Codimension 2 bifurcation of twisted double homoclinic loops

Abstract

A local active coordinates approach is employed to obtain bifurcation equations of twisted double homoclinic loops. Under the condition of one twisted orbit, we obtain the existence and uniqueness and of the 1–1 double homoclinic loop, 2–1 double homoclinic loop, 2–1 right homoclinic loop, 1–1 large homoclinic loop, 2–1 large homoclinic loop and 2–1 large period orbit. For the case of double twisted orbits, we obtain the existence or non-existence of 1–1 double homoclinic loop, 1–2 double homoclinic loop, 2–1 double homoclinic loop, 2–2 double homoclinic loop, 2–1 large homoclinic loop, 1–2 large homoclinic loop, 2–2 large homoclinic loop, 2–2 right homoclinic loop, 2–2 large homoclinic loop, 2–2 left homoclinic loop and 2–2 large period orbit. Moreover, the bifurcation surfaces and their existence regions are given. Besides, bifurcation sets are presented on the 2 dimensional subspace spanned by the first two Melnikov vectors.