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

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.

Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or codimension 3 surfaces, are obtained.

HETERODIMENSIONAL CYCLE BIFURCATION WITH ORBIT-FLIP

The local moving frame approach is employed to study the bifurcation of a degenerate heterodimensional cycle with orbit-flip in its nontransversal orbit. Under some generic hypotheses, we provide the conditions for the existence, uniqueness and noncoexistence of the homoclinic orbit, heteroclinic orbit and periodic orbit. And we also present the coexistence conditions for the homoclinic orbit and the periodic orbit. But it is impossible for the coexistence of the periodic orbit and the persistent heterodimensional cycle or the coexistence of the homoclinic loop and the persistent heterodimensional cycle. Moreover, the double and triple periodic orbit bifurcation surfaces are established as well. Based on the bifurcation analysis, the bifurcation surfaces and the existence regions are located. An example of application is also given to demonstrate our main results.