The area-preserving maps of Chapter 3 have given us a clear picture of the complex behavior in conservative systems as they undergo a transition to chaos. These maps, however, are somewhat special. For example, the whisker map describes the stochastic layer in the separatrix region of the Duffing oscillator, the standard map describes a small neighborhood of the stochastic layer, and the quadratic maps describe the neighborhood of isolated islands. In practice, we are often confronted with a physical system whose Hamiltonian we are given, and then we must determine as much as possible about its global dynamics. We have to ask: What regions of the phase space might undergo a to transition chaos and for what parameter values does it happen? For such systems, it is usually not possible to construct an area-preserving map analytically, but there is still a great deal we can learn about the global behavior by working directly with the Hamiltonian.
KeywordsPhase Space Global Property Stable Manifold Primary Resonance Nonlinear Resonance
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