Time Dependent Nonlinear Oscillations with Many Periodic Solutions

Abstract

We investigate the local structure of nonlinear operators defining time dependent nonlinear oscillations. We explicitly describe the local structure of such operators using as models stable mappings between finite-dimensional spaces. This structure theorem allows us to prove the existence of many periodic solutions for such nonlinear oscillations. We determine the maximum number of such periodic solutions occurring near equilibrium. As one consequence we deduce that there exist oscillations with polynomial nonlinear terms of degree $2n$ for which there exist $(n + 1)^2 $ periodic solutions.