A canonical system of two differential equations with periodic coefficients and the Poincaré–Denjoy theory of differential equations on a torus

A. I. Perov

Voronezh State University, Faculty of Applied Mathematics, Informatics and Mechanics, Voronezh

Abstract: The passage from Cartesian to polar coordinates in a canonical system with periodic coefficients gives rise to a nonlinear differential equation whose right-hand side is periodic in time and the polar angle and thus this equation can be regarded as a differential equation on a torus. In accord with Poincaré–Denjoy theory, the behavior of a solution to a differential equation on a torus is characterized by the rotation number and some homeomorphic mapping of a circle onto itself. We study connections of strong stability (instability) of a canonical system, including the membership in the nth stability (instability) domain, with the rotation number and fixed points of this mapping.

Keywords: canonical system, Floquet multiplier, domains of strong stability, Poincaré–Denjoy theory, rotation number.

UDC: 517.926

Received: 27.03.2008

 English version: Siberian Mathematical Journal, 2010, 51:2, 301–312

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