On a codimension $2$ bifurcation of heteroclinic orbits

A codimension-two resonant bifurcation from a heteroclinic cycle with complex eigenvalues

Robust heteroclinic cycles between equilibria lose stability either through local bifurcations of their equilibria or through global bifurcations. This paper considers a global loss of stability termed a ‘resonant’ bifurcation. This bifurcation is usually associated with the birth or death of a nearby periodic orbit, and generically occurs in either a supercritical or subcritical manner. For a ...

A Survey On Homoclinic And Heteroclinic Orbits

The study of homoclinic and heteroclinic orbits has a long history. This paper surveys some of the recent advances in this topic. We present some of the main results obtained in recent years, and in addition we indicate some possible research directions and some problems for further studies.

A Codimension-3 Bifurcation Problem

In this paper, a bifurcation phenomenon which arises from the interaction of static and dynamic modes in the vicinity of a compound critical point of a nonlinear autonomous system is considered. The critical point is characterized by a double zero and a pair of pure imaginary eigenvalues of the Jacobian. A set of simplified differential equations is obtained by applying the unification techniqu...

Symmetric periodic orbits near a heteroclinic loop

In this paper we consider vector fields in R that are invariant under a suitable symmetry and that posses a “generalized heteroclinic loop” L formed by two singular points (e and e−) and their invariant manifolds: one of dimension 2 (a sphere minus the points e and e−) and one of dimension 1 (the open diameter of the sphere having endpoints e and e−). In particular, we analyze the dynamics of t...

Heteroclinic orbits of semilinear parabolic equations