BIFURCATIONS OF 2-2-1 HETERODIMENSIONAL CYCLES UNDER TRANSVERSALITY CONDITION

Abstract

Bifurcations of generic 2-2-1 heterodimensional cycles connecting to three saddles, in which two of them have two-dimensional unstable manifolds, are studied by setting up a local moving frame. Under a certain transversal condition, we firstly present the existence, uniqueness and noncoexistence of a 3-point heterodimensional cycle, 2-point heterodimensional or equidimensional cycle, 1-homoclinic cycle and 1-periodic orbit bifurcated from the 3-point heterodimensional cycle, and the bifurcation surfaces and bifurcation regions are located when the u-component [Formula: see text] of the vector [Formula: see text] under the Poincaré mapping [Formula: see text] is nonzero. Conversely, we obtain some sufficient conditions such that the bifurcation of a 2-fold 1-periodic orbit occurs and a 1-periodic orbit coexists with the surviving heterodimensional cycle, showing some new bifurcation behaviors different from the well-known equidimensional cycles.