Confidence tori in the analysis of stochastic 3D-cycles

Mathematics and Computers in Simulation

Volumn 80, Issue 2, 2009, Pages 256-269

  Ryashko, L ^a   Bashkirtseva, I ^a   Gubkin, A ^a   Stikhin, P ^a  

^a URAL FEDERAL UNIVERSITY   (Russian Federation)

Author keywords

Cycles; Roessler model; Stochastic sensitivity; Stochastic systems

Indexed keywords

3D MODELS; CYCLES; DISCRETE SETS; EFFECTIVE ALGORITHMS; NON-LINEAR DYNAMIC SYSTEMS; PERIOD DOUBLING BIFURCATION; ROESSLER MODEL; SPATIAL ANALYSIS; STOCHASTIC SENSITIVITY; STOCHASTIC SENSITIVITY ANALYSIS; TRANSITION TO CHAOS;

DYNAMICAL SYSTEMS; SENSITIVITY ANALYSIS; STOCHASTIC SYSTEMS; THREE DIMENSIONAL;

STOCHASTIC MODELS;

EID: 70349761251     PISSN: 03784754     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matcom.2009.06.026     Document Type: Article

References (45)

1. Arecchi F.T., Badii R., and Politi A. Generalized multistability and noise-induced jumps in a nonlinear dynamical system. Phys. Rev. A 32 (1985) 402-408
2. Arnold L., Bleckert G., and Schenk-Hoppe K. The Stochastic Brusselator: Parametric Noise Destroys Hopf Bifurcation in: Stochastic dynamics (1999), Springer, New-York pp. 71-92
3. Baras F. The Stochastic Analysis of Limit Cycle Behavior, Stochastic Dynamics in: Lecture Notes in Physics, vol. 484 (1997), Springer-Verlag pp. 167-178
4. Bashkirtseva I.A., and Ryashko L.B. Sensitivity analysis of the stochastically and periodically forced Brusselator. Physica A 278 (2000) 126-139
5. Bashkitseva I.A., and Ryashko L.B. Sensitivity analysis of stochastically forced Lorenz model cycles under period-doubling bifurcations. Dynam. Syst. Appl. 11 (2002) 293-309
6. Bashkitseva I.A., and Ryashko L.B. Stochastic sensitivity of 3D-cycles. Math. Comput. Simul. 66 (2004) 55-67
7. Bashkitseva I.A., and Ryashko L.B. Sensitivity and chaos control for the forced nonlinear oscillations. Chaos Solitons Fract. 26 (2005) 1437-1451
8. Berglund N., and Gentz B. Geometric singular perturbation theory for stochastic differential equations. J. Diff. Equat. 191 (2003) 1-54
9. Berglund N., and Gentz B. Noise-induced Phenomena in Slow-Fast Dynamical Systems. A Sample-paths Approach Probability and its Applications (2005), Springer
10. Crutchfield J.P., and Huberman B.A. Fluctuations at the onset of chaos. Phys. Lett. A 77 (1980) 407-410
11. Crutchfield J.P., Nauenberg M., and Rudnick J. Scaling for external noise at the onset of chaos. Phys. Rev. Lett. 46 (1981) 933-935
12. Dembo M., and Zeitouni O. Large Deviations Techniques and Applications (1995), Jones and Bartlett Publishers, Boston
13. Fedotov S., Bashkirtseva I., and Ryashko L. Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition. Phys. Rev. E 66 (2002) 066310
14. Fedotov S., Bashkirtseva I., and Ryashko L. Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo. Physica A 342 (2004) 491-506
15. Franzoni L., Mannella R., McClintock P., and Moss F. Postponement of Hopf bifurcations by multiplicative colored noise. Phys. Rev. A 36 (1987) 834-841
16. Freidlin M.I., and Wentzell A.D. Random Perturbations of Dynamical Systems (1984), Springer, New York
17. Gammaitoni L., Hanggi P., Jung P., and Marchesoni F. Stochastic resonance. Rev. Mod. Phys. 70 (1998) 223-287
18. Gao J.B., Hwang S.K., and Liu J.M. When can noise induce chaos?. Phys. Rev. Lett. 82 (1999) 1132-1135
19. Gassmann F. Noise-induced chaos-order transitions. Phys. Rev. E 55 (1997) 2215-2221
20. Horsthemke W., and Lefever R. Noise-induced Transitions (1984), Springer, Berlin
21. Ibrahim R.A. Parametric Random Vibration (1985), John Wiley and Sons, New York
22. Infeld E., and Rowlands G. Nonlinear Waves, Solitons and Chaos (2000), Cambridge University Press
23. Keizer J., Fox R.F., and Wagner J. On the amplification of molecular fluctuations nonstationary systems: hydrodynamic fluctuations for the Lorenz model. Phys. Lett. A 175 (1993) 17-22
24. Kurrer C., and Schulten K. Effect of noise and perturbations on limit cycle systems. Physica D 50 (1991) 311-320
25. Landa P.S., and McClintock P.V.E. Changes in the dynamical behavior of nonlinear systems induced by noise. Phys. Rep. 323 (2000) 1-80
26. Lefever R., and Turner J. Sensitivity of a Hopf bifurcation to external multiplicative noise. In: Horsthemke W., and Kondepudi D.K. (Eds). Fluctuations and Sensitivity in Equilibrium Systems (1984), Springer, Berlin 143-149
27. Lefever R., and Turner J. Sensitivity of a Hopf bifurcation to multiplicative colored noise. Phys. Rev. Lett. 56 (1986) 1631-1634
28. Leung H.K. Stochastic Hopf bifurcation in a biased van der Pol model. Physica A 254 (1998) 146-155
29. Malick K., and Marcq P. Stability analysis of noise-induced Hopf bifurcation. Eur. Phys. J. 36 (2003) 119-128
30. Mangel M. Small fluctuations in systems with multiple limit cycles. SIAM J. Appl. Math. 38 (1980) 120-138
31. Matsumoto K., and Tsuda I. Noise-induced order. J. Stat. Phys. 31 (1983) 87-106
32. McDonnell M.D., Stocks N.G., Pearce C.E.M., and Abbott D. Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization (2008), Cambridge University Press
33. Naeh T., Klosek M.M., Matkowsky B.J., and Schuss Z. A direct approach to the exit problem. SIAM J. Appl. Math. 50 (1990) 595-627
34. Namachchivaya N.Sri. Hopf bifurcation in the presence of both parametric and external stochastic excitations. J. Appl. Mech. 110 (1988) 923-930
35. Ott E., Yorke E.D., and Yorke J.A. A scaling law: how an attractor's volume depends on noise level. Physica D 16 (1985) 62-78
36. Pontryagin L.S., Andronov A.A., and Witt A.A. On the statistical treatment of dynamical systems. Z. Eksp. Teor. Fiz. 3 (1933) 165-180 (in Russian)
37. Roessler O.E. An equation for continuous chaos. Phys. Lett. A 57 (1976) 397-398
38. Roy R.V. Asymptotic analysis of first passage problem. Int. J. Non-Linear Mech. 32 1 (1997) 173-186
39. Ryagin M., and Ryashko L. The analysis of the stochastically forced periodic attractors for Chua's circuit. Int. J. Bifurcat. Chaos 14 (2004) 3981-3987
40. Schenk-Hoppe K.R. Bifurcation scenarios of the noisy Duffing-van der Pol oscillator. Nonlinear Dynam. 11 (1996) 255-274
41. Smelyanskiy V.N., Dykman M.I., and Maier R.S. Topological features of large fluctuations to the interior of a limit cycle. Phys. Rev. E 55 (1997) 2369-2391
42. Soong T.T., and Grigoriu M. Random Vibration of Mechanical and Structural Systems (1993), RTL Prentice-Hall, Englewood Cliffs, NJ
43. Stratonovich R.L. Topics in the Theory of Random Noise (1963), Gordon and Breach, New York
44. Woo M., Neider J., and Davis T. OpenGL Programming Guide (1997), Addison-Wesley Developer's Press, Reading, MA
45. Zippelius A., and Lucke M. The effect of external noise in the Lorenz model of the Bernard problem. J. Stat. Phys. 24 (1981) 345-358