ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Ryashko L. B., Bashkirtseva I. A., Stihin P. V. Stochastical sensitivity оf cycles оf Roessler system in transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 6, pp. 32-47. DOI: 10.18500/0869-6632-2003-11-6-32-47

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Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
517.925.42:531.36
DOI: 
10.18500/0869-6632-2003-11-6-32-47

Stochastical sensitivity оf cycles оf Roessler system in transition to chaos

Autors: 
Ryashko Lev Borisovich, Ural Federal University named after the first President of Russia B.N.Yeltsin
Bashkirtseva Irina Adolfovna, Ural Federal University named after the first President of Russia B.N.Yeltsin
Stihin Pavel Viktorovich, Ural Federal University named after the first President of Russia B.N.Yeltsin
Abstract: 

The response problem оf limit cycles for stochastically forced Roessler system is considered. For stochastical sensitivity analysis two approaches are used: empirical (based оn direct numerical simulation) and theoretical (based оn quasipotential function). The possibilities are demonstrated to describe stochastic bundles spatial orientation аnd scatter form using scatter ellipses. The increase of Roessler system sensitivity to external disturbances in the period-doubling bifurcation zone under transition to chaos is shown.

Key words: 
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Acknowledgments: 
The work was carried out with partial financial support of the RFBR grant (№ 04-01-96098ural).
Reference: 
  1. Pontryagin LS, Andronov AA, Witt AA. On the statistical consideration of dynamic systems. Sov. Phys. JETP. 1933;3(3):165-180 (in Russian).
  2. Stratonovich RL. Selected Topics in the Theory of Fluctuations in Radio Engineering. Moscow: Sov. Radio; 1961. 660 p. (in Russian).
  3. Rytov SM. Introduction to Stochastic Radiophysics. Moscow: Nauka; 1976. 404 p. (in Russian).
  4. Bolotin VV. Random Vibrations of Elastic Systems. Moscow: Nauka; 1979. 336 p. (in Russian).
  5. Dimentberg MF. Nonlinear Stochastic Problems of Mechanical Vibrations. Moscow: Nauka; 1980. 368 p. (in Russian).
  6. Neimark YI, Landa PS. Stochastic and Chaotic Oscillations. Springer; 1992. 512 p.
  7. Soong TT, Grigorin M. Random Vibration of Mechanical and Structural Systems. New Jersey: RTR Prentice-Hall, Englewood Cliffs; 1993. 402 p.
  8. Smelyanskiy VN, Dykman МI, Maier RS. Topological features of large fluctuations to the interior of а limit cycles. Physical Review Е. 1997;55(3):2369-2391. DOI: 10.1103/PhysRevE.55.2369.
  9. Landa PS, McClintock PVE. Changes in the dynamical behavior оf nonlinear systems induced by noise. Physics Reports. 2000;323(1):1-80. DOI: 10.1016/S0370-1573(99)00043-5.
  10. Sinai YG. Stochasticity of dynamic systems. In: Gaponov-Grekhov AV, editor. Nonlinear waves. Moscow: Nauka; 1979. P. 192-212 (in Russian).
  11. Kifer Y. Attractors via random perturbations. Commun. Math. Phys. 1989;121(3):445-455. DOI: 10.1007/BF01217733.
  12. Kopeikin AS, Vadivasova TE, Anischenko VS. Peculiarities of the process of establishing a probability measure on chaotic attractors in the Lorentz and Ressler systems taking into account fluctuations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):65–77 (in Russian).
  13. Ventzel AD, Freidlin MI. Fluctuations in Dynamic Systems Under the Influence of Small Random Disturbances. Moscow: Nauka; 1979. 424 p. (in Russian).
  14. Day MV. Regularity of boundary quasi-potentials for planar systems. Applied Mathematics and Optimization. 1994;30(1):79-101. DOI: 10.1007/BF01261992.
  15. Naeh T, Klosek MM, Matkowsky BJ, Schuss Z. A direct approach to the exit problem. STAM Journal Appl. Math. 1990;50(2):595-627. DOI: 10.1137/0150036.
  16. Neiman AB. Application of cumulant analysis to study bifurcations of dynamic systems perturbed by external noise. Izvestiya VUZ. Applied Nonlinear Dynamics. 1995;3(3):8-21 (in Russian).
  17. Mil'Shtein GN, Ryashko LB. A first approximation of the quasipotential in problems of the stability of systems with random non-degenerate perturbations. Journal of Applied Mathematics and Mechanics. 1995;59(1):47-56. DOI: 10.1016/0021-8928(95)00006-B.
  18. Bashkirtseva IA, Ryashko LB. Quasipotential method in analyzing the sensitivity of self-oscillations to stochastic disturbances. Izvestiya VUZ. Applied Nonlinear Dynamics. 1998;6(5):19-27 (in Russian).
  19. Bashkirtseva IA, Ryashko LB. Quasipotential method in the study of local stability of limit cycles to random influences. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(6):104—113 (in Russian).
  20. Bashkirtseva IA, Ryashko LB. Sensitivity analysis оf stohastically forced Lorenz model cycles under period-doubling bifurcations. Dynamic systems and applications. 2002;11:293-309.
  21. Roessler OE, Wegman K. Chaos in Zhabotinski reaction. Nature. 1978;271:89-90. DOI: 10.1038/271089a0.
  22. Kuznetsov SP. Dynamic Chaos. (Ser. Modern Theory of Oscillations and Waves). Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  23. Demidovich BP. Lectures on the Mathematical Theory of Stability. Moscow: Nauka; 1967. 472 p. (in Russian).
  24. Kuznetsov AP, Kapustina JV. Scaling properties at transition to chaos in model maps in the presence of noise. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(6):78–87 (in Russian).
  25. Crutchfield J, Nauenberg M, Rudnick J. Scaling for external noise at the onset оf chaos. Phys. Rev. Lett. 1981;46(14):933-935. DOI: 10.1103/PhysRevLett.46.933.
  26. Ryashko LB. The stability of stochastically perturbed orbital motions. Journal of Applied Mathematics and Mechanics. 1996;60(4):579-590. DOI: 10.1016/S0021-8928(96)00073-1.
Received: 
25.05.2003
Accepted: 
11.11.2003
Available online: 
06.12.2023
Published: 
31.12.2003
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