Global analysis of a generalized Nosé-Hoover oscillator

In this study, the generalized Nosé-Hoover oscillator is analyzed rigorously. We show that all trajectories not belonging to the two one-dimensional invariant manifolds (one is a straight line and the other is a unit circle) must traverse a two-dimensional unit disk infinitely many times transversally along the same direction in the state space. Consequently, all of the trajectories not located in the straight line invariant manifold have at least one ω-limiting point.