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

For citation:

Ryashko L. B. Quasi-potential method for 2-torus stochastic sensitivity analysis. Izvestiya VUZ. Applied Nonlinear Dynamics, 2006, vol. 14, iss. 1, pp. 38-54. DOI: 10.18500/0869-6632-2006-14-1-38-54

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 138)
Article type: 

Quasi-potential method for 2-torus stochastic sensitivity analysis

Ryashko Lev Borisovich, Ural Federal University named after the first President of Russia B.N.Yeltsin

On the basis of quasi-potential method the stationary distribution of random trajectories in a vicinity of toroidal manifolds of stochastically forced nonlinear systems is investigated. For the quasi-potential approximation the quadratic form defined by some matrix function is used. This function named stochastic sensitivity function characterizes the response of considered system on random disturbances. Construction of this function is reduced to the decision of a boundary problem for linear differential matrix equation. For 2-torus in three-dimensional space a constructive decision of this problem is given. Construction of stochastic sensitivity function is reduced to the decision of some functional equation. Efficiency of the presented results is shown on the example. 

Key words: 
  1. Arnol'd VI. Additional chapters of the theory of ordinary differential equations. Moscow: Nauka; 1978. 304 p. (In Russian).
  2. Neymark YuI. Integral manifolds of differential equations. Radiophysics and Quantum Electronics. 1967;10(3):321–334.
  3. Gurtovnik AS, Neimark YuI. On the question of the stability of quasiperiodic motions. Differ. Uravn. 1969;5(5):824–832.
  4. Samoilenko AM. Elements of the mathematical theory of multi-frequency oscillations. Invariant tori. Moscow: Nauka; 1987. 304 p. (In Russian).
  5. Kolesov AYu, Mishchenko EF. Existence and stability of the relaxation torus. Russian Math. Surveys. 1989;44(3):204–205. DOI: 10.1070/RM1989v044n03ABEH002128.
  6. Kolesov AYu. On the existence and stability of a two-dimensional relaxational torus. Math. Notes. 1994;56(6):1238–1243. DOI: 10.1007/BF02266691.
  7. Ryashko LB. Lyapunov Functions Technique For Stability Analysis And Stabilization Of Invariant 2-Torus. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(4-5):140–154.
  8. Ryashko LB. Exponential mean square stability of stochastically forced 2-torus. Nonlinearity. 2004;17(2):729–742. DOI: 10.1088/0951-7715/17/2/021.
  9. Pontryagin LS, Andronov AA, Witt AA. On statistical consideration of dynamic systems JETP. 1933;3(3):165–180.
  10. Stratonovich RL. The Selected Questions of the Fluctuations Theory in a Radio Engineering. New York: Gordon and Breach; 1967. 560 p.
  11. Rytov SM. Introduction to stochastic radiophysics. Moscow: Nauka; 1976. (In Russian).
  12. Bolotin VV. Random oscillations of elastic systems. Moscow: Nauka; 1979. 335 p.(In Russian).
  13. Dimentberg MF. Nonlinear Stochastic Problems of Mechanical Vibrations. Moscow: Nauka; 1980. 368 p. (in Russian).
  14. Neimark YI, Landa PS. Stochastic and Chaotic Oscillations. Dordrecht: Springer; 1992. 500 p. DOI: 10.1007/978-94-011-2596-3.
  15. Soong TT, Grigoriu M. Random vibration of mechanical and structural systems. New Jersey: RTR Prentice–Hall, Englewood Cliffs; 1993. 402 p.
  16. Ventzel AD, Freidlin MI. Fluctuations in dynamic systems under the influence of small random disturbances. Moscow: Nauka; 1979. 424 p. (In Russian).
  17. Day MV. Regularity of boundary quasi-potentials for planar systems. Applied Mathematics and Optimization. 1994;30(1):79–101. DOI: 10.1007/BF01261992.
  18. Naeh T, Klosek MM, Matkowsky BJ, Schuss Z. A direct approach to the exit problem. SIAM Journal Appl.Math. 1990;50(2):595–627. DOI: 10.1137/0150036.
  19. 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.
  20. Bashkirtseva IA, Isakova MG, Ryashko LB. Asymptotic expansion of the quasipotential for a stochastically perturbed nonlinear oscillator. Differ. Equ. 1999;35(10):1334–1340.
  21. 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).
  22. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of stochastically forced Lorenz model cycles under period-doubling bifurcations. Dynamic Systems and Applications. 2002;11(2):293–309.
  23. Krasnoselsky MA, Lifshits EA, Sobolev AV. Positive linear systems. Moscow: Nauka; 1985. 255 p. (In Russian).
Short text (in English):
(downloads: 86)