Abstract: We consider a one-parameter family \(f_\mu\) of two-dimensional diffeomorphisms such that for \(\mu=0\) the diffeomorphism \(f_0\) has a transversal homoclinic orbit to a nonhyperbolic fixed point of arbitrary finite order \(n\geq 1\) of degeneracy, and for \(\mu>0\) the fixed point becomes a hyperbolic saddle. The goal of the paper is to give a complete description of structure of the set \(N_\mu\) of orbits from a sufficiently small fixed neighborhood of the homoclinic orbit. The main result of the work is a complete description for the set \(N_\mu\) of orbits entirely lying in a neighborhood of the homoclinic structure. It was shown that for \(\mu\geq 0\) the set \(N_\mu\) is hyperbolic (for \(\mu=0\) it is nonuniformly hyperbolic) and the dynamical system \(f_\mu\bigl|_{N_\mu}\) (the restriction of \(f_\mu\) to \(N_\mu\)) is topologically conjugate to some nontrivial subsystem of the topological Bernoulli scheme of two symbols. Thus, we generalize the classical result of Lukyanov and Shilnikov, obtained by them for the case when the fixed point is a nondegenerate saddle-node (\(n=1\)). In addition, we obtained new effective formulas for iterations of one-dimensional maps (maps in the restriction to the central manifold of the diffeomorphism \(f_\mu\)). These formulas are derived using some modification of the well-known methods of qualitative theory, such as the methods of embedding a map to a flow and the Shilnikov cross-maps method.
For citation: Gordeeva, O. V. and Gordeev, V. E. On the Structure of Neigborhood of Homoclinic Orbit to a Nonhyperbolic Fixet Point, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 19-34 (in Russian). DOI 10.46698/p1879-1111-4332-k
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