Abstract
Noise effects on the phase lockings and bifurcations in the sinusoidally forced van der Pol relaxation oscillator are investigated. Deterministic (noise-free) one-dimensional Poincaré mapping is extended to the iteration of the operator defined by a stochastic kernel function. Stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator. In particular, a new method which uses spectra (eigenvalues and eigenfunctions) of the operator to analyze stochastic bifurcations intensively is proposed.
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REFERENCES
L. Arnold, Random dynamical systems, in Dynamical Systems, R. Johnson, ed. (Springer, Berlin, 1995).
L. Arnold and P. Boxler, Stochastic bifurcation: Instructive examples in dimension one, in Diffusion Processes and Related Problems in Analysis, Vol. II: Stochastic Flows, M. Pinsky and V. Wihstutz, eds. (Birkhäuser, 1992), pp. 241-255.
F. R. Gantmacher, The Theory of Matrices, Vol. II (Chelsea, New York, 1959).
L. Glass and M. C. Mackey, From Clocks to Chaos, The Rhythms of Life(Princeton University Press, Princeton, New Jersey, 1988).
J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications(Springer, Berlin, 1987).
J. Grasman and M. J. W. Jansen, Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology, J. Math. Biol. 7:171-197 (1979).
J. Grasman and J. B. T. M. Roerdink, Stochastic and chaotic relaxation oscillations, J. Stat. Phys. 54:949-970 (1989).
J. Guckenheimer, Dynamics of the van der Pol equation, IEEE Trans. CAS-27:983-989 (1980).
W. Horsthemke and R. Lefever, Noise-Induced Transitions, Theory and Applications in Physics, Chemistry, and Biology(Springer-Verlag, Berlin, 1984).
T. Kapitaniak, Chaos in Systems with Noise(World Scientific, Singapore, 1988).
A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics(Springer, Berlin, 1994).
F. Moss and P. V. E. McClintock (eds.), Noise in Nonlinear Dynamical Systems, Vols. 1–3 (Cambridge University Press, Cambridge, 1987).
N. Provatas and M. C. Mackey, Asymptotic periodicity and banded chaos, Physica 53D:295-318 (1991).
T. Tateno, S. Doi, S. Sato, and L. M. Ricciardi, Stochastic phase-lockings in a relaxation oscillator forced by a periodic input with additive noise—A first-passage-time approach, J. Stat. Phys. 78:917-935 (1995).
J. B. Weiss and E. Knoblock, A stochastic return map for stochastic differential equations, J. Stat. Phys. 58:863-883 (1990).
K. Wiesenfeld and F. Moss, Stochastic resonance and the benefits of noise: From ice ages to crayfish and SQUIDs, Nature 373:33-36 (1995).
K. Xu, Bifurcations of random differential equations in dimension one, Random Comp. Dynam. 1:277-305 (1993).
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Doi, S., Inoue, J. & Kumagai, S. Spectral Analysis of Stochastic Phase Lockings and Stochastic Bifurcations in the Sinusoidally Forced van der Pol Oscillator with Additive Noise. Journal of Statistical Physics 90, 1107–1127 (1998). https://doi.org/10.1023/A:1023271109747
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DOI: https://doi.org/10.1023/A:1023271109747