Bifurcations of Twisted Fine Heteroclinic Loop for High-Dimensional Systems

In the paper, under twisted conditions, we consider the bifurcation problem of the fine heteroclinic loop with two hyperbolic critical points for high-dimensional systems. By using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits as the local coordinate system in the small tubular neighborhood of the heteroclinic loop, we construct the Poincaré maps and obtain the bifurcation equations. Then, by considering the small nonnegative solutions of the bifurcation equations, we get the main results of the reservation of the heteroclinic orbits, the existence and existence regions, the coexistence and coexistence regions of the 1-homoclinic loop, 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the bifurcation surfaces and graphs are given.