Bifurcation from heteroclinic orbits with semi-hyperbolic equilibria
The author studies the problem of bifurcation from a heteroclinic orbit joining two semihyperbolic equilibria \(E_ 1\) and \(E_ 2\) in \(\mathbb{R}^ n\). The orbit is assumed to lie in the intersection of the unstable manifold of \(E_ 1\) and the stable manifold of \(E_ 2\). The author assumes that the equilibria undergo transcritical bifurcation when codimension one surfaces \(M_ 1\) and \(M_ 2\), respectively, in the parameter space \(\mathbb{R}^ k\) are crossed. He shows that there are Melnikov-type conditions that govern the existence of a heteroclinic orbit from an equilibrium near \(E_ 1\) to one near \(E_ 2\). The proofs use exponential trichotomies.
scientific article; zbMATH DE number 711482
scientific article