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

In this paper, we consider an equivalent form of the Nosé–Hoover oscillator, x′=y,y′=−x−yz, and z′=y2−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.