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

Abstract

Consider an autonomous ordinary differential equation in Rn{{\mathbb{R}}}^{n}, which has a heteroclinic loop. Assume that the heteroclinic loop consists of two degenerate heteroclinic orbits γ1{\gamma }_{1}, γ2{\gamma }_{2} and two saddle points with different Morse indices. The degenerate heteroclinic orbit in the sense that variational equation along the heteroclinic orbit γi{\gamma }_{i} has di{d}_{i} (di>1{d}_{i}\gt 1, i=1i=1, 2) linearly independent bounded solutions. By the different Morse indices and di{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.

Keywords

• heteroclinic connections
• lin’s method
• exponential dichotomy
• bifurcation
• 34c23
• 34c25
• 34c37