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Lyapunov exponent and rotation number of two-dimensional linear stochastic systems with small diffusion. (English) Zbl 0641.60065
The Lyapunov exponent (growth rate) \(\lambda\) and rotation number \(\alpha\) of all \(2\times 2\) systems \[ dx=Ax dt+\epsilon \sum^{r}_{i=1}B_ k x\circ dW^ k \] (except for nilpotent A) are computed for small \(\epsilon\). On basis of available formulas which represent those numbers as expectation of a simple function on the 1- dimensional projective space with respect to an invariant measure on that space, \(\lambda\) and \(\alpha\) are expanded in powers of \(\epsilon^ 2\) by means of singular perturbation and large deviation theory.
How white noise effects the radial and rotational behavior can be read off from the coefficients of the expansion, given in terms of A and B. Depending on \((A_ 1,B_ 1,...,B_ r)\) noise may e.g. stabilize or destabilize the system \(\dot x=Ax\). The results include simple formulas for the random oscillator.
Reviewer: V.Wihstutz

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
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