Predictions of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems

In this paper, the mechanism of chaotic motion in nonlinear Hamiltonian systems is discussed based on the KAM theory and resonance overlap criterion. The internal resonances and the corresponding chaotic motions are determined analytically for weak interactions. A numerical method based on the energy spectrum is presented for prediction of quasi-periodic and chaotic motions in nonlinear Hamiltonian systems. The presented numerical method can be applied to integrable, nonlinear Hamiltonian systems with many degrees of freedom. A 2-DOF integrable, nonlinear Hamiltonian system is investigated as an example for demonstration of the procedure to numerically determine the chaotic motion in nonlinear Hamiltonian systems. Finally, the Poincare mapping surfaces of chaotic motions for such nonlinear Hamiltonian systems are illustrated. The phase planes, displacement surfaces (or potential domains), and the velocity surfaces (or kinetic energy domains) for the chaotic and quasi-periodic motions are illustrated. The analytical estimates of regular and chaotic motions in nonlinear Hamiltonian systems need to be further investigated. The mathematical theory should be developed for a better prediction of chaotic and quasi-periodic motions in nonlinear Hamiltonian systems with many degrees of freedom.