The Periodic Solutions Of Impulsive Differential Equations And Rayleigh Equations

Impulsive differential equations and Rayleigh equations are two important models of ordinary differential equations. Attetions are paled to study the periodic solutions of these two equations and correlated problems.The characteristic of this article is the integration of functional method and geometry. In the abstract functional frame, we project the solutions onto phase-plane, analysis their geometry character. Based on this geometry point, we get the priori estimate necessary for coincidence degree, and then get periodic solutions of Rayleigh equations which increase on one side, and the periodic solutions of Rayleigh equations under the signed conditions.With regard to the study of periodic solutions of impulsive equations, most results were gotten under the conditions that the impulse is sublinear, and there aren't any results about infinite periodic solutions. In this article, impulsive equations are regarded as the composition of the flow and the maps, we study the the geometry character of Poincare map on phaseplane, when the impulse is homeomorphism, we put out the analytic definition of twist angle arosed by impulse ingeniously, when the impulse is polynomial homeomorphism, we get infinite periodic solutions by Poincare-Birkhoff twist theorem. As for the more general case, that is to say, the impulse isn't homeomorphism, we improve the Poincare-Birkhoff twist theorem, and get partial twist theorem, by this, we study the linear impulsive differential equations that one of the impulse terms is linear and degenerative. In the end, we get a disturbance lemma of impulsive differential equations, based on this lemma, we study the periodic solutions of nonconserative impulsive differential equations.