Bifurcations of Double Homoclinic Loops with Inclination Flip and Nonresonant Eigenvalues

Abstract

In this work, bifurcation analysis near double homoclinic loops with $W^s$ inclination flip of $Γ_1$ and nonresonant eigenvalues is presented in a four-dimensional system. We establish a Poincaré map by constructing local active coordinates approach in some tubular neighborhood of unperturbed double homoclinic loops. Through studying the bifurcation equations, we obtain the condition that the original double homoclinic loops are persistent, and get the existence or the nonexistence regions of the large 1-homoclinic orbit and the large 1-periodic orbit. At last, an analytical example is given to illustrate our main results.