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

Masatoshi Shiino; Keiji Okumura

doi:10.3934/proc.2013.2013.685

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2013, Volume 2013: 685-694. Doi: 10.3934/proc.2013.2013.685 open access

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

Masatoshi Shiino^1, and
Keiji Okumura^2,

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

Received: August 31, 2012

Revised: March 31, 2013

Published: October 2013

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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.

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Mathematics Subject Classification: Primary: 82C31, 34C15; Secondary: 34D06.

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