Oscillations on one dimensional time dependent center manifolds: algebraic curves approach

Abstract

We consider a one dimensional time dependent ODE of degree n ≥2 as the restriction of an arbitrary nonautonomous ODE to the associated one dimensional center manifold. Then, we present an algorithm for computing time dependent algebraic curves of m-th degree with m ≤n−1 . This computation leads us to a m dimensional time dependent bifurcation equation. We determine oscillatory behaviors of the system with the help of the bifurcation equation. Finally, we complete the method for a general parametric planar system and find periodic solutions. The method can be applied for a wide range of nonautonomous systems and does not have restrictions of classical methods such as the Poincaré map.