American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents
  • PROC Home
  • This Issue
  • Next Article
    Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D
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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[5]

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

[6]

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

[7]

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

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[2]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure and Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[3]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic and Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[4]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[5]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[6]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[7]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011
[8]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[9]

Young-Pil Choi, Samir Salem. Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition. Kinetic and Related Models, 2019, 12 (3) : 573-592. doi: 10.3934/krm.2019023
[10]

Seung-Yeal Ha, Jinwook Jung, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. A mean-field limit of the particle swarmalator model. Kinetic and Related Models, 2021, 14 (3) : 429-468. doi: 10.3934/krm.2021011
[11]

Hongjie Dong, Yan Guo, Timur Yastrzhembskiy. Kinetic Fokker-Planck and Landau equations with specular reflection boundary condition. Kinetic and Related Models, 2022, 15 (3) : 467-516. doi: 10.3934/krm.2022003
[12]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic and Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
[13]

William F. Thompson, Rachel Kuske, Yue-Xian Li. Stochastic phase dynamics of noise driven synchronization of two conditional coherent oscillators. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2971-2995. doi: 10.3934/dcds.2012.32.2971
[14]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[15]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[16]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[17]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[18]

Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics and Games, 2021, 8 (4) : 381-402. doi: 10.3934/jdg.2021013
[19]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic and Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385
[20]

Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks and Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013

 Impact Factor: 

Tools

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy