We study the noise-induced synchronization in a system of particles moving in Fahy–Hamann potential [S. Fahy, D.R. Hamann,Phys.Rev.Lett. 69, 761 (1992)] and subjected to generalized Langevin forces. We investi- gate the synchronization dependence on system’s parameters and on mem- ory range. The results show that while in general memory acts against synchronization, for intermediate memory ranges the opposite effect can be observed. Generally the synchronization transition is found to depend on memory range, temperature and dissipation in the system. PAC

Cieśla, Michał

Longa, Lech

Acta Physica Polonica B

Jagiellonian Univeristy Repository

Vol. 38 (2007) ACTA PHYSICA POLONICA B No 5NOISE-INDUCED SYNCHRONIZATIONIN THE FAHY–HAMANN MODEL∗Michał Cieśla, Lech LongaM. Smoluchowski Institute of Physics, Department of Statistical Physicsand Mark Kac Complex Systems Research Center, Jagellonian UniversityReymonta 4, 31-059 Kraków, Poland(Received February 23, 2007)We study the noise-induced synchronization in a system of particlesmoving in Fahy–Hamann potential [S. Fahy, D.R. Hamann,Phys. Rev. Lett.69, 761 (1992)] and subjected to generalized Langevin forces. We investi-gate the synchronization dependence on system’s parameters and on mem-ory range. The results show that while in general memory acts againstsynchronization, for intermediate memory ranges the opposite eﬀect canbe observed. Generally the synchronization transition is found to dependon memory range, temperature and dissipation in the system.PACS numbers: 05.45.Xt, 05.45.Jn1. IntroductionChaotic systems shows extreme sensitivity to even a minor perturbationof initial conditions. Two trajectories very close to each other usually di-verge exponentially in time, which makes long-time evolution unpredictable.That is, identical systems evolving with different initial conditions will notsynchronize their trajectories. The situation may change, however, whenthe systems get coupled e.g. by a common signal. Even a random resettingof velocities in an ensemble of particles with different initial conditions canforce synchronization, as shown by Fahy and Hamann (FH) in 1992 [1].Since then coherence of dynamical processes has become an active fieldof research with examples found in physical, biological, chemical and socialsystems [2–9]. At least four types of synchronization scenarios have beenidentified [10–13] of which the synchronization of identical systems coupledby a common noise has received much attention due to its relative simplicityand importance [13, 14].∗ Presented at the XIX Marian Smoluchowski Symposium on Statistical Physics,Kraków, Poland, May 14–17, 2006.(1719)1720 M. Cieśla, L. LongaUsually we describe chaotic systems by Markovian-type of equations,but realistic time evolution, due to collective effects is, in general, non-Markovian, which inevitably requires memory (time-delayed interactions) tobe taken into account [15–18]. Systematic studies of isolated, time-delayedinteractions were initialized by Schuster and Wagner [19]. They consideredtwo coupled phase oscillators and found multi-stability of synchronized so-lutions. Since then, delayed interactions have been analyzed in differentcontexts [20–23]. The most interesting recent development in this regardare studies showing (a) enhancement of neural synchrony in a network ofcoupled oscillators involving time delays [21] and (b) demonstration of syn-chronization in an ensemble of coupled, chaotic logistic maps with randomdelay times [23].In this paper we demonstrate a constructive influence of memory on thenoise-induced synchronization. To this end we consider generalized Langevindynamics of Fahy–Hamann particles, for which the limit of vanishing mem-ory is well understood.2. ModelWe explore influence of memory on the noise-induced synchronizationby generalizing the Fahy–Hamann studies [1]. Our system comprises of twoidentical, independent particles in a two dimensional potential well given byV (x1, x2) =sin 2pix12pix1+sin 2pix22pix2+(x21 + x22)216pi2, (1)which is sketched in Fig. 1. The motion of the i-th particle (i = 1, 2) isgoverned by the generalized Langevin equation:Fig. 1. Fahy–Hamann potential.Noise-Induced Synchronization in the Fahy–Hamann Model 1721mx¨iα(t) = −∂V (xi1, xi2)∂xiα−mγt∫Γ (t− t′) x˙iα(t′) dt′ + ξ(t) , (2)where m is the particle’s mass, γ is the friction constant, also playingthe role of memory intensity parameter, T is the absolute temperatureand kB is the Boltzmann constant. The particles evolve subject to dif-ferent initial conditions. They are coupled by the common noise ξ, whichis a Γ -correlated stochastic force with zero mean and correlations obeyingfluctuation-dissipation theorem〈〈 ξ(t)ξ(t′) 〉〉 = 2mγkBT Γ (t− t′) . (3)In what follows we restrict ourselves to the exponentially correlated noiseby choosing Γ (t − t′) = e−λ(t−t′), where 1/λ is the memory range. Doubleangular brackets denote averaging over a noise realization.3. SimulationsWe analyze trajectories of the particles by integrating numerically theequations of motion (2) with the help of stochastic version of the Euleralgorithm. Discretization of the equations (2) entails re-scaling the noisestrength by a factor 1/√∆t, where ∆t is the time step. The exponentiallycorrelated noise is generated from uniformly distributed random numbersthrough Ornstein–Uhlenbeck process. Finally, the integral present in (2) isdiscretized according the schemet∫dt′ x˙iα(t′) exp [−λ(t− t′)] =N∑k=0x˙iα(t− k∆t) exp (−λk∆t) . (4)The number N of stored past velocities was chosen equal toN =⌊− ln pλ∆t⌋, (5)where p = 10−5 is the assumed accuracy of the approximation used in eval-uating integral (4). In the limit of λ → ∞ the dynamics becomes reducedto the ordinary Langevin dynamics without memory (N = 0).In simulations we determined the maximal Lyapunov exponent, Λ, asfunction of memory range (1/λ) and γ. Λ was estimated from relativechanges in the distance between the two initially nearby trajectoriesΛ =〈1t− t0 lnd(t)d(t0)〉, (6)1722 M. Cieśla, L. Longawhere d stands for distance at time t and 〈. . .〉 is the averaging over trajec-tory. We used a natural system of units provided by the potential parame-ters: energy εu = V (0, 0)− Vmin ≈ 2.41, time tu =√3, measuring curvatureof the potential at origin and length lu = 1 giving the period of the oscillatingpart of the potential. Also we set the particle’s mass to m = 1.Simulations were carried out according to the following scheme: first ofall we selected initial conditions for a particle at random following Boltzmanndistribution at a given temperature T . Then, the particle was thermalizedfor about t = 500 time steps. Initial position of the second particle waschosen at random, close to the thermalized position of the first particle. Thedistance in positions between the particles did not exceed 10−6 at start, andin velocities was limited by 106 tu. Averaging procedure was performed alonga trajectory over 103 subsequent time steps for 104 different trajectories.Further increase in the number of the time steps and the trajectories didnot lead to systematic changes in Λ.4. ResultsIn the absence of memory the system was originally studied by Fahyand Hamann [1] using Andersen thermostat. The results unambiguouslyshowed that trajectories were exponentially convergent to a common, mas-ter trajectory after a transient period. The same phenomenon has beenreported for an ordinary Langevin dynamics (without memory) applied toa one-dimensional Lennard–Jones chain [27]. Recently Uberuaga and co-workers [9] re-analyzed the synchronization in Andersen and Langevin ther-mostats and suggested that this synchronization should occur for a widerange of potentials and temperatures. Their observation agree with our un-derstanding of the noise-induced synchronization as being an averaged effectof contraction and expansion of the distances between nearby trajectoriesduring their time evolution [14, 25, 27, 28]. The questions arise as whetherAndersen thermostat and Langevin dynamics lead to the same predictionsfor our model and how synchronization is affected by the range of memory.A typical simulation for the memoryless system shows, as expected, syn-chronization of the trajectories (Λ < 0, Fig. 2). Generally the distancebetween two trajectories decays exponentially after the transient period un-til fluctuations of the order of numerical precision are reached. Inset showsthe distance behavior at short times. The oscillations arise as a result ofmotion around local potential minimum. A systematic study shows that Λchanges with temperature and friction constant γ, as shown in Fig. 3. Theresults are summarized in Fig. 4, which shows regions of synchronization(Λ < 0) and chaos (Λ > 0) in (temperature, γ) plane.Noise-Induced Synchronization in the Fahy–Hamann Model 17230 100 200 300 400time1e-401e-321e-241e-161e-081distance10 15 201e-060.0011Fig. 2. Time evolution of the distance between two trajectories. System parametersare: T = 0.1, λ→∞ and γ = 1.0. Inset shows the evolution at short times.1 2 3temperature-101maximal Lyapunov exponent γ = 0.5γ = 1.0γ = 1.5γ = 2.0Fig. 3. Maximal Lyapunov exponent Λ versus temperature T at diﬀerent frictionconstant γ for memoryless system (λ→∞).0 1 2 3 4temperature0.60.811.21.41.61.82γpositive exponentnegative exponentFig. 4. Regions of positive and negative maximal Lyapunov exponent for Fahy–Hamann system without memory (λ→∞).1724 M. Cieśla, L. LongaGenerally synchronization is observed for large frictions, which stays inagreement with our earlier results for a different system [27]. Temperaturedependence shows that Λ reaches it’s maximum at mid-temperature range.In this regime particle trajectories easily penetrate local maxima of the po-tential, whereas at low temperatures they stay for a long time in local min-ima. On the other hand, for high temperatures the local harmonic structurecan be neglected in comparison with the harmonic (x21 + x22)2 boundaries.1 2 3 4temperature00.51maximal Lyapunov exponent γ = 0.5γ = 1.0γ = 1.5γ = 2.0Fig. 5. Maximal Lyapunov exponent Λ versus temperature T for diﬀerent frictionconstant γ and short memory (1/λ = 0.01).Finally, in Fig. 5 we show maximal Lyapunov exponent against tempera-ture for different values of friction constant and for memory range 1/λ = 0.01(N = 5). It is clearly seen that for this memory range no synchroniza-tion occurs. A simple intuitive explanation would be that memory addsadditional degrees of freedom, which should make harder for the system tosynchronize. Unexpectedly this argumentation fails for longer memory. Cal-culations of Λ as function of N ∼ 1/λ show at least three regimes, Fig. 6.In the first regime, corresponding to a short memory range, we observe0 0.05 0.1 0.15 0.2memory range-6-4-202maximal Lyapunov exponentT = 4.0T = 2.0T = 1.0T = 0.5T = 0.2Fig. 6. Maximal Lyapunov exponent Λ against memory 1/λ ≤ 0.2 (N ≤ 100) fordiﬀerent temperatures with friction constants γ = 1.0.Noise-Induced Synchronization in the Fahy–Hamann Model 1725de-synchronization of the system by memory. However, after reaching max-imum Λ drops down and for intermediate memory range synchronization isconsiderably enhanced. The strongest synchronization conditions are metfor 1/λ ≈ 0.14, where Λ approaches minimum. Interestingly, the positionsof the extremes are practically independent on γ. Further details are givenelsewhere [30].5. SummarySummarizing, our results show a possibility of having the constructiveinfluence of memory on the noise-induced synchronization. To our knowl-edge the only other observation pointing to the same conclusion is an en-hancement of neural synchrony in the Hindmarsh–Rose neural network withtime-delays [21]. Please note that the effect is quite counterintuitive for,in the first place, we would expect that memory by introducing extra di-mensions [29] should act in just the opposite way, i.e. make synchronizationmore difficult [22,23]. Though the proposed models are relatively simple, ourobservations to date suggest that this enhancement of synchrony by memoryis a general phenomenon, occurring for a wide class of memory profiles.This work was supported by Grant N202 169 31/3455 of the PolishMinistry of Science and Higher Education, and by the EC Marie CurieHost Grant for Transfer of Knowledge “COCOS” (contract MTKD-CT-20040517186). 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Noise-induced synchronization in the Fahy-Hamann model

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