Bifurcation Complexity from Orbit-Flip Homoclinic Orbit of Weak Type

In this paper, we study the unfolding of a three-dimensional vector field having an orbit-flip homoclinic orbit of weak type. Such a homoclinic orbit is a degenerate version of the so-called orbit-flip homoclinic orbit. We show the existence of inclination-flip homoclinic orbits of arbitrary higher order bifurcating from the unperturbed system. Our strategy consists of using the local moving coordinates method and blow up in the parameter space. In addition, the numerical existence of the orbit-flip homoclinic orbit of weak type is presented based on the truncated Taylor expansion and the numerical computation for both the strong stable manifold and unstable manifold.

Bifurcations Near the Weak Type Heterodimensional Cycle

The aim of this paper is to show the bifurcation phenomena near the weak type heterodimensional cycle when the orbit flip and inclination flip occur simultaneously in its nontransversal heteroclinic orbit. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the persistence of heterodimensional cycles, the coexistence of the heterodimensional cycle and periodic orbits or homoclinic orbits, and the existence of bifurcation surfaces of homoclinic orbits or the periodic orbits are discussed under small perturbations. Moreover, an example is given to show the existence of the system which has a heterodimensional cycle with orbit flip and inclination flip.

Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria

The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orbits and the existence of two or three heteroclinic orbits.

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

NONRESONANT BIFURCATIONS OF HETEROCLINIC LOOPS WITH ONE INCLINATION FLIP

Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.