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A shooting approach to layers and chaos in a forced Duffing equation. (English) Zbl 1025.34015

Summary: We study equilibrium solutions to the problem \[ u_t= \varepsilon^2 u_{xx}-u^3+\lambda u+\cos x,\quad u_x(0,t)= u_x(1,t)=0. \] Using a shooting method, we find solutions for all nonzero \(\varepsilon\). For small \(\varepsilon\), we add to the solutions found by previous authors, especially Angenent, 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 \(\lambda \geq\lambda_0= {3\over 2^{2/3}}\) and \(0<\varepsilon \leq{1\over 4}\). We also consider the bifurcation of solutions as \(\lambda\) increases.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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References:

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