Abstract
The present work is concerned with the behavior of the second bifurcation of a Hopf bifurcation system excited by white-noise. It is found that the intervention of noises induces a drift of the bifurcation point along with the subtantial change in bifurcation type.
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References
G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, New York (1977).
H. Haken, Synergetics, Springer-Verlag, Berlin (1977).
R. Graham, Stochastic methods in nonequilibrium thermodynamics, in L. Arnold et al. eds., Stochastic Nonlinear Systems in Physics, Chemistry and Biology, Berlin, Springer-Verlag (1981), 202–212.
C. Meunier and A. D. Verga, Noise and bifurcation, J. Stat. Phys., 50, 1–2 (1988), 345–375.
N. Sri Namachchivaya, Stochastic bifurcation, Appl. Math. & Compt., 38 (1990), 101–159.
L. Arnold, Lyapunov exponents of nonlinear stochastic systems, Nonlinear Stochastic Dynamic Engrg. Systems, F. Ziegler and G. I. Schueller eds., Springer-Verlag, Berlin, New York (1987), 181–203.
R. Z. Khasminskii, Necessary and sufficient conditions for the asymptotic stability of linear stochastic systems, Theory Prob. & Appl., 12 1 (1967), 144–147.
R. Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff. Alphen aan den Rijn, the Netherlands, Rockville, Maryland, USA (1980).
L. Arnold and V. Wihstutz, eds., Lyapunov exponents, Proc. of a Workshop, held in Bremen, November 12–15, 1984, Springer-Verlag, Berlin, Heidelberg (1986).
S. T. Ariaratnam and W. C. Xie, Lyapunov exponent and rotation number of a two-dimensional nilpotent stochastic system, Dyna. & Stab. Sys., 5, 1 (1990), 1–9.
S. T. Ariaratnam, D. S. F. Tam and W. C. Xie, Lyapunov exponents of two-degree-of-freedom linear stochastic systems, Stochastic Structural Dynamics 1, Y. K. Lin and I. Elishakoff eds., Springer-Verlag, Berlin (1991), 1–9.
N. Sri Namachchivaya and S. Talwar, Maximal Lyapunov exponent and rotation number for stochastically perturbed co-dimension two bifurcation, J. Sound & Vib., 169, 3 (1993), 349–372.
L. Arnold and W. Kliemann, Qualitative theory of stochastic systems, Prob. Anal. and Related Topics, A. T. Bharucha-Reid eds. Academic Press, New York, Lindon. 3 (1983), 1–79.
Z. Schuss, Theory and Applications of Stochastic Differential Equations, John Wiley & Sons, New York (1980).
K. Ito and H. P. McKean Jr., Diffusion Processes and Their Sample Paths, Springer-Verlag, New York (1965).
S. Karlin and H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York (1981).
F. Kozin and S. Prodromou, Necessary and sufficient conditions for almost sure sample stability of linear Ito equations, SIAM J. Appl. Math., 21 (1971), 413–424.
R. R. Mitchell and F. Kozin, Sample stability of second order linear differential equations with wide band noise coefficients, SIAM. J. Appl. Math., 27 (1974), 571–605.
K. Nishoka. On the stability of two-dimensional linear stochastic systems, Kodai Math. Sem. Rep., 27 (1976), 211–230.
L. Z. Xu, W. Z. Chen, The Asymptotic Analysis Methods and Its Applications, Defence Industries Press (1991). (in Chinese)
Nicolis and I. Prigogine, Exploring Complexity, Sichuan Education Press, Chengdu (1986). (Chinese version)
Liu Xianbin, Bifurcation behavior of stochastic mechanics system and its variational method. Ph.D. Thesis, Southwest Jiaotong University (1995). (in Chinese)
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Xianbin, L., Qiu, C. & Dapeng, C. On the two bifurcations of a white-noise excited Hopf bifurcation system. Appl Math Mech 18, 835–845 (1997). https://doi.org/10.1007/BF00133341
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DOI: https://doi.org/10.1007/BF00133341