Mechanism of the non-monotonic dynamical transition

Explain the mechanism by which the trigonometric Nosé–Hoover oscillator transitions from conservative-like dynamics at small coupling, to strongly dissipative dynamics with chaotic attracting sets at intermediate coupling, and back to conservative-like dynamics at large coupling.

Background

The numerical results show a non-monotonic dependence of the phase-space organization on the coupling parameter $a$. For small $a$, regular and conservative-like structures dominate; at intermediate values, chaotic attracting sets and strong average contraction emerge; and for larger $a$, regular structures reappear and attracting structures gradually disappear.

The paper reports that additional numerical experiments suggest persistence of this behavior for very large $a$, but does not provide an analytical explanation of the transition. The authors explicitly identify understanding this mechanism as an unresolved problem.