A Shooting Approach to Layers and Chaos in a Forced Duffing Equation

1 abstract We study equilibrium solutions for the problem u t = ε 2 u xx − u 3 + λu + cos x u x (0, t) = u x (1, 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 by 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 ≥ λ 0 = 3 2 2/3 and 0 < ε ≤ 1 4. We also consider the problem of bifurcation of solutions as λ increases from 0. This is the first of a series of papers studying the existence of bounded solutions of the equation ε 2 u ′′ = u 3 − λu + cos t, (1) where λ and ε are positive parameters. This equation is a standard model in the theory of nonlinear oscillations, one of several which have been called a forced Duffing equation in the literature. With the signs shown it is often referred to as the equation of a " soft spring. " A comprehensive reference to the early theory of (1) is [NM], which gives a detailed account of the results obtained by classical perturbation methods, such as averaging or multi-scale techniques. More recent efforts have used dynamical systems concepts to establish results about chaotic behavior of one sort or another. An important reference is by Angenent, Mallet-Paret, and Peletier [AMPP], who studied stable steady states for a reaction-diffusion equation u τ = ε 2 u xx + f (x, u) (2) with boundary conditions u x (0, τ) = u x (L, τ) = 0 (3) for a class of functions f which were cubic in the state variable u. Equation (1) is obtained from (2) (with f (x, u) = u 3 − λu + cos x) by setting x = t and assuming that u is independent of τ. While the specific form (1) is not mentioned in [AMPP], the methods there are easily 1