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

HAL Id: jpa-00210008

https://hal.archives-ouvertes.fr/jpa-00210008

Submitted on 1 Jan 1985

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Jumping particle model. A study of the phase space of a non-linear 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 non- linear dynamical system below its transition to chaos. Journal de Physique, 1985, 46 (5), pp.681-686.

�10.1051/jphys:01985004605068100�. �jpa-00210008�

Jumping particle ^{model. A} study ^{of the} phase space of a non-linear

dynamical system below its transition to chaos (+)

Pi. Piera0144ski (*), ^{Z. Kowalik} and M. Franaszek

Institute of Molecular Physics, Smoluchowskiego ^{17, 60-179 Poznan, Poland} (Reçu ^{le 16 mai} 1983, révisé le 28 novembre 1984, accepti ^{le 3} janvier 1985)

Résumé.

On observe directement les cascades de bifurcations d’un système dynamique

linéaire. Les résultats

expérimentaux ^sont comparés

prédictions ^de l’application ^de Zaslavskij-Rachko. ^{On montre} la coexistence d’un certain nombre de modes individuels.

Abstract.

Bifurcation cascades in

non-linear dynamical system ^are directly ^observed. Experimental ^{results are} compared ^with predictions ^{of the} Zaslavskij-Rachko mapping. Experimental evidence for coexistence of

number of individual modes is provided.

Classification Physics ^Abstracts

03.20

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 expe- rimental study ^{of the} phase space of a non-linear dyna-

mical system, ^whose simplicity ^{enables one to} interpret

in plain ^{terms all} phenomena ^{which appear in the sys-}

tem 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 ^realiza-

tion of a modification of the Fermi-Ulam thought experiment [4]. ^{The model} (see Fig. 1) consists of a

(+) Work ^{carried out under} Project ^MR-I.9.

(*) Present address : Universite Paris-Sud, Laboratoire de Physique ^des Solides, ^Batiment 510, 91405 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 above-mentioned scenarios. For

instance, ^recent systematic ^{studies of the} Rayleigh-Benard instability performed by ^M. Dubois, ^P. Berg6 and V. Cro- quette [10] ^{with the}

^of

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

biperiodic ^oscilla-

tion.

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 sine-shaped signal ^{from an audio} generator ^{at a} frequency ^{v - 101 : 102 Hz.}

Complex ^AC signals ^{from the} microphone, ^which

«listens to the vibrating surface-steel sphere ^colli-

sion sounds, ^are shaped ^{into identical} pulses ^which

modulate the electron beam intensity (Z-coordinate

of the beam). Thus, each collision is marked on the

oscilloscope ^screen by ^a brighter spot. ^The recording system ^is provided by ^{a saw-tooth} generator synchro-

nized by the main audio generator. ^The piece-wise

linear signal ^from the saw-tooth generator controls the Y-coordinate of the oscilloscope ^{beam; thus} (at ^an appropriate choice of the synchronization phase) ^the

Y-coordinate follows the actual phase 9=0 [mod ² n]

(where, 0 = 2 nvt, t being ^{the real} time) ^{of the surface} velocity function - A sin 0. As a result, ^{the Y-coor-}

dinate of the bright spot which marks the Oi ^collision

moment is proportional ^to 0-i. 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 comple- tely ^the sequence.

To record behaviour of a jumping ^{mode versus the}

vibration amplitude ^A, the X-coordinate 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 http://dx.doi.org/10.1051/jphys:01985004605068100

682

Fig. ^1.

Experimental set-up. 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 para- meters 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 con-}

sists of a sequence of equidistant (in time) jumps

between consecutive maxima of the surface displace-

ment 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

⁺ 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} equi-

distant 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 perturba-

tion (e. g. ^a knock at the loudspeaker membrane) may

drive the particle trajectory ^{from the attractor of the}

Fig. ^2.

Bifurcation diagrams of the basic jumping ^{modes :} M(1), M(2), M(3); a, b,

experimental recordings; a’, b’, ^c’

theoretical plots calculated for k

0.86. 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} propor-

tionally 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 sine-shaped

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,

- Fig. ^3.

^M(1,3) modulation mode;

^{overall view.}

Experimental recording obtained for decreasing A ; b, c, b’,

c’

experimental ^and theoretical plots ^{of the M(1,3) mode}

under decreasing ^and increasing ^vibration amplitude. ^Direc-

tions of the A sweep

indicated by ^arrows in the b’ and c’

plots. Theoretical curves have been calculated for k

0.86.

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 diffe- rent 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 persua- sion (knocking ^{at the} loudspeaker membrane) ^forces

it to do so. As shown in the figure ^{the new} mode denot- ed by ^{M(1,3) can be seen as a} period-three ^modulation

of the basic M(1) mode. Moments of consecutive colli- sions 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 bifurcat- ed on its own. Figure ³ explains clearly why ^{we do not}

observe the particle ^{to enter the M(1,3) solution} spon-

taneously. ^{As seen in the} figure, branches of the mode appear in ^a jump-like ^{manner and an} energetic ^fluctua-

tion is needed to cross the _gap. (The ^{M(2,3) mode has}

been also observed.)

4. Theoretical.

As shown before [5], ^the jumping particle model may be approximated ^well by ^a dissipative (i.e. ^area- contracting) ^{version of} the standard mapping :

which originating from the old Fermi-Ulam accelera- tion problem [4] has been modified in a number of ways [6] ^to cope with a whole variety ^{of different} pro- blems in non-linear 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

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

0.86.

The simplicity of equation (4) ^{enables one to simulate} numerically any jumping mode and its evolution ver- sus A. Consequently, the bifurcation diagram ^of any M(a) mode can be easily ^obtained. Figures 2a’, b’, ^c’

present ^such diagrams calculated for k

0.86. 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 ^discre- pancies ^{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 ^record- ing, ^{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 experi-

ment in which the collision surface provides ^momen-

tum 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} perform- ing ^{should answer the} question ^soon.

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.

Feigenbaum 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(,’) ^7.15.

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 ^loud- speaker (diameter of the membrane 0

25 _{cm, pre-}

viously 4>

¹⁰ cm) ^{and a} smaller steel sphere (dia-

meter 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 area-preserving 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

A/2) (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 _pre-

viously (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 commensu- rate 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 area-contracting ^{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 com- mensurate modulations survive the process. Every- thing 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 manag-

figure [5],

not been observed; apparently, ^at this level of dissi-

pation 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 demons-}

trated [8]. ^{Due to the audio} frequency range in which the bifurcating jumping ^{modes are located, consecu-}

tive bifurcations can actually be heard. This is of educa- tional value, ^{since a} few of the first period doubling

threshold can be determined without _any sophisticated equipment enabling ^{one to} calculate both the dissipa-

tion 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 univer- sal aspects ^{of the} period doubling behaviour. In parti- cular, ^{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

4.67.

The coincidence of the experimental ^{value of} 61

^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 stimulat-

ing discussions. A _generous grant ^of computer ^time from the Institute of Nuclear Physics ^and Technology, AGH, Krakow, ^is gratefully acknowledged. ^In parti- cular, ^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 Phy- sique ^des Solides, Orsay, and Service de Physique ^du

Solide et de Resonance Magn6tique, Gif-sur-Yvette, for their -tiorts in keeping ^our scientific contacts alive.

(2) We

grateful ^{to P.} Berge ^{for this} apt ^term.

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, ^Mos-

cow (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)

2052.

[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) ^L-295.