# zbMATH — the first resource for mathematics

Rotation numbers of periodic orbits in the Hénon map. (English) Zbl 0729.58032
It is known that for an area-contracting diffeomorphism f: $${\mathbb{R}}^ 2\to {\mathbb{R}}^ 2$$ with more than one attractor, the boundaries between respective basins of attraction can be extremely complicated, e.g. can be fractal and contain infinitely many unstable periodic orbits. C. Grebogi, E. Ott and J. Yorke [Physica D 24, 243-262 (1987; Zbl 0613.58018)] have distinguished certain orbits of the basin boundary by being accessible by a path from the interior of the basin. For connected and simply connected basins of attraction the diffeomorphism f acts on the set S of accessible points as if they were on a circle, and so one can associate a rotation number. Although in a one-parameter family of invertible circle maps the rotation number varies continuously with the parameter, this factor is not valid for f/S. The purpose of the paper under review is to study changes in the accessible rotation numbers as map parameters vary. The main result is the following.
Theorem. Let $$f_{\lambda}$$ be a $$C^ 3$$-family of real analytic, area- contracting orientation-preserving maps of the plane. Suppose that a periodic saddle p forms a rotary homoclinic tangency at a point $$q\in orb(p)$$ when $$\lambda =\lambda_*$$ (i.e. there is a tangency of the unstable manifold of q with the stable manifold of an adjacent point from orb(p)). Then there exists a sequence $$\lambda_ n\to \lambda_*$$ and a sequence of saddles $$p_ n\to q$$ such that each saddle $$p_ n$$ has a rotary homoclinic tangency at $$\lambda_ n.$$
A numerical investigation of the Henon map as a particular case is given, too.

##### MSC:
 37B99 Topological dynamics 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
Full Text:
##### References:
 [1] Alligood, K., Tedeschini-Lalli, L., Yorke, J.: Metamporphoses: Sudden jumps in basin boundaries. Commun. Math. Phys. (to appear) · Zbl 0746.58053 [2] Alligood, K., Yorke, J.: Accessible saddles on fractal basin boundaries. Preprint · Zbl 0767.58023 [3] Aronson, D., Chory, M., Hall, G., McGehee, R.: Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer assisted study. Commun. Math. Phys.83, 303-354 (1982) · Zbl 0499.70034 · doi:10.1007/BF01213607 [4] Birkhoff, G.D.: Sur quelques courbes fermees remarquables. Bull. Soc. Math. France60, 1-26 (1932) · Zbl 0005.22002 [5] Cartwright, M.L., Littlewood, J.E.: Some fixed point theorems. Ann. Math.54, 1-37 (1951) · Zbl 0058.38604 · doi:10.2307/1969308 [6] Gavrilov, N., Silnikov, L.: On the three dimensional dynamical systems close to a system with a structurally unstable homoclinic curve. I. Math. USSR Sbornik17, 467-485 (1972); II. Math. USSR Sbornik19, 139-156 (1973) · Zbl 0255.58006 · doi:10.1070/SM1972v017n04ABEH001597 [7] Grebogi, C., Ott, E., Yorke, J.: Basin boundary metamorphoses: changes in accessible boundary orbits. Physica24 D, 243-262 (1987) · Zbl 0613.58018 [8] Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0515.34001 [9] Hammel, S., Jones, C.: Jumping stable manifolds for dissipative maps of the plane. Preprint · Zbl 0686.58008 [10] Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys.50, 69-78 (1976) · Zbl 0576.58018 · doi:10.1007/BF01608556 [11] Hockett, K., Holmes, P.: Josephson’s junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets. Ergod. Th. Dynam. Syst.6, 205-239 (1986) · Zbl 0582.58020 [12] Newhouse, S.: Diffeomorphisms with infinitely many sinks. Topology13, 9-18 (1974) · Zbl 0275.58016 · doi:10.1016/0040-9383(74)90034-2 [13] Robinson, C.: Bifurcation to infinitely many sinks. Commun. Math. Phys.90, 433-459 (1983) · Zbl 0531.58035 · doi:10.1007/BF01206892
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.