Existence Problem of Invariant Torus Particle Motion in Rotating Nonlinear Dynamical Systems

In order to study whether the invariant torus of integrable Hamiltonian systems is retained under small perturbations, we established the Hamiltonian equations in polar coordinates. Using the first integral of the energy conservation equation, the transformation of the second-order state variable from a system with two degrees of freedom into a system with a single degree of freedom was analysed. Secondly, based on the Kolmogorov-Arnold-Moser (KAM) theorem, the existence of invariant tori in the perturbed system was confirmed. Finally, numerical simulations were performed to elucidate the analysis. The results show that the time history curve of the system is periodic, the phase portrait is dense, and the Poincaré map is a closed curve. The system is quasi-periodic, and the invariant torus of the integrable Hamiltonian system is shown to still exist under small perturbations. Moreover, the closed curve Poincaré mapping corresponds to the KAM invariant closed curve.