, we choose an embedding strategy X t = (Y t? ; Y t?2 ; : : : ; Y t?d), where is called the lag and d the embedding dimension, and assume that the time series can be modeled by Y t = F (X t ) + t () for some scalar function F and i.i.d. Gaussian random variates t. When t = for all t, equation () denes a discrete dynamical system on embedding space, X t 7! (F (X t ); Y t? ; : : : ; Y t?(d?)): (2) If the time series Y t is generated by a dynamical system, then Takens theorem implies that for d suciently large (2) contains equivalent dynamics to the generating system integrated over some time step T. Given a time series we can approximate this dynamical system using 4
a model ^F in place of F. Our aim is to estimate ^F and analyze the dynamical system dened by ^F, assuming that in the neighbourhood of the tted data something of the local and global properties of the original system are contained in it. We chose to construct radial basis models [7] for ^F. There are a number of problems with this approach. How does one correctly determine d and to retain local and global properties of the ow? How are estimates of eigenvalues and eigenspaces aected by changes in d and? If d is not the same as the original system, then does it make sense to evaluate the eigenvalues of xed points? These important questions are beyond the scope of this paper. We will adopt standard methods, which for this study seem to give reasonable results. To build a model from the experimental data we choose the lag to be the nearest integer to one quarter of the average inter-peak period of the time series []. The false nearest neighbours algorithm suggests that for this lag a three dimensional embedding is sucient []. Since the experimental system has a monotone drift in parameters over time we assume a model () for the time series where X t = (Y t? ; Y t?2 ; Y t?3 ; t); (3) and scales t to have roughly the same range as Y t for a time series, which is necessary to avoid problems building a radial basis model. We obtained an optimal model ^F for F using a radial basis model composed of cubic radial basis functions with chaperons [7] as centres and incorporating linear basis functions. The dynamical system it denes can be written (x; y; z) 7! ( ^F (x; y; z; ); x; y); (4) where is a bifurcation parameter. The xed points of (4) are of the form (x; x; x) where x is a solution to x = ^F (x; x; x; ). Furthermore, the eigenvalues and eigenspaces of the linearized system at a xed point are given by the Jacobian B @ @ ^F =@x @ ^F =@y @ ^F =@z C A : (5) The discrete dynamical system (4) is supposed to result from integrating a ow over some time period T. Fixed points of the map correspond to xed points of the ow, and likewise there is a correspondence of invariant eigenspaces. If T is small, the relationship between the real part of an eigenvalue of an eigenspace of the ow and the real part of an eigenvalue of the corresponding eigenspace of the map is Re() = exp(re()t ). For all values of in the range of the experiment the model system has two xed points; one in the negative quadrant, which we label A, and the other in the positive quadrant, which we label B. (The two xed points we label A and B appear to correspond to upper-planar and nonplanar xed points, respectively, in the analysis of Bajaj and Johnson.) Fixed point A, because of its location, we thought to be implicated as the possible location of a Shil'nikov mechanism. Table lists the xed points and eigenvalues of our radial basis model calculated for four values of. The logarithm of the absolute value of the eigenvalues of the map should be proportional to the real parts of the eigenvalues in the corresponding ow. The constant of proportionality is indeterminate, but the Shil'nikov theorem only requires calculation of the ratio from which this constant cancels. The values of calculated for table suggest, 5
Fixed Point Location x 2. A -.69-38 :257 :75i.47 B.66-36?:27 :876i.67 A -.69-2 :294 :765i.42 B.63-288?:2 :882i.33 A -.69-8 :3 :749i.44 B.72-39?:224 :872i 2. A -.69-23 :3 :699i.57 B.69-32?:26 :855i Table : Fixed points and eigenvalues for radial basis model of a Shil'nikov bifurcation. The locations of the xed points are (x; x; x) because a simple time-delay embedding is used. The eigenvalues are and 2, and =? log(j 2 j)=j log(j j)j. and gure 3 conrms, that always very much less than unity for this experiment, which is consistent with the chaotic side of the Shil'nikov mechanism. We have now established the existence of a suitable xed point for a Shil'nikov mechanism. In order to nally infer a Shil'nikov mechanism, it remains to be shown that the dynamical system has, or is close to having, a suitable homoclinic orbit. In the map such an orbit should correspond to the unstable manifold of the xed point just identied. The approximate location of the unstable manifold can be traced by the technique of elastic-imaging [8]; in this technique the extension of the unstable manifold is traced by repeated images of a initial short line segment from the xed point in the direction of the unstable eigenspace. Figure 4 shows the unstable manifold when it is followed once around attractor. Notice that although the images start out fairly smooth they begin to undulate because small errors in map are amplied by the dierential rotation rate of the map. Figure 5 shows the unstable manifold when it is followed twice around the attractor. Stretching and folding of map forms a loop out of the undulations creating a \lock-stitching" manifold around attractor. Note the loop of manifold at the far left and identify its next image, where it has become a sharp, pendulous spike of the manifold. In subsequent iterates this spike rapidly heads for innity, while other parts of the manifold continue around the attractor; an immediate implication of this is the unstable manifold of the xed point intersects its stable manifold, and hence the map has a homoclinic orbit. Unfortunately, this does not correspond to the ow having a homoclinic orbit. The unstable manifold of the ow should be identied with that of the map, but the structure of the map's unstable manifold just described cannot occur in a ow, because a trajectory cannot make a transverse intersection with a stable manifold. The structure seen in the map's unstable manifold must be a artifact of small errors in the approximation being amplied by iteration. That such a problem arises is expected. However, we can appeal to the shadowing lemma [4] to suggest that because the approximate map has a homoclinic orbit, the unstable manifold of the ow must be very close to having a homoclinic orbit. This is sucient for an application of the Shil'nikov theorem. Closer analysis of the unstable manifold of the map implies that the homoclinic orbit passes on the correct side of the stable manifold to generate asymmetric periodic and chaotic motions. The same behaviour of the 6
.58 Delta for S4 data.56.54.52 Delta.5.48.46.44.42.4.2.4.6.8..2.4.6.8.2 Bifurcation parameter Figure 3: Plotted here is Shil'nikov's value for the model system as function of the bifurcation parameter. The value of is always much less than unity in the experimental parameter range. unstable manifold was observed for all values of the bifurcation parameter in the range of interest. Finally, we observe that the unstable manifold of the xed point is intimately associated with the location of the periodic and chaotic motions: indeed, the unstable manifold wraps around the attractor in a way one should expect in a Shil'nikov bifurcation. We can infer from the above evidence that the transition between periodic and apparently chaotic motions seen in the experimental time series is the product of a Shil'nikov mechanism, and that, modulo intrinsic noise, the complex motions are chaotic motions. However, the authors do not nd the evidence % convincing. It would have been better to have found an homoclinic orbit. We suspect that the parameter range of the experimental system is too far from where the homoclinic orbit exists to conclusively infer its presence. 4 Modeling a Poincare map We now analyze the local bifurcations in greater detail and establish that there is a perioddoubling cascades which include a period three window, although none of this is immediately 7
unstable manifold ( step).5.5.5.5 x3 x2 -.5 -.5 - - -.5 -.5 -.5 - -.5.5.5 x Figure 4: The unstable manifolds of the reconstructed map followed once around the attractor. Note how small errors in the map are amplied by the dierential rotation on the attractor. This is an unavoidable and expected instability. observable in the data. Using an interactive 3-D viewer it was found that the embedded time series seems to lie on a two-dimensional ribbon with a half twist and the suggestion of a fold. The system is rather like the Rossler system but with a half twist. The half twist occurs at the bottom of gure 2 and the fold at the top of the attractor, within an arc of about one fth of the attractor length. The two-dimensional ribbon structure suggests modeling the system with a one-dimensional Poincare return map. If this were to fail, then we would try a two-dimensional return map. As will be seen, a one-dimensional model is very successful. A Poincare return map can be obtained directly from a time series of a ow, in some instances, by simply constructing a time series of suitable local minima or local maxima. If _x = f(x) and y = g(x) is a scalar measurement, then _y = Dg(x) _x = Dg(x)f(x) and local extrema of y occur where Dg(x)f(x) =. If the readout function g(x) is non-degenerate, then Dg(x) is everywhere non-vanishing. Consequently, Dg(x)f(x) = denes codimension submanifolds on which the vector eld f(x) is transverse, in fact normal, except where f(x) =. In practice, one needs a time series sampled frequently enough to accurately determine local extrema, and one must take care to ensure that local extrema all live on the same piece of submanifold and that the vector eld is crossing in the same direction in all cases. For the The investigators used SceneViewer, an Open Inventor application available on Silicon Graphics and other platforms. 8
unstable manifold (2 step).5.5.5.5 x3 x2 -.5 -.5 - - -.5 -.5 -.5 - -.5.5.5 x Figure 5: The unstable manifolds of the reconstructed map followed twice around the attractor. The amplication of the undulations results in part of the unstable manifold of the xed point intersecting its unstable eigenspace. This implies there is a homoclinic orbit in the map and that the ow is close to having a homoclinic orbit. system considered we chose local minima y t such that y s > y t for all js? tj < T =2 where T is the approximate period of the time series (e.g., T =2 is the rst minimum of the autocorrelation function.) By the above methods we obtained two scalar time series t n, the time of the nth local minimum and the system state at the minimum z n are realizations of a stochastic process Z n where = y(t n ). We then assumed that the z n Z n+ = F (Z n ; ) + n ; (6) is a bifurcation parameter and n are i.i.d. Gaussian random variates. Our aim is to model F (z; ). We assume that the bifurcation parameter is a monotonic function of time and hence dene n = t n and build a model of z n+ = ^F (z n ; n ) + n. The scale parameter is chosen so that the range of n is the same as that of z n, because this scaling is advantageous to the radial basis model. The function F (z; ) was modeled with a radial basis model using cubics with chaperons as centres and linear functions. Interactive viewing of this surface ^F (z; ) showed it to be unimodal for xed, very nearly quadratic in z and very nearly linear in ; gure 6 is a two dimensional projection of the surface. Figure 7 shows a numerically computed bifurcation diagram of ^F which is seen to have a period doubling bifurcation, which only 9
vaguely corresponds to an equivalent plot of the experimental data shown in gure 8. Notable is the lack of a period doubling cascade and period three window in the experimental data and we will now search for reasons why these are not seen. D model with bifurcation parameter.25.3.35 Next stat value.4.45.5.2.55.6.6.55.5.45.4.35.3 State value. Bifurcation Figure 6: A projection from three dimensions of the embedded return map data with bifurcation parameter dependence (circles) and the surface t of the return map model (dots). Figure 9 shows the empirically derived expectation of z n for the system (6) when there is a small dynamical noise component and no simulation of measurement noise. The dynamical noise chosen had a standard deviation = :3, which is half of the standard deviation of the residual errors of the tted model. The choice of was somewhat arbitrary. Using the full standard deviation of the residuals gave too great a dispersion, which indicates some measurement error is present, for example, from the method of calculating the z n. On the other hand, we expect some signicant level of dynamical noise because additive measurement error would give a uniform blurring of the bifurcation diagram, but this is not consistent with the dispersion of the experimental data. A maximum likelihood estimate of the relative contributions of measurement and dynamical noise could be obtained through boot-strapping, but we have not done this. As can be seen, the period three orbits of the model system are suciently sensitive to perturbation that this small amount of dynamic noise is enough to almost completely mask their existence. The expectation of the state variable z n shown in gure 9 agrees well with the experimental sample shown in gure 8, although the model predicts a more prominent central ridge in the period three window and a hollow above it. There are four possible sources of discrepancy between predicted expectation of the state variable and apparent experimental
.3 Bifurcation diagram of D model.35 State value.4.45.5.55.2.4.6.8..2.4.6.8.2 Bifurcation parameter Figure 7: A numerically determined bifurcation diagram of the model (6). It is seen to be a classic period doubling bifurcation complete with period three window. distribution: small errors in the model to which the location of the middle point of the period three orbit is sensitive; the assumption that the stochastic terms in (6) are normally distributed; the assumption that the experimental system attains its asymptotic state; and additional blurring due to measurement error that accounts for the remaining residual component. With regard to the rst source of discrepancy, when our algorithms to generate models were run repeatedly to generate many models and their bifurcation diagrams compared it was found that they diered only slightly, but most notably in the onset of the period three window and the location of the middle point of the period three orbit. This could partly account for the noted discrepancies. With regard to the second source of discrepancy, the assumption of normality of errors is most likely false. There are several sources of error in the experimental time series z n, notably errors in determining local minima, which are likely to be nearly normally distributed, and the intrinsic dynamical noise of the system. The latter is really an accumulation of small perturbations through a complete cycle of the ow. The system certainly attens these errors onto the two-dimensional sheet of its attractor and this implies that on return to the Poincare section there is a larger perturbation across the sheet than perpendicular to the sheet. This would imply greater levels of noise than the model might predict and furthermore this blurring would be more noticeable of an sensitive period three orbit than it would be of a more stable period two orbit. The third source of discrepancy arises because gure 9 shows the asymptotic distribution of states given noise, whereas in g-
.3 Bifurcation diagram of D model.35.4 State value.45.5.55.6.2.4.6.8..2.4.6.8.2 Bifurcation parameter Figure 8: A plot a the time series zn where time axis has been rescaled to be bifurcation parameter. If the system were noise free and the model accurate, then this plot should resemble gure 7. It does so only vaguely; there is no clear period doubling or period three window. ure 8 the bifurcation parameter is constantly increasing and the system does not necessary attain its asymptotic distribution for any parameter value, in particular it is unlikely to do so in sensitive regions such as near bifurcation points and in the period-three window. The nal source of discrepency might account for some of the small disprepencies, but does not account for the main disrepencies because it is a uniform blurring. 5 Conclusions It has been argued through the use of minimum description length models that the apparently chaotic motions of a vibrating string are due to a Shil'nikov mechanism, and in particular a folding of phase space that this mechanism induces. The folding results in an essentially onedimensional return map. A model of this return map predicts a period doubling route to chaos that includes a period three window within the parameter range of the experiment. However, simulation of the model return map suggests that the dynamical noise of the experimental system is sucient to mask the period doubling and the sensitive period three window from direct observation. 2
.3 Noise driven bifurcation density (sigma=.3).35.4 State value.45.5.55.6.2.4.6.8..2.4.6.8.2 Bifurcation parameter Figure 9: The empirically calculated expectation of the state variable zn for the model system when the noise has a standard deviation of :3. The density plot shows the asymptotic probability of Zn falling in a narrow range when is held xed. References [] H. D. I. Abarbanel and M. B. Kennel. Local false nearest neighbors and dynamical dimensions from observed chaotic data. Technical report, Department of Physics, University of California, San Diego, 992. [2] A. K. Bajaj and J. M. Johnson. On the amplitude dynamics and crisis in resonant motion of stretched strings. Phil. Trans. R. Soc. Lond., 338A:{4, 992. [3] R. Brown. Orthonormal polynomials as prediction functions in arbitrary phase space dimensions. Technical report, Institute for Nonlinear Science, University of California at San Diego, 992. [4] J. D. Farmer and J. J. Sidorowich. Optimal shadowing and noise reduction. Physica D, 47:373{392, 99. [5] P. Glendinning and C. Sparrow. Bifurcations near homoclinic orbits. In XVIth International Congress of Theoretical and Applied Mechanics. 984. [6] P. Glendinning and C. T. Sparrow. Local and global behavior near homoclinic orbits. J. Stat. Phys., 35:645{697, 983. 3
[7] K. Judd and A. I. Mees. A model selection algorithm for nonlinear time series. Physica D, in press, 994. [8] K. Judd, A. I. Mees, K. Aihara, and M. Toyoda. Grid imaging for a two-dimensional map. International Journal of Bifurcation and Chaos, ():97{2, 99. [9] A. I. Mees and C. T. Sparrow. Some tools for analyzing chaos. Proceedings IEEE, 75:58{7, 987. [] T. Molteno and N. B. Tullaro. J. Sound Vib., 37:327, 99. [] T. Sauer, J. A. Yorke, and M. Casdagli. Embedology. J. Stat. Phys., 65:579{66, 992. 4