A. A. GUBKIN
Russia, 620083, Yekaterinburg, str. Lenina st., 51,
Urals state university,
Department of mathematics and mechanics
L. B. RYASHKO
Russia, 620083, Yekaterinburg, str. Lenina st., 51,
Urals state university,
Department of mathematics and mechanics
The paper is devoted to the stability analysis for stochastic
differential equations with periodic coefficients under parametric
random disturbances. Such systems are basic mathematical models
for many real oscillatory processes. Multiplicative kind of
disturbances makes their analysis more difficult.
In the systems with random disturbances one can study different
kinds of stability. The kind considered here is the stability in
mean squares. In the basis of the method developed in this paper
lies the spectral criterion, allowing to reduce the question on
stability to evaluating of spectral radius of some positive
operator. For evaluating the spectral radius a simple iterative
method is suggested.
The main theoretical result in the paper is the proof of iterative
method convergence, obtained with the help of theory of positive
operators. Quite simple sufficient conditions of convergence allow
to use the method for wide class of systems analysis.
The suggested method in the difficult task of stability domains
construction is more effective in comparison to traditionally used
method of second moments.
The work was supported by RFFI grant 04-01-96098ural.