Existence and uniqueness of heteroclinic orbits for the equation λu‴ + u′ = f(u)

Abstract

SynopsisIffis a continuous even function which is decreasing on (0,∞) and such that±α are its only zeros and are simple, then in three-dimensional phase spacethe unstable manifold of the equilibrium u = −α and the stable manifold of u = α are both two dimensional. If λ<0 it is shown that there is a unique bounded orbit of the equation λu‴ + u′ = f(u), and that this is a heteroclinic orbit joining these two equilibria. Other results on the existence and uniqueness of heteroclinic orbits are also established when f is not even and when f is not monotone on (0, ∞).