Abstract
For relaxation oscillators stochastic and chaotic dynamics are investigated. The effect of random perturbations upon the period is computed. For an extended system with additional state variables chaotic behavior can be expected. As an example, the Van der Pol oscillator is changed into a third-order system admitting period doubling and chaos in a certain parameter range. The distinction between chaotic oscillation and oscillation with noise is explored. Return maps, power spectra, and Lyapunov exponents are analyzed for that purpose.
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References
J. Argémi and B. Rossetto,J. Math. Biol. 17:67–92 (1983).
P. Collet and J.-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems(Birkhäuser, Basel, 1980).
J. P. Crutchfield and J. D. Farmer,Phys. Rep. 92:45–82 (1982).
W. Ebeling, H. Herzel, and E. E. Sel'kov, inProceedings 16th FEBS Congress, Part C(VNU Science Press, 1985), pp. 443–49.
H. Engel-Herbert, W. Ebeling, and H. Herzel, inTemporal Order, L. Rensing and N. I. Jaeger, eds. (Springer-Verlag, Berlin, 1985), pp. 144–152.
J. D. Farmer, E. Ott, and J. A. Yorke,Physica 7D:153–180 (1983).
C. W. Gardiner,Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer-Verlag, Berlin, 1983).
J. Grasman,Asymptotic Methods for Relaxation Oscillations and Applications (Springer-Verlag, New York, 1987).
H. Herzel, Stabilization of Chaotic Orbits by Random Noise, Preprint, Humboldt University Berlin (1987).
T. Kapitaniak,Chaos in Systems with Noise (World Scientific, Singapore, 1988).
S. Karlin and H. M. Taylor,A First Course in Stochastic Processes (Academic Press, New York, 1975).
J. Kottalam, B. J. West, and K. Lindenberg,J. Stat. Phys. 46:119–133 (1987).
E. N. Lorenz,J. Atmos. Sci. 20:130–141 (1963).
R. Lozi, Modèles mathématiques qualitatifs simples et consistants pour l'étude dequelques systèmes dynamiques expérimentaux, Thesis, University of Nice (1983).
K. Matsumoto and I. Tsuda,J. Stat. Phys. 31:87–106 (1983).
K. Matsumoto,J. Stat. Phys. 34:111–127 (1984).
R. M. May,Nature 216:459–467 (1976).
E. F. Mishchenko and N. K. Rosov,Differential Equations with Small Parameters and Relaxation Oscillations (Plenum Press, New York, 1980).
H. Oka and H. Kokubu,Jpn. J. Appl. Math. 2:495–500 (1985).
D. Ruelle and F. Takens,Commun. Math. Phys. 20:167–192 (1971).
W. M. Schaffer, S. Ellner, and M. Kot,J. Math. Biol. 24:479–523 (1986).
R. Shaw,Z. Naturforsch. 36a:80–112 (1981).
S. Smale,Bull. Am. Soc. 73:747–817 (1967).
F. Takens, Transitions from periodic to strange attractors in constrained equations, Report ZW-8601, University of Groningen, The Netherlands (1986).
S. Ushiki and R. Lozi, Confinor and anti-confinor in constrained “Lorenz” system, preprint, University of Nice (1986).
A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano,Physica 16D:285–317 (1985).
A. Zippelius and M. Lücke,J. Stat. Phys. 24:345–358 (1981).
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Grasman, J., Roerdink, J.B.T.M. Stochastic and chaotic relaxation oscillations. J Stat Phys 54, 949–970 (1989). https://doi.org/10.1007/BF01019783
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DOI: https://doi.org/10.1007/BF01019783