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,
The chaotic phase synchronization transition is studied in connection with the zero Lyapunov exponent. We propose a hypothesis that it is associated with a switching of the maximal finite-time zero Lyapunov exponent, which is introduced in the framework of a large deviation analysis. A noisy sine circle map is investigated to introduce this hypothesis and it is tested in an unidirectionally coupled Rössler system by using the covariant Lyapunov vector associated with the zero Lyapunov exponent.
Within the context of the Brownian ratchet model, a molecular rotary system was studied that can perform unidirectional rotations induced by linearly polarized ac fields, and produce positive work under loads. The model is based on the Langevin equation for a particle in a two-dimensional (2D) three-tooth ratchet potential of threefold symmetry.The performance of the system is characterized by the coercive torque, i.e., the strength of the load competing with the torque induced by the ac driving field, and the energy efficiency in force conversion from the driving field to torque. We propose a master equation for coarse-grained states, which takes into account boundary motion between states, and develop a kinetic description to estimate mean angular momentum (MAM) and powers relevant to the energy balance equation. The framework of analysis incorporates several 2D characteristics, and is applicable to a wide class of models of smooth 2D ratchet potential. We confirm that the obtained expressions for MAM, power, and efficiency of the model can predict qualitative behaviors. We also discuss the usefulness of the torque/power relationship for experimental analyses, and propose a characteristic for 2D ratchet systems.
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