Sur L’Existence et le Comportement de Cycles de Certaines Équations Différentielles Non Linéaires (Solutions Periodiques de Systemes Autonomes)

Abstract

This paper deals with the properties of cycles and of their frequency $\omega $ for some nonlinear differential equations depending on a parameter $\lambda $ (second order differential equation, coupling of two second order equations, and third order equation). The use of inequalities from the theory of series and of integrals permits us to establish some norms of the solutions as a function of $\omega $, to relate $\lambda $ and $\omega $, and to study the dependence of the norms on $\lambda $ and $\omega $. The equations, once they have been put in the form of a transformation on a function space, allow us, through the use of the Leray–Schauder theorem, to prove, on the basis of results for small $| \lambda |$, the existence of cycles for all values of the parameter. We sketch what happens to the cycles as $| \lambda | \to \infty $.