Homoclinic and heteroclinic bifurcations in vector fields

An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin’s method, Shil’nikov variables and homoclinic centre manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinite-dimensional systems, are reviewed briefly.