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

By Sergeev

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16), property a) of condition 3. showed that condition 2 implies condition 3. Thus, we have Finally, suppose that condition 3 is fulfilled. , Ar} (here A i = ~ E i --- ~i' J = i . . . 5~ The p r o o f (A), i=1 ..... 8 is complete. 9 This theorem yields a criterion for the stability of the lowest exponent k = I) and also a criterion for the stability of all exponents (for k = n). 9. 's) if and only if there exists an integrally separated decomposition E ( A ) =El (~ E~, and COROLLARY 1 1 . 1 0 (s'ee a l s o [17, The exponents 4]).

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. , 16, No. 3, 438-448 (]980). I.

3. 13) we may assume t h a t for each j = l, 2, Ixj+l qOI = Ixi(tj) I (this can be achieved by successively multiplying each solution xj+1, j = 1, 2, ... 3). 11), we have xi+l (t j) = h (xi (ti), xj+x (t j), (pj), ] = 1, 2 .... 11~ Now for each j = I, 2, ... we rotate at time tj solution xj in the direction of the vector xj+1(tj) by the angle ~j. 12 ~ , solution xj hits the vector xj+~(tj) at time tj. Equation B obtained in this way has a solution b which satisfies the conditions b (6-0 =xj (tj_l) for all j = I, 2, .