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Extra info for Contribution to the theory of Lyapunov exponents for linear systems of differential equations

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K. P. Persidskii, "On the stabil~ty of motion in the first approximation," Mat. , 40, No. 3, 284-292 (1933). I. G. Petrovskii, Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (]970). M. M. Postnikov, Lectures on Geometry. llnd Semester. Linear Algebra and Differential Geometry [in Russian], Nauka, Moscow (1979). I. N. Sergeev, "Exact upper bounds of mobility for the Lyapunov exponents of a system of differential equations and the behavior of exponents under perturbations that tend to zero at infinity," Differents.

1. The normal decomposition E(A) = Elg:~... @Ea, of the space of solutions of equation A ~ is integrally separated if and only if every equation B ~ (A) admits a normal orthogonalizable decomposition with the same characteristics. 2. Suppose all Lyapunov exponents of equation A ~ are distinct. 6). 6~ In the case n = 2 a stronger statement, which requires no constraints on the Lyapunov exponents, holds true. 3. Let A ~ [ (R 2). Then the diagonalizability of all equations equivalent to the existence of an integrally separated decomposition B ~ (A) is E(A) =L@N, d i m L = d i m N = 1.

Functionals A i and are residual. 10 ~ . of space ~ . Proof. The maximal (minimal) i-th exponent is upper Pick an arbitrary equation A ~ . (lower) semicontinuous at any point The upper semicontinuity of functional %max I at the point A is verified as follows. 1, for every ~ > 0 there is an ~ > 0 sup ~ (B)< ~ x (A) + 6. p(A,B)<~ But t h e n f o r e v e r y e q u a t i o n C from the r ~max (C) • of equation sup A ~ (B )NK ~ ~ , tA , ) + 6, _ ~m p(B,C)

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### Contribution to the theory of Lyapunov exponents for linear systems of differential equations by Sergeev

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