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Lyapunov exponents as functions of a parameter. (English. Russian original) Zbl 0705.34058

Math. USSR, Sb. 65, No. 2, 369-384 (1990); translation from Mat. Sb., Nov. Ser. 137(179), No. 3(11), 364-380 (1988).
It is well-known that Lyapunov exponents can be discontinuous (and upper semi-discontinuous) functions of system’s parameters. The author shows that they are upper semi-continuous functions “as a rule”.
Let \(\dot z=A(t,\mu)z\) be the system in variations of the system \(\dot x=f(t,x,\mu)\) for the solution \(x_{\mu}(t)\), \(0\leq t<\infty\), \(\mu \in (\mu_ 1,\mu_ 2)\) and some natural suppositions are fulfilled. Let \(\lambda_ 1(\mu)\geq...\geq \lambda_ n(\mu)\) be the characteristic exponents of the system \(\dot z=A(t,\mu)z\). Then: (I) The functions \(\lambda_ j(\cdot)\) belong to the second Baire class. (II) They are continuous functions on some dense subset \(\Gamma \subset (\mu_ 1,\mu_ 2)\) of \(G_{\delta}\)-type. (III) The set of all points of upper semi-discontinuity of all functions \(\lambda_ j(\cdot)\) contains some set of the second category.
Some more general theorems are also proved.
Reviewer: V.Yakubovich

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
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