Stochastic bifurcation in Duffing-Van der Pol oscillators

Physica A: Statistical Mechanics and its Applications

Volumn 338, Issue 3-4, 2004, Pages 319-334

  He, Qun ^a   Xu, Wei ^a   Rong, Haiwu ^b   Fang, Tong ^a  

^a NORTHWESTERN POLYTECHNICAL UNIVERSITY   (China)

^b Foshan Inst of Sci and Technology   (China)

Author keywords

Duffing van der Pol oscillators; Global analysis; Stochastic bifurcation

Indexed keywords

BIFURCATION (MATHEMATICS); MAPPING; NONLINEAR SYSTEMS; PROBABILITY; TOPOLOGY;

GLOBAL ANALYSIS; STOCHASTIC ATTRACTORS; STOCHASTIC BIFURCATION;

RANDOM PROCESSES;

EID: 2942525861     PISSN: 03784371     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physa.2004.01.067     Document Type: Article

References (25)

1. Arnold L. Random Dynamical Systems. 1998;Springer, Berlin.
2. P. Baxendale, Asymptotic behavior of stochastic flows of diffeomorphisms, in: K. Ito, T. Hida (Eds.), Stochastic Processes and Their Applications, Lecture Notes in Mathematics, Vol. 1203, Springer, Berlin, Berlin Heidelberg, New York, 1986, pp. 1-19.
3. Meunier C., Verga A.D. Noise and bifurcation. J. Stat. Phys. 50(2):1988;345-375.
4. Crauel H., Flandoli F. Additive noise destroys a pitchfork bifurcation. J. Dyn. Differential Equations. 10:1998;259-274.
5. Wolf A., Swift J.B., Swinney H.L., Vastano J.A. Determining Lyapunov exponents from a time series. Physica D. 16:1985;285-317.
6. 3=0 Int. J. Non-linear Mech. 15:1980;449-458.
7. Guckenheimer J., Holmes P. Nonlinear Oscillation Dynamical Systems and Bifurcation of Vector Fields. 1983;Springer, Berlin.
8. Namachchivaya N.S. Stochastic bifurcation. Appl. Math. Comput. 38(2):1990;101-159.
9. Schenk-Hoppe K.R. Bifurcation scenarios of the noisy Duffing-van der Pol oscillator. Nonlinear Dyn. 11:1996;255-274.
10. To C.W., Li D.M. Degenerated Hopf bifurcations in a stochastically disturbed non-linear system. J. Sound Vib. 219(2):1999;359-369.
11. Leung H.K. Stochastic Hopf bifurcation in a biased van der Pol model. Physica A. 254(1):1998;146-155.
12. Ariaratnam S.T. Some illustrative examples of stochastic bifurcation. Thompson J.M.T., Bishop S.R. Nonlinearity and Chaos in Engineering Dynamics (IUTAM Symposium 1994). 1994;267-274 Wiley, New York.
13. Hsu C.S. A theory of cell to cell mapping dynamical systems. ASME J. Appl. Mech. 47:1980;931-939.
14. Tongue B.H., Gu K.Q. A higher order method of interpolated cell mapping. J. Sound Vib. 125:1988;169-179.
15. Hsu C.S. A generalized theory of cell to cell mapping for nonlinear dynamical system. ASME J. Appl. Mech. 53:1981;695-701.
16. Sun J.Q., Hsu C.S. The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. J. Appl. Mech. 57:1990;1018-1025.
17. Sun J.Q. Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method. J. Sound Vib. 180:1995;785-795.
18. Guder R., Kreuzer E. Using generalized cell mapping to approximate invariant measures on compact manifolds. Int. J. Bifurcation Chaos. 7:1997;2487-2499.
19. Hsu C.S. Global analysis of dynamical systems using posets and digraphs. Int. J. Bifurcation Chaos. 5:1995;1085-1118.
20. Jianxue Xu, Ling Hong Generalized cell mapping digraph method for global analysis. Acta Mech. Sinica. 31:1999;724-730. (in Chinese).
21. Ling Hong, Jianxue Xu Crises and chaotic transient by the generalized cell mapping digraph method. Phys. Lett. A. 262:1999;361-375.
22. Ling Hong, Jianxue Xu Further study of interior crises in systems of ordinary differential equations. Acta Phys. Sinica. 49:2000;1228-1234. (in Chinese).
23. Ling Hong, Jianxue Xu Discontinuous bifurcations of chaotic attractors in forced oscillator by generalized cell mapping digraph (GCMD) method. Int. J. Bifurcation Chaos. 11:2001;723-736.
24. Ling Hong, Jianxue Xu A chaotic saddle in a Wada fractal basin boundary. Acta Mech. Sinica. 34:2002;136-141. (in Chinese).
25. Zhu W.Q., Lu M.Q., Wu Q.T. Stochastic jump and bifurcation of a Duffing oscillator under narrow-band excitation. J. Sound Vib. 165:1993;285-304.