Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena

Pages: 685 - 694, Issue special, November 2013

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Masatoshi Shiino - Department of Applied Physics, Faculty of Science, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-0033, Japan (email)
Keiji Okumura - FIRST, Aihara Innovative Mathematical Modelling Project, Japan Science and Technology Agency, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan (email)

Abstract: Synchronization phenomena occurring as a result of cooperative ones are ubiquitous in nonequilibrium physical and biological systems and also are considered to be of vital importance in information processing in the brain. Those systems, in general, are subjected to various kinds of noise. While in the case of equilibrium thermodynamic systems external Langevin noise is well-known to play the role of heat bath, few systematic studies have been conducted to explore effects of noise on nonlinear dynamical systems with many degrees of freedom exhibiting limit cycle oscillations and chaotic motions, due to their complexity. Considering simple nonlinear dynamical models that allow rigorous analyses based on use of nonlinear Fokker-Planck equations, we conduct systematic studies to observe effects of noise on oscillatory behavior with changes in several kinds of parameters characterising mean-field coupled oscillator ensembles and excitable element ones. Phase diagrams representing the dependence of the largest and the second largest Lyapunov exponents on the noise strength are studied to show the appearance and disappearance of synchronization of limit cycle oscillations.

Keywords:  Nonlinear Fokker-Planck equations, exactly solvable mean-field models, nonequilibrium phase transitions, noise-induced synchronization, noise-induced chaos and limit cycles, control of attractors.
Mathematics Subject Classification:  Primary: 82C31, 34C15; Secondary: 34D06.

Received: September 2012;      Revised: April 2013;      Published: November 2013.