Electron. J. Differential Equations, Vol. 2016 (2016), No. 240, pp. 1-12.

Sensitivity analysis of stochastically forced quasiperiodic self-oscillations

Irina Bashkirtseva, Lev Ryashko

We study a problem of stochastically forced quasi-periodic self-oscillations of nonlinear dynamic systems, which are modelled by an invariant torus in the phase space. For weak noise, an asymptotic of the stationary distribution of random trajectories is studied using the quasipotential. For the constructive analysis of a probabilistic distribution near a torus, we use a quadratic approximation of the quasipotential. A parametric description of this approximation is based on the stochastic sensitivity functions (SSF) technique. Using this technique, we create a new mathematical method for the probabilistic analysis of stochastic flows near the torus. The construction of SSF is reduced to a boundary value problem for a linear differential matrix equation. For the case of the two-torus in the three-dimensional space, a constructive solution of this problem is given. Our theoretical results are illustrated with an example.

Submitted May 30, 2016. Published August 31, 2016.
Math Subject Classifications: 34D35, 37H10, 34K14.
Key Words: Stochastic differential equations; tori; stochastic sensitivity; quasipotential.

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Irina Bashkirtseva
Ural Federal University
Pr. Lenina 51, Ekaterinburg 620083, Russia
email: irina.bashkirtseva@urfu.ru
Lev Ryashko
Ural Federal University
Pr. Lenina 51, Ekaterinburg 620083, Russia
email: lev.ryashko@urfu.ru

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