The Existence of Invariant Tori and Quasiperiodic Solutions of the Nosé–Hoover Oscillator

Abstract

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, $x^{\prime }=y,y^{\prime }=-x-yz,\mathrm{and}{z}^{\prime }=y^2-a$ , where $a$ is a positive real parameter. Under a series of transformations, it is transformed into a 2-dimensional reversible system about action-angle variables. By applying a version of twist theorem established by Liu and Song in 2004 for reversible mappings, we find infinitely many invariant tori whenever $a$ is sufficiently small, which eventually turns out that the solutions starting on the invariant tori are quasiperiodic. The discussion about quasiperiodic solutions of such 3-dimensional system is relatively new.