On the structure of separatrix-swept regions in singularly-perturbed Hamiltonian systems

. In this paper, we study the invariant structures in separatrix-swept regions of adiabatic (singularly-perturbed) planar Hamiltonian systems. Our main result is that lobe area is 0(1) asymptotically. This result has important consequences for determining the structure in the region complementary to which Kruskal's adiabatic invariance theory applies, for transport in many applications, and for attempts at proving the existence of an 0(1)-sized stochastic or chaotic region in applications. We use adiabatic Melnikov perturbation theory to approximate an exact action-theoretic result which is a generalization of results in (16].

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