Abstract
The phenomenon of stochastic resonance (SR) is investigated for chaotic systems perturbed by white noise and a harmonic force. The bistable discrete map and the Lorenz system are considered as models. It is shown that SR in chaotic systems can be realized via both parameter variation (in the absence of noise) and by variation of the noise intensity with fixed values of the other parameters.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. V. Turaev, L. P. Shilnikov, and S. V. Gonchenko, On models with non-rough Poincaré homoclinic curves,Physica D (1992).
L. P. Shilnikov, Bifurcation theory and turbulence, inNonlinear and Turbulent Processes (Gordon and Breanch, Harward Academic Publishers, New York, 1984), Vol. 2, pp. 1627–1635.
V. S. Anishchenko,Complicated Oscillations in Simple Systems (Nauka, Moscow, 1990).
Yu. Kifer, Attractors via random perturbations,Commun. Math. Phys. 121:445–455 (1989).
A. R. Bulsara, W. C. Schieve, and E. W. Jacobs, Homoclinic chaos in systems perturbed by weak Langevin noise,Phys. Rev. A 41:668–681 (1990).
V. S. Anishchenko and A. B. Neiman, Structure and properties of chaos in presence of noise, inNonlinear Dynamics of Structures, R. Z. Sagdeevet al., eds. (World Scientific, Singapore, 1991), pp. 21–48.
V. S. Anishchenko and M. A. Safonova, Bifurcations of attractors in presence of fluctuations,J. Tech. Phys. Lett. 57:1931–1943 (1987).
W. Horsthemke and R. Lefever,Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology (Springer-Verlag, Berlin, 1984).
V. S. Anishchenko and M. A. Safonova, Noise induced exponential extension of phase trajectories in the regular attractors neighborhood,J. Tech. Phys. Lett. 12:740–744 (1986).
W. Ebeling and L. Schimansky-Geier, Transition phenomena in multidimensional system—Models of evolution, inNoise in Nonlinear Dynamical Systems, F. Moss and P. V. E. McClintock, eds. (Cambridge University Press, Cambridge, England, 1989), Vol. 1, pp. 279–304.
V. S. Anishchenko and A. B. Neiman, Noise induced transition in Lorenz model,J. Tech. Phys. Lett. 17:43–47 (1990).
R. Benzi, A. Sutera, and A. Vulpiani, The mechanism of stochastic resonance,J. Phys. 14A:L453-L457 (1981).
S. Fauve and F. Heslot, Stochastic resonance in a bistable system,Phys. Lett. 97A:5–8 (1983).
B. McNamara, K. Wiesenfeld, and R. Roy, Observation of stochastic resonance in a ring laser,Phys. Rev. Lett. 60:2625–2629 (1988).
B. McNamara and K. Wiesenfeld, Theory of stochastic resonance,Phys. Rev. 39A:4854–4869 (1989).
P. Jung and P. Hanggi, Stochastic nonlinear dynamics modulated by external periodic forces,Europhys. Lett. 8:505–511 (1989).
C. Presila, F. Marchesoni, and L. Gammaitoni, Periodically time-modulated bistable systems: Nonstationary statistical properties,Phys. Rev. 40A:2105–2113 (1989).
P. Jung and P. Hanggi, Resonantly driven Brownian motions: Basic conceptions and exact results,Phys. Rev. 41A:2977–2988 (1990).
T. Zhou, F. Moss, and P. Jung, Escape-time distributions of a periodically modulated bistable system with noise,Phys. Rev. 42A:3161–3169 (1990).
F. Moss, Stochastic resonance, in Rate Processes in Dissipative Systems: 50 Years after Kramers, P. Hanggi and J. Troe, eds.,Ber. Bunsenges. Phys. Chem. (1991).
F. Moss, Stochastic resonance: From the ice ages to the monkey's ear, Department of Physics, University of Missouri at St. Louis, St. Louis, Missouri (1992).
H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions,Physica 7:284–304 (1940).
F. T. Arecci, R. Badii, and A. Politi, Low-frequency phenomena in dynamical systems with many attractors,Phys. Rev. 29A:1006–1009 (1984).
V. S. Anishchenko, Interaction of attractors. Intermittency of the chaos-chaos type,J. Tech. Phys. Lett. 10:629–633 (1984).
V. S. Anishchenko and A. B. Neiman, Increasing of correlations time under chaos-chaos intermittency,J. Tech. Phys. Lett. 13:1063–1066 (1987).
G. Schuster,Deterministic Chaos (Physik-Verlag, Weinheim, 1984).
Yu. L. Klimontovich,Wave and Fluctuation Processes in Lasers (Nauka, Moscow, 1974).
H. M. Ito, Ergodicity of randomly perturbed Lorenz model,J. Stat. Phys. 35:151–158 (1984).
V. V. Bykov and A. L. Shilnikov, On boundaries of Lorenz attractor existence, inMethods of Qualitative Theory and Bifurcations (Gorky University, Gorky, 1989), pp. 151–159.
V. S. Anishchenko and A. B. Neiman, Dynamical chaos and color noise,J. Tech. Phys. Lett. 16:21–25 (1990).
M. I. Rabinovich, Stochastic oscillation and turbulence,Uspekhi Fiz. Nauk 125:123–168 (1978).
V. S. Anishchenko,Dynamical Chaos in Physical Systems. Experimental Investigation of Self-Oscillating Circuits (Teubner-Texte, Leipzig, 1989).
L. Gammaitoni, F. Marchesoni, E. Menichella-Saetta, and S. Santucci, Stochastic resonance in bistable systems,Phys. Rev. Lett. 62:349–352 (1989).
L. Gammaitoni, E. Menichella-Saetta, S. Santucci, F. Marchesoni, and C. Fresilla, Periodically modulated bistable systems: Stochastic resonance,Phys. Rev. 40A:2114–2119 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Anishchenko, V.S., Neiman, A.B. & Safanova, M.A. Stochastic resonance in chaotic systems. J Stat Phys 70, 183–196 (1993). https://doi.org/10.1007/BF01053962
Issue Date:
DOI: https://doi.org/10.1007/BF01053962