On dynamical properties of diffeomorphisms with homoclinic tangencies
- Gonchenko, Sergey V.
- Shilnikov, Leonid P.
- Turaev, Dmitry
2010 Mathematics Subject Classification
- 37G25 37D45 37C15 37G30 34C20 34C27
- Newhouse regions, Henon map, renormalization, strange attractor, chaos, moduli, stable periodic orbits
We study bifurcations of a homoclinic tangency to a saddle fixed point without non-leading multipliers. We give criteria for the birth of an infinite set of stable periodic orbits, an infinite set of coexisting saddle periodic orbits with different instability indices, non-hyperbolic periodic orbits with more than one multiplier on the unit circle, and an infinite set of stable closed invariant curves (invariant tori). The results are based on the rescaling of the first-return map near the orbit of homoclinic tangency, which is shown to bring the map close to one of four standard quadratic maps, and on the analysis of the bifurcations in these maps.