Heterodimensional Cycle Bifurcation With Orbit-Flip

Abstract

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.

Bifurcation Analysis of Homoclinic Flips at Principal Eigenvalues Resonance

“…Zhang in [10,11] continued to research on these problems and gave some theoretical proofs of the existence of -periodic orbit and -homoclinic orbit and also their existence regions via the method initially established in [18]. Besides these the flip heterodimensional cycles have also attracted attentions nowadays, see [16]. In this paper, we extend the method to fit a higher codimension case of 3 flips with resonant.…”

“…More recently, the flip of heterodimensional cycles or accompanied by transcritical bifurcation is got attention, see [15][16][17], the double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for the homoclinic orbit and the periodic orbit. But the research is not concerned with multiple flips.…”

Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips

“…By the theory of invariance manifolds and our hypotheses, the system (1.1) can be transformed into a normal form in some neighborhood U of the origin O as (see [2,7,12,13])…”

“…In recent years, the single flip of the heterodimensional cycle or accompanied by transcritical bifurcation is discussed more (see [2,6,7]), the double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for the homoclinic orbit and the periodic orbit are given.…”

Resonant Homoclinic Bifurcations with Orbit Flips and Inclination Flips

“…Recently, the flip of heterodimensional cycle or accompanied by transcritical bifurcation is discussed much (see [11][12][13]). The double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for homoclinic orbits and periodic orbits.…”