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On stochastic stability of Lyapunov exponents of equations of arbitrary order. (Russian) Zbl 0625.34068

The paper deals with the following two equations: \[ (1)\quad x^{(n)}+a_ 1(t)x^{(n-1)}+...+a_ n(t)x=0, \]
\[ (2)\quad y^{(n)}+(a_ 1(t)+c_{\xi,1}(t,\omega))y^{(n-1)}+...+(a_ n(t)+c_{\xi,n}(t,\omega))y=0. \] In (1), \(a_ j(t)\) are deterministic functions while in the perturbed equation (2), \(c_{\xi,j}(t,\omega)\) are stochastic processes depending on the parameter \(\xi\). The problem is to study the Lyapunov numbers of (1) and (2). Representing firstly (1) and (2) in the form of linear systems of the type \(\dot x=A(t)x\) and \(\dot y=(A(t)+C_{\xi}(t,\omega))y,\) the author finds conditions ensuring the almost sure continuity of the Lyapunov numbers of (2) as \(\xi\to 0\) and their closeness to the Lyapunov numbers of (1). Some particular cases are considered which are of independent interest.
Reviewer: J.M.Stoyanov

MSC:

34F05 Ordinary differential equations and systems with randomness
34D05 Asymptotic properties of solutions to ordinary differential equations
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