The numerical prediction of chaos and quasi-periodic motion on the homoclinic surface of a two-degree-of-freedom (2-DOF) nonlinear Hamiltonian system is presented through the energy spectrum method. For weak interactions, the analytical conditions for chaotic motion in such a Hamiltonian system are presented through the incremental energy approach. The Poincaré mapping surfaces of chaotic motions for this specific nonlinear Hamiltonian system are illustrated. The chaotic and quasi-periodic motions on the phase planes, displacement subspace (or potential domains), and the velocity subspace (or kinetic energy domains) are illustrated for a better understanding of motion behaviors on the homoclinic surface. Through this investigation, it is observed that the chaotic and quasi-periodic motions almost fill on the homoclinic surface of the 2-DOF nonlinear Hamiltonian system. The resonant-periodic motions for such a system are theoretically countable but numerically inaccessible. Such conclusions are similar to the ones in the KAM theorem even though the KAM theorem is based on the small perturbation.

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