Stochastic Model of Chaotic Phase Synchronization. I

Abstract

As a stochastic model of chaotic phase synchronization (CPS), a phase dynamics driven by dichotomous Markov noise is proposed. The master equation is analytically solved for the rotation number and the phase diffusion constant as the characteristics of CPS. An approximate analysis based on the projection operator method is also carried out. Through this analysis, it is revealed that CPS phenomenon can be faithfully described by the present stochastic model. §1. IntroductionSynchronization is a ubiquitous phenomenon in systems consisting of units exhibiting oscillation, and it is observed in physical, chemical, and biological systems, as well as in electrical and mechanical systems. 1) Over the last 340 years, since Huygens' experiment, many studies have been carried out on coupled limit cycle oscillator systems. Kuramoto found that phase variables play an important role in the synchronization-desynchronization phenomenon in coupled limit cycle oscillator systems, and he proposed a phase model for coupled limit cycle oscillator systems. 2) In 1983, two of the present authors first reported a mutual synchronization phenomenon in a coupled oscillator system consisting of identical chaotic oscillators and its breakdown. 3) This phenomenon is called complete synchronization. Then, ten years ago, Rosenblum, Pikovsky and Kurth reported chaotic phase synchronization (CPS), which is observed in coupled non-identical chaotic oscillators possessing periodic dynamics. 4) These types of chaotic synchronization are observed typically as the coupling strengths among oscillators are changed.In order to study the ubiquity of such phenomena, we need standard dynamical models of synchronization. As is well known, the phase model is a standard model of synchronization phenomena in coupled limit cycle systems. Recently, we proposed a mapping model as a possible standard model of CPS phenomena. 5) The validity of the phase model in a coupled limit cycle system results from the facts that the amplitude can be adiabatically eliminated. However the amplitude cannot be eliminated in CPS, because the amplitude is chaotic, or "noisy" in some sense and it is almost independent of the appropriately defined phase. Considering this point,

Stochastic Model of Chaotic Phase Synchronization. II

“…The results obtained here can be confirmed with the analytic expressions for the rotation number and the phase diffusion constant obtained in Ref. 9). Then, the asymmetric peak of the phase diffusion constant due to critical enhancement of chaotic fluctuations 12), 13) is also studied using a scaling analysis.…”

“…For example, anomalous scaling behavior at the CPS transition has been proved, 3)-6) and stochastic models, in which a chaotically fluctuating force due to amplitude fluctuations is modeled by a stochastic force with finite amplitude, have been proposed as useful in the study of the CPS transition. 7)- 9) In a previous paper, 9) we propose an analytically solvable stochastic model of CPS. In this paper, as a continuation of the previous paper, we present a complementary study of the same model employing approximate methods that are useful for obtaining a physical understanding of the stochastic model of CPS and also the CPS transition itself.…”

“…A similar study based on a general approximate method for time correlation functions 11) is carried out in Ref. 9). This paper is organized as follows.…”

“…Adiabatic approximation for phase slips As in Ref. 9), here we examine the properties of the stochastic phase dynamics described by the equation…”

Dichotomous noise models of gene switches

“…21 Dichotomous noise generally breaks detailed balance in the circuits and thus creates non-equilibrium steady states which cannot always be described by quasi-equilibrium fluctuation statistics. Dichotomous noise driven phenomena include robust phase synchronization, 22,23 stochastic hypersensitivity, 24,25 enhanced stochastic resonance, 26 hysteresis, 27 and patterning. 28,29 Brute force simulation of the full master equation for genetic networks has already yielded many insights.…”