On integrability of the Nosé–Hoover oscillator and generalized Nosé–Hoover oscillator

Abstract

The Nosé–Hoover system is a basic primitive model for the molecular dynamics simulations, which describes the equilibrium characterized by canonical distributions at a constant temperature. Its simplest form, called the Nosé–Hoover oscillator (NH), is a three-dimensional quadratic polynomial system that admits both regular invariant tori and chaotic trajectories. Recently, a similar system, called the generalized Nosé–Hoover oscillator (GNH), was constructed to show the coexistence of invariant tori and topological horseshoe for time-reversible systems in [Formula: see text]. This paper aims to study the integrability of both NH and GNH models. We show that (i) in the case of [Formula: see text], both NH and GNH models are integrable by quadratures and the general solutions are given; (ii) in the case of [Formula: see text] the non-existence of either global analytic first integrals or Darboux first integrals of two models are discussed and a complete characterization of Darboux polynomials and exponential factors are given; (iii) both NH and GNH models are not rationally integrable in an extended Liouville sense except for several parameter values by an extended Morales–Ramis theory; (iv) the reduced systems of NH and GNH models at the Poincaré compactification balls are integrable, which yields complete descriptions of dynamics at infinity for NH and GNH models. Interestingly, the topological structure of NH and GNH models at infinity strongly depends on the sign of [Formula: see text]. These results may help us better understand the complex and rich dynamics of nonlinear time-reversible systems.