A theorem on the first heteroclinic tangency in two-dimensional maps. Orientation-preserving cases

Abstract

Using the properties of the Jordan curve, the following theorem on the heteroclinic tangency in orientation-preserving two-dimensional maps is proved: LetT _μ :R ² →R ² be a one-parameter family ofC ¹ diffeomorphisms andJ=DetDT _μ be such that 0<J⩽1 or 1⩽J<∞. LetW ⁿ _u be the unstable manifold of a hyperbolicn-cycle andW ^m _s the stable manifold of a hyperbolicm-cycle. Suppose that forμ<μ _c ,W ⁿ _u andW ^m _s have no common points, and that forμ>μ _c ,W ⁿ _u andW s/m have a transversal heteroclinic point. Then atμ=μ _c ,W ⁿ _u andW ^m _s are in the first asymptotic heteroclinic tangency except for the following three cases: (1)n=m; both cycles are without reflection. (2)m=2n; then- andm-cycles are with and without reflection, respectively; (3)n=2m; then- andm-cycles are without and with reflection, respectively.