Analytical results on the existence of periodic orbits and canard-type invariant torus in a simple dissipative oscillator

Abstract

In this paper we consider a simple dissipative oscillator, determined by a two-parameter three-dimensional system of ordinary differential equations, obtained from the Nosé–Hoover oscillator by adding a small anti-damping term in its third equation. Based on numerical evidence, complex dynamics of this system was presented in a recent paper, such as the coexistence of periodic orbits, chaotic attractors and a stable invariant torus. Here we analytically prove the existence of a small periodic orbit from which a stable invariant torus bifurcates near the origin of the dissipative oscillator. We also show that the oscillations near the torus present a kind of relaxation oscillation behavior, like canard-type oscillations, commonly found in singularly perturbed systems. The obtained results extend and provide analytical proofs for some dynamical properties of the considered system, which were numerically described in the literature.