Codimension 3 Non-resonant Bifurcations of Rough Heteroclinic Loops with One Orbit Flip*

Abstract

Heteroclinic bifurcations in four dimensional vector fields are investigated by setting up a local coordinates near a rough heteroclinic loop. This heteroclinic loop has a principal heteroclinic orbit and a non-principal heteroclinic orbit that takes orbit flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit and 1-periodic orbit are studied. The existence of the two-fold or three-fold 1-periodic orbit is also obtained.

Homoclinic flip bifurcations accompanied by transcritical bifurcation

“…According to Liouville's formula, det Φ(T ) = 0 implies det Φ(−T ) = 0, so we must have φ 12 = 0. Similar to that of [13,28], the remaining can be proved.…”

“…In recent years, many papers focus on the following three different codimension two bifurcations: resonant leading eigenvalues, an inclination-flip condition or an orbit-flip condition. The first possibility was worked out in [11], the second in [12], and the third in [13,14]. But the corresponding problems with nonhyperbolic equilibrium are rarely investigated (see [15]) where the bifurcation of the inclination-flip homoclinic orbit associated to a saddle-node singularity was studied.…”

Nonresonant Bifurcations of Heteroclinic Loops With One Inclination Flip

“…The following two lemmas can be shown in almost the same way as given in [Shui & Zhu, 2006], so we omit their proofs.…”