On two-dimensional area-preserving mappings with homoclinic tangencies. (English. Russian original) Zbl 1041.37033

Gonchenko, S. V.; Shilnikov, L. P.

Dokl. Math. 63, No. 3, 395-399 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 378, No. 6, 727-732 (2001).

The authors consider \(C^r\)-smooth (\(r\geq 3\)) two-dimensional area-preserving diffeomorphisms \(f\) having a saddle fixed point \(O\) with multiplicators \(\lambda\) and \(\lambda^{-1}\), where \(|\lambda|<1\), and a nontransversal homoclinic orbit \(\Gamma\) at whose points the stable and unstable invariant manifolds of the saddle \(O\) have quadratic tangency. The case of \(\lambda>0\) is investigated with the aim to study the structure of the set \(N\) of trajectories lying entirely in a small neighborhood of the closure of a homoclinic orbit \(\Gamma\).

Reviewer: Boris V. Loginov (Ul’yanovsk)

MSC:

37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
37C25	Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
37D05	Dynamical systems with hyperbolic orbits and sets
37G25	Bifurcations connected with nontransversal intersection in dynamical systems
37G30	Infinite nonwandering sets arising in bifurcations of dynamical systems

Keywords:

two-dimensional area-preserving mapping; homoclinic orbit; structure of the set of trajectories