A shooting approach to layers and chaos for a forced Duffing equation

Abstract

We study equilibrium solutions for the problem $u_t=ε^2 u_{xx} -u^3 +λu -cos(t)$,$u_x(0,t)=u_x(L,t)=0$. Using a shooting method we find solutions for all non-zero $ε.$ For small $ε$ we add to the solutions found by previous authors, especially Angennent, Mallet-Paret and Peletier, and Hale and Sakamoto, and also give new elementary ode proofs of their results. Among the new results is the existence of internal layer-type solutions. Considering the ode satisfied by equilibria, but on an infinite interval, we obtain chaos results for $λ\geq λ_{0}=\frac{3}{2^{2/3}}$ and $0<ε\leq {1/4}.$ We also consider the problem of bifurcation of solutions as $λ$ increases from $0.$