Analytical determination of the bifurcation thresholds in stochastic differential equations with delayed feedback

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 82, Issue 5, 2010, Pages

  Gaudreault, Mathieu ^a   Drolet, François ^a,b   Viñals, Jorge ^a  

^a MCGILL UNIVERSITY   (Canada)

^b FPINNOVATIONS   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYTICAL DETERMINATION; ANALYTICAL EXPRESSIONS; CHARACTERISTIC TIME; COMBINED EFFECT; CONDITIONAL AVERAGE; CONDITIONAL PROBABILITIES; DELAY TIME; DELAYED FEEDBACK; DYNAMICAL VARIABLES; GOVERNING EQUATIONS; MODEL EQUATIONS; MULTIPLE-SCALE EXPANSION; NUMERICAL INTEGRATIONS; OSCILLATORY SOLUTIONS; PITCHFORK BIFURCATIONS; SLOWLY VARYING AMPLITUDES; STOCHASTIC BIFURCATION; STOCHASTIC DIFFERENTIAL DELAY EQUATIONS; STOCHASTIC DIFFERENTIAL EQUATIONS;

DIFFERENTIAL EQUATIONS; FOKKER PLANCK EQUATION; INTEGRATION; NONLINEAR EQUATIONS; NONLINEAR FEEDBACK; NUMERICAL METHODS; STOCHASTIC MODELS; STOCHASTIC SYSTEMS;

HOPF BIFURCATION;

EID: 78651348648     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.82.051124     Document Type: Article

References (26)

1. J. Belair and M. Mackey, J. Dyn. Differ. Equ. 1, 299 (1989). 10.1007/BF01053930
2. M. B. Elowitz and S. Leibler, Nature (London) 403, 335 (2000). 10.1038/35002125
3. J. Hasty, J. Pradines, M. Dolnik, and J. Collins, Proc. Natl. Acad. Sci. U.S.A. 97, 2075 (2000). 10.1073/pnas.040411297
4. D. Bratsun, D. Volfson, L. Tsimring, and J. Hasty, Proc. Natl. Acad. Sci. U.S.A. 102, 14593 (2005). 10.1073/pnas.0503858102
5. T. Galla, Phys. Rev. E 80, 021909 (2009). 10.1103/PhysRevE.80.021909
6. S. Guillouzic, I. L'Heureux, and A. Longtin, Phys. Rev. E 59, 3970 (1999). 10.1103/PhysRevE.59.3970
7. M. C. Mackey and I. G. Nechaeva, Phys. Rev. E 52, 3366 (1995). 10.1103/PhysRevE.52.3366
8. J. Lei and M. C. Mackey, SIAM J. Appl. Math. 67, 387 (2007). 10.1137/060650234
9. M. Gaudreault, F. Lepine, and J. Viñals, Phys. Rev. E 80, 061920 (2009). 10.1103/PhysRevE.80.061920
10. R. Graham and A. Schenzle, Phys. Rev. A 25, 1731 (1982). 10.1103/PhysRevA.25.1731
11. F. Drolet and J. Viñals, Phys. Rev. E 57, 5036 (1998). 10.1103/PhysRevE.57.5036
12. M. San Miguel and R. Toral, Instabilities and Nonequilibrium Structures VI, edited by, E. Tirapegui,,, J. Martinez,, and, R. Tiemann, (Kluwer Academic, The Netherlands, 2000).
13. H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1987).
14. T. D. Frank, Phys. Rev. E 72, 011112 (2005). 10.1103/PhysRevE.72.011112
15. R. Shao and Y. Chen, Physica A 388, 977 (2009). 10.1016/j.physa.2008.12. 001
16. H. Zhang, W. Xu, Y. Xu, and D. Li, Physica A 388, 3017 (2009). 10.1016/j.physa.2009.04.032
17. R. D. Driver, Ordinary and Delay Differential Equations (Springer-Verlag, New York, 1977).
18. T. Saaty, Modern Nonlinear Equations (Dover Publications, New York, 1981).
19. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 2nd ed. (Springer-Verlag, Berlin, 1997).
20. M. M. Klosek, Acta Phys. Pol. B 35, 1387 (2004).
21. M. M. Klosek and R. Kuske, Multiscale Model. Simul. 3, 706 (2005). 10.1137/030601375
22. R. Kuske, in IUTAM Symposium on Nonlinear Stochastic Dynamics, edited by, N. S. Namachchivaya, and, Y. K. Lin, (Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003), p. 492.
23. P. H. Baxendale, in IUTAM Symposium on Nonlinear Stochastic Dynamics, edited by, N. S. Namachchivaya, and, Y. K. Lin, (Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003), p. 492.
24. P. H. Baxendale, Stochastic Proc. Appl. 113, 235 (2004). 10.1016/j.spa.2004.05.001
25. F. Baras, M. M. Mansour, and C. van der Broeck, J. Stat. Phys. 28, 577 (1982). 10.1007/BF01008325
26. M. Gaudreault, J. M. Berbert, and J. Viñals, Phys. Rev. E (to be published).