Trigonometric Nosé--Hoover oscillator: chaos, periodic orbits and integrability

Abstract: We introduce a trigonometric version of the Nosé-Hoover oscillator in which the quadratic mechanical terms and unbounded thermostat coupling are replaced by bounded trigonometric functions. This formulation replaces the harmonic potential by a pendulum-type potential and confines the thermostat interaction to a bounded periodic form. The resulting two-parameter system is naturally defined on the three-dimensional torus and reduces near the origin, to leading order, to the classical polynomial Nosé-Hoover model. We investigate its global dynamics using Poincaré sections, bifurcation diagrams, Lyapunov spectra, Kaplan-Yorke dimensions, and the Lyapunov Integrability Test (LIT). The numerical results reveal the coexistence of regular and chaotic dynamics and characterize changes in dissipative behavior across the parameter plane. We then analyze the two limiting cases associated with the parameter axes. For $a=0$, we construct two functionally independent first integrals on regular domains, whereas for $b=0$ the dynamics reduces to a family of two-dimensional systems on invariant tori, which are analyzed using Darboux polynomials and exponential factors. First-order averaging near the intersection of these integrable limits yields periodic solutions bifurcating from unperturbed periodic orbits and an obstruction to regular $C<sup>1$ first integrals in their neighborhoods. Independently, differential Galois theory applied to the normal variational equation, together with the Ayoul-Zung and Li-Shi criteria, excludes meromorphic $B$-integrability and non-constant meromorphic first integrals near a particular non-equilibrium phase curve for $ab\neq0$. Thus, despite retaining the local structure of the classical Nosé-Hoover oscillator, its trigonometric counterpart exhibits markedly different global dynamics and integrability.