Pathology in dynamical systems. III. Analytic Hamiltonians

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This paper presents practical hypotheses for proving the existence of nondegenerate (i.e., transverse) homoclinic and heteroclinic orbits to hyperbolic periodic orbits, and the attendant embeddings of the Smale horseshoe mapping, in real analytic Hamiltonian systems with two degrees of freedom. The results are applied to the H&on-Heiles Hamiltonian in Section 5. The paper extends results of [l, 21, w h ere analyticity was not assumed. Moreover, due to difficulties in proving the hyperbolicity of the relevant periodic orbits, subsequently overcome in [16], these earlier papers only investigated approximate embeddings of a “topological” horseshoe mapping, and at the expense of a considerable amount of technical detail. Here the topological technicalities have been substantially reduced, and the true embedding is obtained. With the exception of the verification of certain hypotheses in the example of Section 5, this work can be read independently of [l, 21. Section 1 discusses how nondegenerate homoclinic orbits can be obtained from “topologically” nondegenerate (seminondegenerate) ones, and provides a complete proof of an assertion in [7]. Section 2 then details conditions for proving the existence of topologically nondegenerate heteroclinic and homoclinic orbits. In Section 3 the results of the first two sections are examined in the context of real analytic Hamiltonians. In particular, a simplification of a set of hypotheses of [2] is given, which hypotheses must be verified when applying