Bifurcations of Heteroclinic Orbits

Abstract

Thesearchfortravelingwavesolutionsofasemilineardiffusionpartialdifferen- tialequationcanbereducedtothesearchforheteroclinicsolutionsoftheordinarydifferential equation ¨ u − c ˙ u + f (u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ1 as t →− ∞and u(t) → γ2 as t →∞ where γ1 ,γ 2 are zeros of f . We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result thatforlarge cthereisauniquepositiveheteroclinicorbitfrom0to1when f (0) = f (1) = 0 and f (u )> 0f or 0< u < 1. When there are more zeros of f , there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < − θ< 0 <θ < 1a ndf is odd. The heteroclinic orbit that tends to 1 as t →∞ starts at one of the three zeros, −θ,0 ,θ as t →− ∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.