VOLUME 82, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 15 MARCH 1999

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Experimental Verification of Noise Induced Attractor Deformation

Martin Diestelhorst,1 Rainer Hegger,2 Lars Jaeger,2,* Holger Kantz,2 and Ralf-Peter Kapsch1,2

1Fachbereich Physik, Martin-Luther-Universität Halle-Wittenberg, Friedemann-Bach-Platz 6, 06108 Halle, Ger2Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, German

(Received 29 October 1998)

A periodically driven nonlinear electric resonance circuit with a ferroelectric inside the capacitanceshows the recently discovered effect of noise induced attractor deformations. By exposing the systemto dynamical noise we observe and quantitatively characterize the effect of attractor elongation dueto perturbations at homoclinic tangencies. The analysis of the experimental observations fits perfectlywith theoretical predictions. [S0031-9007(99)08661-5]

PACS numbers: 05.45.Ac, 05.40.Ca, 07.50.Ek

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Noise is ubiquitous in experiments and even, by noislike round-off errors, in numerical simulations. Sinceinfinitesimal perturbations are known to grow exponentially fast in deterministic chaotic systems, noise intefering with the dynamics (dynamical noise) can be ofconsiderable relevance in real systems. Thus the pioneing works on chaos in deterministic systems were sofollowed by research on the interaction between nonlinedynamics and stochastic processes. One relevant resugiven by the shadowing lemma of Bowen [1] and Anoso[1]: In hyperbolic systems, in the vicinity of each pseudoorbit generated by permanently perturbing the systemstate there is a solution of the unperturbed system (showing). The distance between this solution and the psedoorbit can be interpreted as measurement noise. Thif the purely deterministic system creates a fractal attrator, this fractality is smeared out on scales below the noilevel, but the attractor remains unchanged on length scalarger than the noise level. Hyperbolicity is given, wheat each point on the attractor the tangent space canuniquely decomposed into stable and unstable subspaand when these are everywhere transverse to each otHowever, almost all known model equations which cabe related to realistic physical systems are nonhyperboso that this theorem does not apply. Recently, it waobserved that different kinds of nonhyperbolicity, amonthem that introduced through homoclinic tangencies, doin fact destroy shadowability [2–4].

Homoclinic tangencies are points on the attractowhere the stable and unstable manifolds are tangento each other. Thus they do not span the tangespace, and perturbations transverse to these manifoldsneither grow nor shrink exponentially but are subject teffects related to the local structure in space. Thisvisualized in Fig. 1: Since distances along the attract(which coincides with the unstable manifold) grow undeforward iteration but shrink along the stable manifold, thlocal curvature at a homoclinic tangency diverges undforward iteration [5]. A continuation of this structureinto the vicinity of the attractor using the concept of thunstable foliation shows that at these points perturbatio

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can be amplified (by factors of 5 or more in typical mapsThus the attractor of the system subject to dynamicnoise is deformed at certain homoclinic tangencies (noinduced attractor deformation) [2].

In short, the predictions of the theory are the followingDynamical noise leads to a fattening of the whole attracat every point, and this is of the order of the noislevel. At the images of certain homoclinic tangencienamely, those where the sum of the curvatures of staand unstable manifold is minimal and which are calleprimary [5,6], an elongation of the attractor occurs. Thelongation is proportional to the noise level (at least fmaps with an almost one-dimensional attractor; for a recgeneralization, see [4]), and the proportionality factincreases monotonically with the number of iterates frothe primary tangency. The primary homoclinic tangenciare marked by a kind of symmetry under time reversPerturbations at a primary tangency are amplified bo

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FIG. 1. Structure of the stable and unstable foliation athomoclinic tangency and its images: The pointsA and B areattracted towards each other along the stable manifold; threpel each other along the arc of the unstable direction.perturbation (the arrow) is thus stretched and moved away frthe attractor (continuous line) (figure taken from [2]).

© 1999 The American Physical Society

VOLUME 82, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 15 MARCH 1999

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under forward and backward iteration. This property wbe exploited below to identify the primary tangencienumerically and is in fact a much better criterion than thone used previously, interpreting the curvatures.

The locations of attractor elongation can be predicted eperimentally by the superposition of attractors for slightdifferent parameter values, since parts of the attracwhere the noise induced elongations occur are also thwhere parameter changes modify the attractor most dmatically. This reflects the fact that nonhyperbolicity alsdestroys structural stability, i.e., the possibility to countebalance parameter changes by a change of coordinates

The experiment [8] was performed at an electric seriresonance circuit, consisting of a linear inductanceL ­ 100 mH and a nonlinear capacitanceCnl formedby a thin plate of ferroelectric Triglycine sulfate (TGS)with 7.31 3 4.66 mm2 electroded area and 0.28 mmthick. The temperature of the TGS capacitor waT ­ 308 K, i.e., below the phase transition temperatuof Tc ­ 322 K. The circuit was driven by a sinusoidavoltage of14.04 Vrms and a frequency of 2369 Hz, so thaswitching of the ferroelectric domains occurred. Undcertain simplifying assumptions one can derive a dampand driven Duffing equation as a model for the dynami[9]. Although the formation of domains inside the TGSand possible relaxation phenomena violate these assutions, there is very good evidence that a second ordordinary differential equation with a periodic driving termis capable of a full description of the observed dynami[10]. Thus a stroboscopic view of the experimentdata represents a two-dimensional section throughthree-dimensional phase space of the system withany projection.

In the experiment, the dielectric displacementD at thecapacitance and its temporal derivativeÙD through thecircuit were measured by the voltages across a large lincapacitance (forD) and a small resistance (forÙD) in serieswith the circuit (see Fig. 2 and Ref. [10]).

Sampling both signals exactly with the driving frequency of the sinusoidal voltage using the externclock mode of the digital oscilloscope Nicolet Pro3and the synchronous output signal of a function genetor HP33120A, which is phase locked with the drivingenerator HP3325B, stroboscopic sections of the attracwere recorded. Phase locking and tuning the phasetween driving voltage and the synchronous signal of tHP33120A allowed the observation of different stroboscopic sections of the same attractor. So it was possito look experimentally for the most promising Poincarsections, where a slight increase of the magnitude ofdriving voltage caused the largest elongation of the attrator. Stochastic noises of adjustable amplitudes and almuniform distribution, generated by a second HP33120generator, were added to the driving voltage and thus cpled into the system. In Fig. 3 we show a series of strboscopic views with increasing noise levels. The effe

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FIG. 2. The nonlinear series resonance circuit including telements for measuring the dielectric displacementD and itstemporal derivativeÙD.

of noise induced attractor deformation can be clearly sein the left parts of the attractor: The attractor is stretchmore than it is fattened. For large noise levels, these elgations are curved, merging for even more noise.

In Fig. 4 we show the elongation of these two arm(extracted from plots of the attractors) as a functioof the noise amplitude. The result is in agreemewith the predicted proportionality between elongatioand noise level. The factor of proportionality will becalled stretching factor in the following. For large noisamplitudes rather large errors are induced by the finnumber of points representing quite thin tails, whichtaken into account by the error bars. The lowest curshows the “fattening” of the attractor due to noise anthus a kind of effective noise level. The upper left armthe attractor grows with a stretching factor of 5.5 (centrcurve) and the lower left with a factor of 7 (upper curvefaster than the attractor gets fatter.

In the stroboscopic view, our data represent a twdimensional map. We can approximate the dynamicslocal linear models [11]

$xn11 ­ An $xn 1 $bn , (1)

whereAn and $bn are designed to minimize the one-steprediction error

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computed for a small neighborhoodUn of $xn. ThematricesAn are identical to the Jacobians at$xn of thestroboscopic map. Define$e1 to be the eigenvector of thesmallest eigenvalue ofCf ­ s

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and $e2 to be the eigenvector of the smallest eigenvalof Cb ­ s

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VOLUME 82, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 15 MARCH 1999

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FIG. 3. Stroboscopic views of the experimental data: Noisfree (a),3% (b), and6% (c) of noise coupled into the dynamics.Noise level in the root mean square sense in relation to tamplitude of the driving voltage. The bars perpendicular to thattractor denote the left endings of the unperturbed attractor.

iteration of the inverse Jacobians). As discussed in [2$e1 converges fast (inl) towards the tangent of thestable manifold at$xn, whereas$e2 converges towards thetangent of the unstable manifold in$xn. At a homoclinictangency both vectors are parallel, or, in other words, tmost expanding direction of

Qnk­n2l Ak is perpendicular

to the attractor. At these points, the largest eigenvalu

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FIG. 4. The elongation (in arbitrary units) of the two left parof the attractor (with respect to the unperturbed attractor shoin Fig. 3a) and its fattening as a function of the noise amplituin percent of the driving voltage (curves from top to bottomlower left arm, upper left arm, fattening).

of Cf and Cb give the squares of the expansion ratof perturbations off the attractor under forward anbackward iteration, respectively. They allow onecharacterize whether a homoclinc tangency is primarypreimage, or an image, and the largest eigenvalue ofCf

predicts (if an image) the noise amplification. Primatangencies are characterized by the fact that the stretcfactor is larger than unity both forward and backwain time, whereas for nonprimary tangencies it is eithforward larger and backward smaller than unity (imagof the primary tangency) or forward smaller and backwalarger than unity (preimages).

In Fig. 5 we show again the experimental stroboscoview of the noiseless attractor together with differecandidates for homoclinic tangencies obtained fromexperimental data. For each point of the time seriescomputed$e1 and $e2 and plotted a filled circle when theangle between them is smaller thanpy100 and if they

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FIG. 5. The attractor together with the numerically detemined homoclinic tangencies.

VOLUME 82, NUMBER 11 P H Y S I C A L R E V I E W L E T T E R S 15 MARCH 1999

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simultaneously fulfill the condition to be primary, i.e.,when both stretching factors are larger than unity. Winclude their two images and one preimage, which bconstruction are also (close to) homoclinic tangencies.this way the information about the dynamics in tangenspace of the noiseless data yields stretching factors of 6the upper and7.2 for the lower arm. This is in reasonableagreement with the values 5.5 and 7 which we extravisually from the noisy attractors and are reported in Fig.

These results prove that the noise amplification dooccur at homoclinic tangencies, it does occur at the firand second images of the primary tangencies, and this no amplification at the primary tangencies or its preimages. Moreover, we do not find any more positionwhere noise induced attractor deformation should occusince the next images of the homoclinic tangencies aall located inside the attractor. This is a result of stronfolding: In a noiseless representation we expectsee a fractal structure with more arms, but these armare extremely close to the prominent lower part of thattractor and cannot be resolved any more. For larnoise levels, also the lower arm merges with the uppone, making it impossible to estimate its length any mo(reason for the truncation of the upper curve in Fig. 4).

To summarize, it is shown that nonhyperbolicity haeffects that are experimentally verifiable. Certain locof nonhyperbolic structure, the images of primary homoclinic tangencies, give rise to elongations of the attractwhen the deterministic dynamics is perturbed by noisMoreover, the theoretical concept of “primary homoclinictangency” becomes relevant for the interpretation of thexperimental findings, since without this concept (anthe theory behind it) one could not understand why thright parts of the attractor are not elongated. In [4]scaling law dmax . e1yD1 was derived. In this law,d

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corresponds to what we call elongation,e is the noiseamplitude, andD1 is the information dimension of the at-tractor. Our Fig. 4 suggests the validity of this law withD1 ø 1, which is in agreement with the fact that the flowattractor is close to two dimensional and thus the strobscopic attractor is almost one dimensional.

*Present address: Olsen & Associates, Zürich, Switzeland.

[1] B. V. Anosov, Proc. Steklov Inst. Math.90, 1 (1967);R. Bowen, J. Differential Equations18, 333 (1975).

[2] L. Jaeger and H. Kantz, Physica (Amsterdam)105D,79–96 (1997).

[3] C. Grebogi, S. M. Hammel, J. Yorke, and T. Sauer, PhysRev. Lett. 65, 1527 (1990); S. Dawson, C. Grebogi,T. Sauer, and J. A. Yorke, Phys. Rev. Lett.73, 1927(1994).

[4] C. G. Schroer, E. Ott, and J. A. Yorke, Phys. Rev. Lett.81,1397–1400 (1998).

[5] F. Giovannini and A. Politi, J. Phys. A24, 1837 (1991).[6] P. Grassberger and H. Kantz, Phys. Lett.113A, 235

(1985).[7] For example, J. Guckenheimer and P. Holmes,Nonlinear

Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (Springer, New York, 1983).

[8] M. Diestelhorst and H. Beige, Ferroelectrics81, 15(1988); E. Brauer, H. Beige, L. Flepp, and M. Klee, PhasTransitions42, 167 (1993).

[9] H. Beige, M. Diestelhorst, R. Forster, and T. KrietschPhase Transitions37, 312 (1992).

[10] R. Hegger, H. Kantz, F. Schmüser, M. DiestelhorstR.-P. Kapsch, and H. Beige, Chaos8, 727–736 (1998).

[11] For example, J. D. Farmer and J. J. Sidorowich, PhyRev. Lett. 59, 845 (1987); J. P. Crutchfield andB. S. McNamara, Complex Systems1, 417 (1987).

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