Invariant Tori and Heteroclinic Invariant Ellipsoids of a Generalized Hopf–Langford System

Abstract

In this paper, the bounded invariant surfaces of a generalized Langford system are discussed. Firstly, by the first integrals of systems restricted in the Poincaré sections of a periodic orbit, the accurate expressions of a heteroclinic orbit, a family of invariant tori and a heteroclinic invariant ellipsoid are given near a periodic orbit. Then, applying the successor functions to compute the periods of periodic orbits for the systems in the Poincaré sections, we present the parameter conditions for the existence of periodic orbits with any periods on these invariant tori. Finally, using the averaging theory and the theory of the Poincaré bifurcation and by determining the monotonicity of the ratio of two Abelian integrals, we give the conditions respectively such that the system has a unique invariant torus and a unique heteroclinic invariant ellipsoid near a zero-Hopf equilibrium.