Interplay between couplings and common noise in phase synchronization: disagreement between global analysis and local stability characterization

arXiv:0809.2398v2 [nlin.SI] 19 Nov 2008 Interplay between couplings and common noise in phase synchronization: disagreement between global analysis and local stability characterization David Garc´ıa-´Alvarez,∗Alireza Bahraminasab, Aneta Stefanovska, and Peter V.E. McClintock Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom (Dated: November 19, 2008) We consider two coupled phase oscillators in the presence of proportional (“common”) and inde- pendent white noises. The global synchronization properties of the system are analytically studied via the Fokker-Planck equation. When the “effective coupling” is big compared to the common noises, the former favors and the latter hinder synchronization. On the contrary, when the coupling is small compared to the proportional noises, we find that the latter induce synchronization, opti- mally when their intensities are big and in the n:m synchronization ratio. Furthermore, in such case a small value of the coupling is better for synchronization. Finally, we show that synchronization, which is a global property, must not be studied via local stability such as with Lyapunov exponents. PACS numbers: 05.45.Xt, 05.10.Gg, 05.10.–a, 05.40.Ca. Synchronization of interacting oscillatory processes is ubiquitous in nature [1]. For any general system of cou- pled oscillators, phase dynamics approach can be usefully applied provided that the intensities of all interactions on the oscillators, such as couplings between oscillators, ef- fect of noise on the oscillators, etc, are small compared to the natural frequencies [2]. In such case, synchronization as an adjustment of rhythms of phase coupled oscillatory processes because of the interaction with each other is well understood. Another more intriguing kind of syn- chronization is the one induced by noise, which is ob- served in many natural and experimental systems, such as lasers [3], neurons [4], or ecological systems [5]. The first theoretical approach in the study of noise-induced synchronization in phase oscillators was by analyzing the Lyapunov exponent [6], but Lyapunov exponent only de- termines the local stability. A well established method of analyzing the global behavior of the system is via the Fokker-Planck equation [7], and synchronization in uncoupled phase oscillators with noise has already been studied using this approach [8, 9]. Still, an analysis that takes into account the two routes to synchronization – interactions between the oscillators (couplings), and common noise –, and studies the in- terplay between each other is lacking. In fact, in most real systems, coupling among the oscillators and the ac- tion of noise are simultaneously present, e.g. [10, 11]. In this letter we systematically investigate for the first time the effect of couplings, independent noises, and pro- portional noises (i.e., common noises but with different intensities) on phase synchronization. In particular, we show analytically and numerically how coupling and com- mon noise compete in achieving synchronization: two dif- ferent routes to synchronization are possible depending on whether the couplings or the common noise prevail. Moreover, the properties of the system when synchro- nized via couplings or by common noise clearly differ – for example, whether increasing the couplings or the common noise results in stronger or weaker synchroniza- tion –, and therefore our results will be useful for taking apart systems in nature synchronized via interaction or via common noise. The global behavior of the system will be studied via Fokker-Planck. After reducing and solving analytically the equation, we will analyze the role of non-common noise, proportional noise, and couplings in synchroniza- tion, and we will show the competence between couplings and common noise. Finally, we will present a case that shows that the Lyapunov exponent is not always a good indicator of synchronization. Let us consider a system of two coupled oscillators in the presence of proportional and independent noises. The equations for the amplitude variables read: ˙xα(t) = Fα(xα(t)) + ǫα Vα(x1(t), x2(t)) + Gα(xα(t)) p Dα ξ(t) + Hα(xα(t)) p Eα ηα(t), (1) for α=1,2. Here xα is the amplitude component of the α-th oscillator, Fα is its individual dynamics, the V ’s are the coupling functions, and the ǫ’s are the coupling intensities. ξ(t) is the common noise, and ηα(t) are the non-common noises. G and H represent the coupling of the oscillators to the noises. ξ(t) and ηα(t) are assumed to be independent Gaussian white noises, with zero mean value and unit intensity (as the actual intensities were taken aside into √Dα and √Eα): ⟨ξ(t) ξ(s)⟩= δ(t −s), ⟨ηα(t) ηβ(s)⟩= δαβ δ(t −s), ⟨ξ(t) ηα(s)⟩= 0. As long as the intensities of the couplings and of the noises remain small, we can make phase dynamics. Ap- plying the phase reduction method to equation (1) we ob- tain the following equations for the phase variables [2, 9]: ˙φα(t) = ωα + ǫα vα(φ1(t), φ2

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