Stochastic sensitivity of 3D-cycles

Authors:
I. A. Bashkirtseva;L. B. Ryashko
Affiliations:
Department of Mathematics, Ural State University, Lenina 51, 620083 Ekaterinburg, Russia;Department of Mathematics, Ural State University, Lenina 51, 620083 Ekaterinburg, Russia
Venue:
Mathematics and Computers in Simulation
Year:
2004

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Abstract

The limit cycles of nonlinear systems under the small stochastic disturbances are considered. The random trajectories of forced system leave the deterministic cycle and form some stochastic bundle around it. The probabilistic description of this bundle near cycle based on stochastic sensitivity function (SSF) is suggested. The SSF is a covariance matrix for periodic solution of linear stochastic first approximation system. This matrix is a solution of the boundary problem for linear matrix differential equation. For 3D-cycles this matrix differential equation on the basis of singular expansion is reduced to the system of three scalar equations only. The possibilities of SSF to describe some peculiarities of stochastically forced Roessler model are demonstrated.