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

الأصل · EN

We study equilibrium solutions for the problem uₜ=ε² uxx -u³ +λu -cos(t),uₓ(0,t)=uₓ(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≥ λ₀=32²/³ and 0<ε≤ 1/4. We also consider the problem of bifurcation of solutions as λ increases from 0.