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