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Stochastic stability of damped Mathieu oscillator parametrically excited by a Gaussian noise. (English) Zbl 1264.34117

Summary: We analyze the stochastic stability of a damped Mathieu oscillator subjected to a parametric excitation of the form of a stationary Gaussian process, which may be both white and coloured. By applying deterministic and stochastic averaging, two Itô’s differential equations are retrieved. Reference is made to stochastic stability in moments. The differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. A necessary and sufficient condition of stability in the moments of order \(r\) is that the matrix \(\mathbf A_r\) of the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. In the case of the first moments (averages), critical values of the parameters are expressed analytically, while for the second moments the search for the critical values is made numerically. Some graphs are presented for representative cases.

MSC:

34F05 Ordinary differential equations and systems with randomness
60H25 Random operators and equations (aspects of stochastic analysis)
34D20 Stability of solutions to ordinary differential equations
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