Integral separation, Lyapunov transformations, and the small change in the direction of the solutions of periodic linear systems of ordinary differential equations

[摘要] First, we discuss the relation between the integral separation of solutions of a linear periodic system x' = A(t)x (1) and the structure of its characteristic multipliers. Second, we investigate the reducibility of (1) by a complex or a real Lyapunov transformation. Third, we prove the property of the small change in the direction of solutions of omega-periodic system (1) as the result of a small perturbation of (1) under the following assumptions: (a) When the characteristic multipliers of (1) are distinct and the perturbation of (1) is admissible. The perturbation C(t) = B(t) - A(t) is called admissible to the system (1), if the characteristic multipliers rho(k) and <(rho)over tilde>(k) of the systems (1) and y' = B(t)y, (2) respectively, have the same argument, i.e., arg rho(k) = arg <(rho)over tilde>(k), k = 1,..., n. In this case, we also say that the perturbed omega-periodic system (2) is admissible to the system (1). (b) When the characteristic multipliers of (1) are real and distinct and the perturbation is arbitrary. (C) 1997 Academic Press.

[发布日期] 1997-07-15