Abstract
A double expansion in powers of the damping coefficient and noise intensity is shown to be a powerful method for obtaining the stationary distribution of systems that after rescaling become weakly damped conservative ones. Systems undergoing Hopf bifurcations belong to this class. As an illustrative example, the generalized van der Pol oscillator is considered around its bifurcation point. A calculation is carried out up to third order in both the noise intensity and the bifurcation parameter (damping coefficient).
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References
A. A. Andronow and C. E. Chaikin,Theory of Oscillations (Princeton University Press, Princeton, 1949).
N. Minorsky,Nonlinear Oscillations (Van Nostrand, New York, 1962).
W. Ebeling,Strukturbildung bei irreversiblen Prozessen (Teubner Verlag, Leipzig, 1976).
G. Nicolis and I. Prigogine,Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).
H. Haken,Synergetics, An Introduction (Springer-Verlag, Berlin, 1977);Advanced Synergetics (Springer-Verlag, Berlin, 1983).
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, Berlin, 1983).
R. L. Stratonovich,Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963).
Z. Schuss,Theory and Applications of Stochastic Differential Equation (Wiley, New York, 1980).
N. van Kampen,Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981).
P. Hänggi and H. Thomas,Phys. Rep. 88:207 (1982).
C. Gardiner,Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983).
H. Risken,The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984).
R. Graham and T. Tél,Phys. Rev. A 35:1328 (1987).
J. K. Cohen and R. M. Lewis,J. Inst. Math. Appl. 3:266 (1967).
A. D. Ventzel and M. I. Freidlin,Russ. Math. Surveys 25:1 (1970); M. I. Freidlin and A. D. Ventzel,Random Perturbations of Dynamical Systems (Springer-Verlag, Berlin, 1984).
R. Graham, inCoherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum Press, New York, 1973);
, inFluctuations, Instabilities and Phase Transitions, T. Riste, ed. (Plenum Press, New York, 1975);
, inStochastic Processes in Nonequilibrium Systems, L. Garrido, P. Seglar, and P. J. Shephard, eds. (Springer-Verlag, Berlin, 1978);
, inStochastic Nonlinear Systems, L. Arnold and R. Lefever, eds. (Springer-Verlag, Berlin, 1981).
R. Kubo, K. Matsuo, and K. Kitahara,J. Stat. Phys. 9:51 (1973).
Yu. Kifer,Math. SSSR Izv. 8:1083 (1974).
D. Ludwig,SIAM Rev. 17:605 (1975).
K. Kitahara,Adv. Chem. Phys. 29:85 (1975).
R. Graham and A. Schenzle,Phys. Rev. A 23:1302 (1981); H. Schmidt, S. W. Koch, and H. Haug,Z. Phys. B 51:85 (1983); P. Talkner and P. Hänggi,Phys. Rev. A 29:768 (1984).
B. J. Matkowsky and Z. Schuss,SIAM J. Appl. Math. 42:822 (1982);Phys. Lett. 95A:213 (1983); E. Ben-Jacob, D. J. Bergmann, B. J. Matkowsky, and Z. Schuss,Phys. Rev. A 26:2805 (1982).
P. Talkner and D. Ryter,Phys. Lett. 88A:163 (1982); inNoise in Physical Systems and 1/f Noise (North-Holland, Amsterdam, 1983); D. Ryter,Physica 130A:205 (1985);142A:103 (1987); P. Talkner,Z. Phys. B 68:201 (1987).
W. G. Faris and G. Jona-Lasinio,J. Phys. A 15:3025 (1982); G. Jona-Lasinio, inTurbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, N. Ghil, R. Benzi, and G. Parisi, eds. (North-Holland, Amsterdam, 1985).
M. Dörfle and R. Graham,Phys. Rev. A 27:1096 (1983).
R. Graham and A. Schenzle,Z. Phys. 52:61 (1983).
R. Graham and T. Tél,Phys. Rev. Lett. 52:9 (1984);
,J. Stat. Phys. 35:729 (1984);
,Phys. Rev. A 31:1109 (1985);
,Phys. Rev. A 33:1322 (1986).
H. Lemarchand and G. Nicolis,J. Stat. Phys. 37:609 (1984).
R. Graham, D. Roekaerts, and T. Tél,Phys. Rev. A 31:3364 (1985); D. Roekaerts and F. Schwarz,J. Phys. A 20:L127 (1987).
A. Schenzle and T. Tél,Phys. Rev. A 32:596 (1986).
H. R. Jauslin,J. Stat. Phys. 42:573 (1986);Physica 144A:179 (1987).
R. Reibold,Z. Phys. B 62:397 (1986).
R. Graham,Europhys. Lett. 2:901 (1986).
P. Hänggi,J. Stat. Phys. 42:105 (1986).
V. Altares and G. Nicolis,J. Stat. Phys. 46:191 (1987); E. Sulpice, A. Lemarchand, and H. Lemarchand,Phys. Lett. 121A:67 (1987).
R. Graham, Macroscopic potentials, bifurcations and noise in dissipative system, preprint (1987).
B. van der Pol,Phil. Mag. 3:65 (1927); M. L. Cartwright and J. E. Littlewood,J. Land. Math. Soc. 20:180 (1945); P. Holmes and D. Rand,Q. Appl. Math. 35:495 (1978); U. Parlitz and W. Lauterborn,Phys. Rev. A 36:1428 (1987).
F. Baras, M. Malek Mansour and C. Van den Broeck,J. Stat. Phys. 28:577 (1982); D. Ryter, P. Talkner, and P. Hänggi,Phys. Lett. 93A:447 (1983); P. Hänggi and P. Riseborough,Am. J. Phys. 51:347 (1983).
L. Schimansky-Geier, A. V. Tolstopyatenko, and W. Ebeling,Phys. Lett. 108A:329 (1985); W. Ebeling and L. Simansky-Geier,Fluid Dyn. Trans. 12:7 (1985).
W. Ebeling, H. Herzel, W. Richert, and L. Schimansky-Geier,Z. Angew. Math. Mech. 66:141 (1986).
H. Haken,Phys. Rev. Lett. 13:329 (1964); H. Risken,Z. Phys. 186:85 (1965); R. Graham,Quantum Statistics in Optics and Solid-State Physics (Springer-Verlag, 1973).
K. H. Hoffman,Z. Phys. B 49:245 (1982).
Lord Rayleigh,Theory of Sound, Vol. I (Dover, New York, 1945).
M. O. Hongler and D. Ryter,Z. Phys. B 31:333 (1978); W. Ebeling and H. Engel-Herbert,Physica 104A:378 (1981).
R. Graham and H. Haken,Z. Phys. 243:289 (1971);245:141 (1971).
H. Risken and K. Voigtlander,J. Stat. Phys. 41:825 (1985).
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Tél, T. On the stationary distribution of self-sustained oscillators around bifurcation points. J Stat Phys 50, 897–912 (1988). https://doi.org/10.1007/BF01019146
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DOI: https://doi.org/10.1007/BF01019146