Periodic or homoclinic orbit bifurcated from a heteroclinic loop for high-dimensional systems

Abstract Consider an autonomous ordinary differential equation in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{n} , which has a heteroclinic loop. Assume that the heteroclinic loop consists of two degenerate heteroclinic orbits <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:math> {\gamma }_{1} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {\gamma }_{2} and two saddle points with different Morse indices. The degenerate heteroclinic orbit in the sense that variational equation along the heteroclinic orbit <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>γ</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {\gamma }_{i} has <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {d}_{i} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:math> {d}_{i}\gt 1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> i=1 , 2) linearly independent bounded solutions. By the different Morse indices and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>d</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:math> {d}_{i} , the heteroclinic loop is a heterodimensional loop, at the same time, it has high codimension in this situation. Applying Lin’s method to the heteroclinic loop, we derived the bifurcation function. The zeros of this function correspond to the conditions under which periodic or homoclinic orbits can bifurcate from the high-codimension heteroclinic loop in the perturbed system.