On Energy Surfaces and the Resonance Web

A framework for understanding the global structure of near-integrable n DOF Hamiltonian systems is proposed. To this aim two tools are developed—the energy-momentum bifurcation diagrams and the branched surfaces. Their use is demonstrated on a few near-integrable 3 DOF systems. For these systems possible sources of instabilities are identified in the diagrams, and the corresponding energy surfaces are presented in the frequency space and by the branched surfaces. The main results of this formulation are theorems which describe the connection between changes in the topology of the energy surfaces and the existence of resonant lower dimensional tori.