ARTICLE IN PRESS
Physica E 40 (2007) 402–405 www.elsevier.com/locate/physe
Chaos–nonchaos phase transitions induced by multiplicative noise in ensembles of coupled two-dimensional oscillators Akihisa Ichiki, Hideaki Ito, Masatoshi Shiino Department of Physics, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8551, Japan Available online 23 June 2007
Abstract We study an analytically tractable model of ensembles of mean-field coupled two-dimensional limit-cycle oscillators to investigate the effects of noise on the dynamic behavior in the thermodynamic limit. The two-dimensional limit-cycle oscillator has nonlinearity only in the coupling part, and two kinds of noise, Langevin noise and noise introduced in the coupling strength, which is regarded as multiplicative noise, are dealt with. Under the influence of noise in the coupling strength, phase transitions involving chaos–nonchaos bifurcations are found to occur for an appropriate choice of the nonlinearity as the noise level is changed, while the usual Langevin noise dose not induce such bifurcations in our model. r 2007 Elsevier B.V. All rights reserved. PACS: 05.40.Ca; 05.45.a; 05.45.Xt Keywords: Coupling noise; Nonlinear Fokker–Planck equation; Chaos–nonchaos phase transition; Chaos in stochastic systems; Noise induced chaos; Exact time dependent solution to nonlinear Fokker–Planck equation
1. Introduction Dynamical behaviors of nonlinear systems subjected to additive or multiplicative noise have been extensively studied and attracted much attention in the fields of statistical physics. Recently effects of external noise on the behaviors of coupled nonlinear oscillators have become a matter of great concern. Various types of bifurcation phenomena including chaos–nonchaos phase transitions have been found. Some of those results imply that noise can assist coupled systems to exhibit non-trivial attractors [1–11], on the contrary to our intuition that the stochasticity of noise deteriorates the order of the system [1]. Studies of bifurcation phenomena of coupled nonlinear systems under the influence of noise have been conducted mostly by means of numerical simulations [8,12–16]. However, resorting to such approaches to understand bifurcation phenomena in the thermodynamic limit, one encounters some difficulties of time-consuming simulations Corresponding author.
E-mail address:
[email protected] (H. Ito). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.06.042
and point broadening in the space of order parameters such that the shape of theoretically obtained attractors are scattered and one cannot distinguish routes to chaos. To avoid such drawbacks of numerical approaches Shiino and Yoshida proposed analytically solvable models taking the form of analog neural networks to confirm that chaos–nonchaos phase transitions induced by noise really occur in the thermodynamic limit on the basis of a mean field approach [1]. The result is of interest for elucidating the role of noise in the bifurcation phenomena exhibited by coupled nonlinear oscillator systems in the thermodynamic limit. However, in the system they dealt with, chaotic behaviors are observed as the intrinsic properties of a single oscillator without noise. In the present paper, we focus on the occurrence of chaos that is caused by noise when the case with a single oscillator has nothing to do with chaotic behaviors. 2. Model and mean field dynamics We consider a system of N elements coupled via nonlinear mean-field interactions, each of which is subjected to independent additive external noise as well as
ARTICLE IN PRESS A. Ichiki et al. / Physica E 40 (2007) 402–405
independent noise in the coupling: 8 N P > > > x_ i ¼ ax xi þ J xij F x ðb1x xj þ b2x yj Þ þ Zxi ðtÞ; > < j N P > > > _ i ¼ ay yi þ J yij F y ðb1y xj þ b2y yj Þ þ Zyi ðtÞ; y > : j
(1)
where xi and yi are the x- and y-components of the oscillator at site i(i ¼ 1; . . . ; N) respectively, ak ; b1k ; b2k ðk ¼ x; yÞ are constants, F k represent nonlinear functions characterizing the nonlinear coupling and Zki the Langevin white noise with its intensity Dk : hZki ðtÞZlj ðt0 Þi ¼ 2Dk dij dkl dðt t0 Þ. We take the mean field coupling strength J kij : k J kij ¼ J¯ ij þ xkij ðtÞ ,
(2)
k where J¯ ij constants and xkij denotes Gaussian white noise with its intensity D~ k introduced in the coupling strength: D E xkij ðtÞxlmn ðt0 Þ ¼ 2D~ k =Ndim djn dkl dðt t0 Þ. As shown below
this noise as multiplicative noise may give rise to the chaos–nonchaos bifurcations. For simplicity we take a k ferromagnetic gauge J¯ ij ¼ J k =N. The deterministic dynamics of the one-body system for Eq. (1) (N ¼ 1) yields only limit cycles or fixed points. In the stochastic system for Eq. (1) with finite N the fine structure of the orbit of averaged physical quantities is, in general, obscured. On the other hand, in the large N limit, the coupled system may exhibit non-trivial attractors even under the influence of noise. However, the Langevin noise Zki solely does not bring about the chaotic behaviors in our mean-field (N ! 1) coupled two-dimensional limit-cycle oscillator system, where nonlinearity appears only in the coupling. We are concerned mainly with studying behaviors of the system driven by noise in the thermodynamic limit N ! 1, where we take advantage of using the nonlinear Fokker–Planck equation (nonlinear FPE) [1,17,18] corresponding to Eq. (1) for describing the temporal evolution of the empirical probability density: qPðt; x; yÞ q ¼ ½ðax x þ J x hF x iÞPðt; x; yÞ qt qx q ½ðay y þ J y hF y iÞPðt; x; yÞ qy 2 2 y q x q þ Deff 2 þ Deff 2 Pðt; x; yÞ, qx qy
ð3Þ
where Dkeff ¼ Dk þ D~ k hF 2k i. Imposing the self-averaging property, we have Z hF k i dx dy F k ðb1k x þ b2k yÞPðt; x; yÞ, Z dx dy F 2k ðb1k x þ b2k yÞPðt; x; yÞ. hF 2k i
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Then it follows the temporal evolution of the order parameters in terms of the moments: 8 d > > hxi ¼ ax hxi þ J x hF x i; > > > dt > > > d > > > > dthyi ¼ ay hyi þ J y hF y i; > > >
> > > d 2 > > > hv i ¼ 2ay hv2 i þ 2Dyeff ; > > dt > > > > d > > : huvi ¼ ðax þ ay Þhuvi; dt where hi denote averages with respect to the probability density Pðt; x; yÞ, and u x hxi, v y hyi. It should be noted that the noise intensity Dk are modified to the effective ones Dkeff as a consequence of the effects of coupling noise. It is also noted that Eq. (4) takes a closed form, which will turn out to suffice to deal with our system for sufficiently large times. Provided that the quantities hF k i and hF 2k i are known as a function of time, Eq. (3) becomes to be a quasi-linear FPE. Then we see that a specific solution of Eq. (3) takes a Gaussian form: 1 1 P~ G ðt; u; vÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp sT CðtÞ1 s , (5) 2 2p det CðtÞ where sT ¼ ðu; vÞ and the correlation matrix CðtÞ has the components cij ðtÞ ¼ hsi sj i ði; j ¼ 1; 2Þ, which are given as the set of solutions of the moment equations (4). One can easily prove an H-theorem that the Hfunctional Z P H½P; PG P ln dx dyX0 PG decreases monotonically with time, where PG ðt; x; yÞ P~ G ðt; x hxi; y hyiÞ. Thus the probability density Pðt; x; yÞ approaches the Gaussian probability density PG ðt; x; yÞ for sufficiently large times [1]. Note that unlike the case of the H-theorem for linear FPEs, the uniqueness of PG is not ensured. Although one can choose an arbitrary bounded function F k ðxÞ, we choose F k ðxÞ ¼ sin x for simplicity in this paper. Because of this choice one can calculate hF k i and hF 2k i analytically using PG ðt; x; yÞ to have 8 2 b1k 2 b22k 2 > > > hF hu hv i ¼ exp i b b huvi i k 1k 2k > > 2 2 > < sinðb1k hxi þ b2k hyiÞ; (6) > 2 2 2 1 2 2 > > hF i ¼ exp ½2b hu i 4b b huvi 2b hv i 1k 2k > k 1k 2k 2 > > : 12 cosð2b1k hxi þ 2b2k hyiÞ: Solving Eq. (4) together with Eq. (6) and the relation Dkeff ¼ Dk þ D~ k hF 2k i, we can investigate the effect of noise on the behaviors of the system in the thermodynamic limit after sufficiently large times.
ARTICLE IN PRESS A. Ichiki et al. / Physica E 40 (2007) 402–405
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3. Nonequilibrium phase transitions induced by noise 0.0008
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0.0006 0.0005 0.4
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Fig. 1. Plot of fixed-point-type and limit-cycle-type attractors of Eq. (4) in the deterministic limit. Filled circles represent stable fixed points, white circles unstable fixed points and filled triangles saddle node. Solid line shows the trajectory of the limit-cycle-type attractor. It is seen that fixed points are located regularly and the linear stability of each fixed point changes alternately.
0
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Fig. 2. Chaotic attractor in the case D~ x ¼ 0:001960. The three-dimensional attractor is projected to hxi-hyi plane in the upper graph. The lower graph illustrates an enlarged portion of the trajectory projected to hxi-hyi plane. The dynamical flow from the bottom left is found to switch between upper and lower branches chaotically.
1.4 1.2 1 0.8 0.6 0.4 0.2 0 −0.2
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Largest Lyapunov Exponent
It is easily seen that in Eq. (4) not the Langevin noise intensity Dk but the coupling noise intensity D~ k may give rise to the occurrence of chaos, because Eq. (4) turn out to be a three-dimensional or four-dimensional system after sufficiently large times only if the coupling noise is introduced to the system. Since we would like to investigate noise-induced chaos–nonchaos bifurcations, we focus on the effect of the coupling noise intensity D~ k to the system. For simplicity we set Dx ¼ Dy ¼ D~ y ¼ 0 below. In this paper we assume ax ¼ 0:5, ay ¼ 1:0, b1x ¼ 0:5, b2x ¼ 1:5, b1y ¼ 12:0, b2y ¼ 1:0, J x ¼ 18:0, and J y ¼ 30:0. With this parameter set, various types of bifurcations including chaos-nonchaos ones are indeed found to occur for appropriate values of D~ x as shown below. First, we consider the case with the deterministic limit ðD~ x ¼ 0Þ. With the set of parameters the system shows, in the two-dimensional hxi-hyi phase space, limit-cycle-type attractors which have six loops (solid line in Fig. 1). In addition Eq. (4) have a number of fixed points, which are arranged regularly and whose linear stability changes alternately (see also Fig. 1). Due to the arrangement of such many fixed points, the system has, roughly speaking, the sensitivity to initial conditions such that it may approach different attractors even if two initial conditions are close to each other. Next we investigate the effects of the coupling noise D~ x applied to the system. The system may exhibit chaotic attractors in the presence of coupling noise as we mentioned before and the chaotic behaviors are indeed observed. We find that the occurrence of chaotic attractors accompanies the change in the number of loops of limit-cycle-type attractors, which decreases as D~ x
0.00196 0.00198 0.002 0.00202 0.00204 0.00206 Strength of Coupling noise of x–component
Fig. 3. Plot of the largest Lyapunov exponents of the attractors (limit cycle and chaos) against D~ x . The graph depicts detail structures of the chaos-nonchaos bifurcations in the case where the number of the loops of the limit cycle switches from three to two.
increases: for D~ x t0:0001715 the limit-cycle-type attractor has six loops, and for D~ x \0:0001715 the system undergoes bifurcations repeatedly between limit cycle and chaos accompanying the decrease of the number of loops, and
ARTICLE IN PRESS A. Ichiki et al. / Physica E 40 (2007) 402–405
finally settles down to fixed-point-type attractors. We give one example of such chaotic attractors in Fig. 2, which appears when the number of loops of the limit-cycle-type attractor switches from 3 to 2. We conduct numerical computations for the largest Lyapunov exponents l of the system as a function of D~ x in Fig. 3.The attractor shown in Fig. 2, is indeed a chaotic one with l ¼ 0:891. The occurrence of the repeated transitions from limit-cycle-type attractor to chaotic one as well as those of the reverse direction are found as D~ x changes. We find that most of the transitions from limit cycle to chaos are of period-doubling type. 4. Conclusions We have shown the occurrence of nonequilibrium phase transitions induced by noise in the mean-field nonlinearly coupled two-dimensional oscillator system in the thermodynamic limit. It should be noted that the phase transitions involve chaos-nonchaos bifurcations which are found to occur only with noise introduced in the coupling strength. In the presence of such noise, we have found that the five variables hxi; hyi; hu2 i; hv2 i; huvi in Eq. (4) exhaustively as well as exactly describe the behaviors of the system for sufficiently large times. This is because, owing to the Htheorem, the empirical probability density as a solution of the nonlinear FPE becomes Gaussian. The occurrence of chaos induced by noise in the thermodynamic limit is considered to be generic phenomena. Details of the study will be reported elsewhere.
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Acknowledgment One of the authors (A.I.) was supported by the 21st Century COE Program at TokyoTech ‘‘Nanometer-Scale Quantum Physics’’ by the Ministry of Education, Culture, Sports, Science and Technology. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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