Jumping particle model. A study of the phase space of a non-linear dynamical system below its transition to chaos

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1 Jumping particle model. A study of the phase space of a nonlinear dynamical system below its transition to chaos Pi. Pierański, Z. Kowalik, M. Franaszek To cite this version: Pi. Pierański, Z. Kowalik, M. Franaszek. Jumping particle model. A study of the phase space of a nonlinear dynamical system below its transition to chaos. Journal de Physique, 1985, 46 (5), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1985 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 On Bifurcation J. Physique 46 (1985) MAI 1985, 681 Classification Physics Abstracts Jumping particle model. A study of the phase space of a nonlinear dynamical system below its transition to chaos (+) Pi. Piera0144ski (*), Z. Kowalik and M. Franaszek Institute of Molecular Physics, Smoluchowskiego 17, Poznan, Poland (Reçu le 16 mai 1983, révisé le 28 novembre 1984, accepti le 3 janvier 1985) 2014 Résumé. observe directement les cascades de bifurcations d un système dynamique non linéaire. Les résultats expérimentaux sont comparés aux prédictions de l application de ZaslavskijRachko. On montre la coexistence d un certain nombre de modes individuels Abstract. cascades in a nonlinear dynamical system are directly observed. Experimental results are compared with predictions of the ZaslavskijRachko mapping. Experimental evidence for coexistence of a number of individual modes is provided. 1. Introduction. Period doubling [1] and intermittency [2] are the two basic phenomena which, as predicted theoretically, were found to bridge chaos and order in a number of experimental systems [3] which display a transition from periodic to chaotic behaviour (1). It is the aim of the present paper to describe an experimental study of the phase space of a nonlinear dynamical system, whose simplicity enables one to interpret in plain terms all phenomena which appear in the system before it enters the region of chaos. 2. ExperimentaL The experimental model we designed, described in detail previously [5], can be seen as a practical realization of a modification of the FermiUlam thought experiment [4]. The model (see Fig. 1) consists of a (+) Work carried out under Project MRI.9. (*) Present address : Universite ParisSud, Laboratoire de Physique des Solides, Batiment 510, Orsay, France. (1) It seems worth reminding, however, that real physical systems may reach the chaotic regime in ways which are different from the two abovementioned scenarios. For instance, recent systematic studies of the RayleighBenard instability performed by M. Dubois, P. Berg6 and V. Croquette [10] with the use of an ingenious technique (which enables one to observe directly the Poincare section of the phase space of the system) show that the hydrodynamical system may reach its chaotic regime via a biperiodic oscillation. steel sphere jumping perpendicularly on a slightly concave, horizontal surface of a lens made to vibrate harmonically by a loudspeaker to the membrane of which the lens is fixed. The loudspeaker is supplied with a sineshaped signal from an audio generator at a frequency v 101 : 102 Hz. Complex AC signals from the microphone, which «listens to the vibrating surfacesteel sphere collision sounds, are shaped into identical pulses which modulate the electron beam intensity (Zcoordinate of the beam). Thus, each collision is marked on the oscilloscope screen by a brighter spot. The recording system is provided by a sawtooth generator synchronized by the main audio generator. The piecewise linear signal from the sawtooth generator controls the Ycoordinate of the oscilloscope beam; thus (at an appropriate choice of the synchronization phase) the Ycoordinate follows the actual phase 90 [mod 2 n] (where, 0 2 nvt, t being the real time) of the surface velocity function A sin 0. As a result, the Ycoordinate of the bright spot which marks the Oi collision moment is proportional to 0i. Obviously, when the collision sequence is periodic (and commensurate with the surface vibration frequency) one observes on the oscilloscope screen a standing set of points the number and distribution of which represents completely the sequence. To record behaviour of a jumping mode versus the vibration amplitude A, the Xcoordinate of the beam is controlled by an AC/DC converter supplying a DC voltage proportional to A. Thus, using a slow sweep Article published online by EDP Sciences and available at

3 Experimental,,, 682 Fig. 1. setup. Description in text. of the amplitude one is able to study the history of a jumping mode versus A. Photographs presented in figures 2 and 3 were obtained using this technique. The intensity of the oscilloscope.image and parameters of the photographic exposure were chosen such that a clear recording of the collision sequence was obtained only when the sequence was periodic; any chaotic sequence of collisions produced an image too dispersed to be recorded by the photographic film. 3. Results. It has been found that in the experimental conditions we created the sphere can easily be put into a number of jumping modes, whose evolution versus A is recorded in figure 2. The simplest mode M( ) recorded in figure 2a consists of a sequence of equidistant (in time) jumps between consecutive maxima of the surface displacement function, i.e. its period T is equal to that of the surface vibration. Consecutive collision moments Oi are located at a phase 6 of the surface vibration such that the amount of momentum gained by the particle during the collisions compensates its losses due to dissipation (inelastic collisions, viscous friction during the particle s flight, etc...). Elementary analysis shows [5], that the M(l) mode can be excited and sustained but at 9 e (7c/2,7r). Consequently the M(l) mode is described by relation : e(l){a) denotes the shift of the collision phase from % n, where it would be located in absence of any dissipation. Obviously, there exists a minimum value A 0 ( ) of the surface vibration amplitude A below which the M(l) mode cannot be sustained. When A + AÓ1) (from above) then e(1 )(A) + n/2, i.e. the collision phase shifts to the point of maximum surface velocity. Below the point the M(l) mode must decay. As seen in figure 2a, the M(1) mode is bounded also from above. Beyond A A11) the mode becomes unstable giving way to its modulated version M(1,2) in which a jump shorter than T is followed by a longer jump. The periodicy in 2 T is preserved. The M(1,2) mode is described by :. where 01, ljj2 > 0. The splitting ql i + ljj 2 increases with A and at Ai 1) the M (1, 2) mode becomes unstable, bifurcating into the second order modulation mode M(1,2,2) the periodicity of which is preserved in 4 T. The story repeats itself once more, but the third period doubling occurring at an Aá1) results in a mode which proves to be on the verge of destruction by noise. The fourth bifurcation has never been observed; instead the system enters the region of chaos. A simple reasoning convinces one that the equidistant jumps of the particle may take place not only between consecutive, but also between every second, every third etc. periods of the surface vibration. Figures 2b and 2c provide evidence for these kinds of equidistant jumping modes. A strong enough perturbation (e. g. a knock at the loudspeaker membrane) may drive the particle trajectory from the attractor of the

4 Bifurcation experimental M(1,3) experimental an 683 Fig. 2. diagrams of the basic jumping modes : M(1), M(2), M(3); a, b, c recordings; a, b, c theoretical plots calculated for k Arrows in the experimental recordings indicate positions of the bifurcation points as predicted by numerical calculations. M(1) mode into basins of its 2 T or 3 T versions : M(2) or M(3). Due to the doubled (or triple) momentum with which the particle arrives, in these modes, at the vibrating surface dissipation losses are proportionally higher and, consequently, the collision phase must be shifted to phases of higher surface velocity, i.e. closer to 9 n12. Figures 2b and 2c indicate clearly that this is the case. Energetic collisions which take place in the high velocity modes perturb considerably the sineshaped motion of the collision surface (generally, the level of noise increases) which destroys some of the more subtle versions of the modes. Thus, as seen in figure 2b second bifurcation vanishes from the diagram of the M(2) mode, while in the case of the M(3) mode (Fig. 2c) even the first bifurcation is hardly visible. Let us point out that diagrams of the M(2 ) and M(3) modes are very difficult to record in a single sweep, modulation mode; a Fig. 3. overall view. Experimental recording obtained for decreasing A ; b, c, b, c and theoretical plots of the M(1,3) mode under decreasing and increasing vibration amplitude. Directions of the A sweep are indicated by arrows in the b and c plots. Theoretical curves have been calculated for k 0.86.

5 684 since one can never start the sweep from the AÓ1) threshold, where the modes decay, but the starting value of A must be located in a safe distance from it. Thus, diagrams presented in figures 2b and 2c have their initial parts cut off. The diagram of the M(1) mode presented in figure 2a has been recorded for decreasing A. This is possible, since sometimes the particle enters the M(1) solution spontaneously. Equidistant jumping modes M(1), M(2), M(3) and their period doubled versions were not the only modes we observed. Figure 3 presents evidence for a different solution. Namely, before the M(1) mode becomes unstable (giving way to its period doubled version M(1,2») another possibility appears. The particle does not take this route spontaneously but a patient persuasion (knocking at the loudspeaker membrane) forces it to do so. As shown in the figure the new mode denoted by M(1,3) can be seen as a periodthree modulation of the basic M(1) mode. Moments of consecutive collisions in this mode are given by where As seen in figure 3c, the new mode follows its own period doubling route, from which we observe only the first step i.e. the M(1,2,3) mode. The Mini,3,..) mode is obviously metastable and, as soon as its period doubling route ends, the particle comes back to the main M(l) mode, which in the meantime has bifurcated on its own. Figure 3 explains clearly why we do not observe the particle to enter the M(1,3) solution spontaneously. As seen in the figure, branches of the mode appear in a jumplike manner and an energetic fluctuation is needed to cross the gap. (The M(2,3) mode has been also observed.) and from above) all equidistant jumping modes M(a) : where a denotes the number of consecutive surface vibration periods over which the particle jumps in the M(a) mode. Figure 4 presents a plot of the r(1) ;.B1)/ A( ) ratio, which can be conveniently used to determine the value of the dissipation parameter k. As indicated in the figure, F (1) 4.06 (found from data recorded in Fig. 2a) leads to k The simplicity of equation (4) enables one to simulate numerically any jumping mode and its evolution versus A. Consequently, the bifurcation diagram of any M(a) mode can be easily obtained. Figures 2a, b, c present such diagrams calculated for k The same value of k has been also used in calculating plots presented in figure 3. As seen in the figures, good qualitative agreement has been obtained in all cases. Two significant discrepancies seem worth pointing out : 1) The difference in splitting of the upper and lower branches of the M(1,2,2) mode is much less distinct in the experimental diagram than in its theoretical counterpart, where it is striking. 2) The two lower branches of the M(1,3) mode apparently cross each other in the experimental recording, while no such effect is seen in the theoretical plot. It seems that the discrepancies may stem from an essential simplification which one makes describing the experimental model by mapping (4). Namely, in the experiment the collision surface truly vibrates while equations of mapping (4) describe a thought experiment in which the collision surface provides momentum to the particle without changing its own position. Whether this is the proper explanation we do not know at present, but numerical calculations we are performing should answer the question soon. 4. Theoretical. area As shown before [5], the jumping particle model may be approximated well by a dissipative (i.e. contracting) version of the standard mapping : which originating from the old FermiUlam acceleration problem [4] has been modified in a number of ways [6] to cope with a whole variety of different problems in nonlinear dynamics. In its dissipative version the mapping is often referred to as the Zaslavskij Rachko mapping [7]. Elementary algebra provides expressions for the A( )(k) and A( )(k) thresholds which limit (from below Fig. 4. r(1)(k) and b(l)(k) plots calculated numerically from equation (5). Arrows indicate the experimental value of r (1) and resulting value of k.

6 A/2) 685 Three consecutive bifurcations seen in figure 2a enable one to determine the first Feigenbaum ratio 6 (1) (A(l) À.B1»)/(À.1) A( )). As shown in figure 4, the ratio depends strongly on k and for k 0.86 (determined from the r(1)(k) dependence) 6(, ) Arrows in figure 2a indicate the values of A at which the second and third bifurcations should take place (position A(1) of the first bifurcation has been used to fix the scale of the theoretical drawings). As seen, the agreement is not bad. The value of k we determined in the present study differs considerably from that determined previously (k 0.2 [5]). The difference is, however, well justified since the experimental system has been redesigned so as to reduce dissipation : we applied a bigger loudspeaker (diameter of the membrane 0 25 cm, previously 4> 10 cm) and a smaller steel sphere (diameter Q 3 mm, previously J 4 mm). Results of the experimental study of the M(1,3) mode we report in the present paper are in agreement with what we intuitively predicted previously [5]. Namely, as A increases in the k 1 areapreserving mapping, the (7r, z) centre of its lji+ 1 (lj) map gives birth to consecutive modulations of the main jumping mode (represented by the centre). The period L of the newly born modulations is given by : L 2 n/arccos (1 (7) thus, as increases, L shortens (e.g. for 1, 2, 3, 4, L 5, 4, 3, 2 respectively). As the (n, n) centre gives birth to a new modulation mode; those born previously (seen in the map as eliptic orbits surrounding the (n, n) centre) move away from it. The fate of a modulation mode depends on whether it is commensurate or not. Those of a low order commensurability soon fix their phase turning into belts of smaller islands which represent secondary modulations imposed on the primary one. For instance, the L 6 mode born at A 1.5 turns into a well visible belt of six islands located on the periphery of the main island (see Fig. 1 b in [5]). When the dissipation is switched on (k 1), the mapping (4) becomes areacontracting and the (J i + 1 «(J i) map loses its periodicity. Elliptic orbits turn into spirals converging onto centres of islands point attractors with their basins are formed. Not all commensurate modulations survive the process. Everything depends on their order, distance from the (n, n) centre, dimensions of their own island and, of course, the level of dissipation. For k 0.86 which we managed to reach under the present experimental conditions, the M (1,6) mode seen in figure 1 b of reference [5], has not been observed; apparently, at this level of dissipation the mode has been destroyed. In general, effects of the dissipation on the landscape of the k 1, ()i+ 1 «() ) maps are far from trivial and require careful analysis. 5. Conclusions. It seems that the o la bille cahotante» (2) model we have presented is one of the simplest physical objects in which the period doubling cascade can be demonstrated [8]. Due to the audio frequency range in which the bifurcating jumping modes are located, consecutive bifurcations can actually be heard. This is of educational value, since a few of the first period doubling threshold can be determined without any sophisticated equipment enabling one to calculate both the dissipation factor k and the first convergence ratio 6"), though due to effects of noise (induced in the system by the jumping itself) accuracy with which the latter value is determined is not high. The model, in general, is not well suited for quantitative studies of the universal aspects of the period doubling behaviour. In particular, there is no chance of reaching such stages of the bifurcation cascade at which the convergence ratio would come close to its universal limit 61D The coincidence of the experimental value of 6 1 we determined previously [5] with the universal limit b 1 must be seen as an effect of high dissipation. On the other hand, the effects of noise are interesting on their own [9], since no doubt they are present in any natural mechanical system which displays the period doubling transition to a chaotic behaviour. Acknowledgments. We are indebted to Prof. J. Malecki for many stimulating discussions. A generous grant of computer time from the Institute of Nuclear Physics and Technology, AGH, Krakow, is gratefully acknowledged. In particular, we thank Prof K. Przewlocki for his support. One of us (P. P.) would like to thank his brother Pawel and other workers from Laboratoire de Physique des Solides, Orsay, and Service de Physique du Solide et de Resonance Magn6tique, GifsurYvette, for their tiorts in keeping our scientific contacts alive. (2) We are grateful to P. Berge for this apt term.

7 686 References [1] FEIGENBAUM, M., J. Stat. Phys. 19 (1978) 25. [2] MANNEVILLE, P., POMEAU, Y., Phys. Lett. 75A (1979) 1. [3] See references [3] to [6] in [5]. [4] See e.g. : ZASLAVSKIJ, G. M., Statistical Irreversibility in Nonlinear systems (in Russian), Nauka, Moscow (1970). [5] PIERA0144SKI, P., J. Physique 44 (1983) 573. [6] LICHTENBERG, A. J., LIEBERMAN, M. A., Physica 10 (1980) 291. [7] ZASLAVSKIJ, G. M., RACHKO, K. R., JETF 76 (1979) [8] CROQUETTE, V., Pour la Science, December (1982). [9] CRUTCHFIELD, J. P., FARMER, J. D., HUBERMAN, B. A., Phys. Rep. 92 (1982) 46. [10] DUBOIS, M., BERGÉ, P., CROQUETTE, V., J. Physique Lett. 43 (1982) L295.