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**Additional resources for Contribution to the theory of Lyapunov exponents for linear systems of differential equations**

**Example text**

V. Fomin, Elements of the Theory of Functions and Functional Analysis, Graylock (]96]). A. M. Lyapunov, General Problem of the Stability of Motion [in Russian], Gostekhizdat, Moscow--Leningrad (1950). V. M. Millionshchikov, "A criterion for the smallness of the modification of the directions of solutions to a linear system of differential equations under small perturbations of the coefficients of the system," Mat. Zametki ~, No. 2, ]73-180 (1968). V. M. Millionshchikov, "Proof of the attainability of central exponents," Sib.

The boundedness Functional Then the following simple ~i is locally bounded at the point A. , n}. 7 ~ a13 < aA + e, aB>/--aa--e, of ~i in any e-neighborhood of equation A. l ~i has at the point A finite upper and lower limits (see [II]): lim e~+O Me, lim m e. 1 ~ shows that the limits are actually finite. 3). We call the numbers ~ a ~ (A) - - lira k~(B), B~A and ~min(A) ~ lira ~ (B), B~A the maximal and respectively the oscillation of the i-th exponent at the point A. 3~ minimal i-th exponent of equation A, and call the difference follow the properties: ~ " (A) < ~ (A) < z?

2) show that solutions x and y lie either between solutions --a and b' or between solutions a and g'. 9)), which contradicts (I0. I0). The theorem is proved. , there are equations A ~ and B ~ D (A) such that A is diagonalizable and B is not. p. the nondiagonalizable equation B becomes diagonalizable. 's preserve a number of important properties of equations (see Secs. 9). 4 below shows, this is not always possible, even if for n = 2. Here is a good place to recall (see [16]) that every equation (and even every equation with integral separateness) can be made diagonalizable by arbitrarily small perturbations.

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

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