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2013, 2013(special): 685-694. doi: 10.3934/proc.2013.2013.685

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, Japan
2. 

FIRST, Aihara Innovative Mathematical Modelling Project, Japan Science and Technology Agency, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan

Received  September 2012 Revised  April 2013 Published  November 2013

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, noise-induced chaos and limit cycles, exactly solvable mean-field models, nonequilibrium phase transitions, noise-induced synchronization, control of attractors..
Mathematics Subject Classification: Primary: 82C31, 34C15; Secondary: 34D0.
Citation: Masatoshi Shiino, Keiji Okumura. Control of attractors in nonlinear dynamical systems using external noise: Effects of noise on synchronization phenomena. Conference Publications, 2013, 2013 (special) : 685-694. doi: 10.3934/proc.2013.2013.685
References:
[1]

R. C. Desai and R. Zwanzig, Statistical mechanics of a nonlinear stochastic model, J. Stat. Phys., 19 (1978), 1-24.  Google Scholar

[2]

D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys., 31 (1983), 29-85.  Google Scholar

[3]

M. Shiino, H-theorem and stability analysis for mean-field models of non-equilibrium phase transitions in stochastic systems, Phys. Lett. A, 112 (1985), 302-306. Google Scholar

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T. D. Frank, "Nonlinear Fokker-Planck Equations," Fundamentals and applications. Springer Series in Synergetics. Springer-Verlag, Berlin, 2005.  Google Scholar

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M. A. Zaks, X. Sailer, L. Schimansky-Geier and A. B. Neiman, Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems, Chaos, 15 (2005), 026117-1-026117-13.  Google Scholar

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H. Hasegawa, Generalized rate-code model for neuron ensembles with finite populations, Phys. Rev. E, 75 (2007), 051904-1-051904-11.  Google Scholar

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T. Kanamaru and K. Aihara, Stochastic synchrony of chaos in a pulse-coupled neural network with both chemical and electrical synapses among inhibitory neurons, Neural Comp., 20 (2008), 1951-1972.  Google Scholar

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M. Shiino and K. Yoshida, Chaos-nonchaos phase transitions induced by external noise in ensembles of nonlinearly coupled oscillators, Phys. Rev. E, 63 (2001), 026210-1-026210-6. Google Scholar

[12]

A. Ichiki, H. Ito and M. Shiino, Chaos-nonchaos phase transitions induced by multiplicative noise in ensembles of coupled two-dimensional oscillators, Physica E, 40 (2007), 402-405. Google Scholar

[13]

M. Shiino and K. Doi, Nonequilibrium phase transitions in stochastic systems with and without time delay: Controlling various attractors with noise, Proceedings of the 2007 IEEE Symposium on Foundations of Computational Intelligence (2007), 100-105. Google Scholar

[14]

K. Okumura and M. Shiino, Analytical approach to noise effects on synchronization in a system of coupled excitable elements, Lecture Notes in Computer Science, 6443 (2010), 486-493. Google Scholar

[15]

K. Okumura, A. Ichiki and M. Shiino, Effects of noise on synchronization phenomena exhibited by mean-field coupled limit cycle oscillators with two natural frequencies, Physica E, 43 (2011), 794-797. Google Scholar

[16]

K. Okumura, A. Ichiki and M. Shiino, Stochastic phenomena of synchronization in ensembles of mean-field coupled limit cycle oscillators with two native frequencies, Europhys. Lett., 92 (2010), 50009-p1-50009-p6. Google Scholar

show all references

References:
[1]

R. C. Desai and R. Zwanzig, Statistical mechanics of a nonlinear stochastic model, J. Stat. Phys., 19 (1978), 1-24.  Google Scholar

[2]

D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys., 31 (1983), 29-85.  Google Scholar

[3]

M. Shiino, H-theorem and stability analysis for mean-field models of non-equilibrium phase transitions in stochastic systems, Phys. Lett. A, 112 (1985), 302-306. Google Scholar

[4]

M. Shiino, Dynamical behavior of stochastic systems of infinitely many coupled nonlinear oscillators exhibiting phase transitions of mean-field type: H theorem on asymptotic approach to equilibrium and critical slowing down of order-parameter fluctuations, Phys. Rev. A, 36 (1987), 2393-2412. Google Scholar

[5]

T. D. Frank, "Nonlinear Fokker-Planck Equations," Fundamentals and applications. Springer Series in Synergetics. Springer-Verlag, Berlin, 2005.  Google Scholar

[6]

L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, Stochastic resonance, Rev. Mod. Phys., 70 (1998), 223-287. Google Scholar

[7]

B. Lindner, J. García-Ojalvo, A. Neiman and L. Schimansky-Geier, Effects of noise in excitable systems, Phys. Rep., 392 (2004), 321-424. Google Scholar

[8]

M. A. Zaks, X. Sailer, L. Schimansky-Geier and A. B. Neiman, Noise induced complexity: From subthreshold oscillations to spiking in coupled excitable systems, Chaos, 15 (2005), 026117-1-026117-13.  Google Scholar

[9]

H. Hasegawa, Generalized rate-code model for neuron ensembles with finite populations, Phys. Rev. E, 75 (2007), 051904-1-051904-11.  Google Scholar

[10]

T. Kanamaru and K. Aihara, Stochastic synchrony of chaos in a pulse-coupled neural network with both chemical and electrical synapses among inhibitory neurons, Neural Comp., 20 (2008), 1951-1972.  Google Scholar

[11]

M. Shiino and K. Yoshida, Chaos-nonchaos phase transitions induced by external noise in ensembles of nonlinearly coupled oscillators, Phys. Rev. E, 63 (2001), 026210-1-026210-6. Google Scholar

[12]

A. Ichiki, H. Ito and M. Shiino, Chaos-nonchaos phase transitions induced by multiplicative noise in ensembles of coupled two-dimensional oscillators, Physica E, 40 (2007), 402-405. Google Scholar

[13]

M. Shiino and K. Doi, Nonequilibrium phase transitions in stochastic systems with and without time delay: Controlling various attractors with noise, Proceedings of the 2007 IEEE Symposium on Foundations of Computational Intelligence (2007), 100-105. Google Scholar

[14]

K. Okumura and M. Shiino, Analytical approach to noise effects on synchronization in a system of coupled excitable elements, Lecture Notes in Computer Science, 6443 (2010), 486-493. Google Scholar

[15]

K. Okumura, A. Ichiki and M. Shiino, Effects of noise on synchronization phenomena exhibited by mean-field coupled limit cycle oscillators with two natural frequencies, Physica E, 43 (2011), 794-797. Google Scholar

[16]

K. Okumura, A. Ichiki and M. Shiino, Stochastic phenomena of synchronization in ensembles of mean-field coupled limit cycle oscillators with two native frequencies, Europhys. Lett., 92 (2010), 50009-p1-50009-p6. Google Scholar

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