Bifurcation tearing in a forced Duffing equation

The structure of solution branches of the equation epsilon(2)u(xx) = u(3) - lambda u + delta g(x), for x is an element of R is studied, for real constants, epsilon, lambda and delta, where g(x) is periodic in x and u satisfies Neumann boundary conditions. For alpha := u(0) bifurcation structure in the alpha-delta plane occurs that for smalls leads to families of singular asymptotic solutions with interior transition layers between outer solutions. In contrast, bifurcation in the alpha-lambda plane produces families of oscillatory solutions that for smalls have large numbers, n, of oscillations between envelope outer solutions. The n label of the solution branch is defined by its local bifurcation at epsilon(2)n(2) = lambda > 0. Further the bifurcation structure on the n branch is classified by the integer n being either even or odd. When n is even the pitchfork bifurcation in the alpha-delta plane leads to connected branches of solutions in the alpha-lambda-delta space. But when n is odd the presence of g(x) leads to cusp catastrophes (for small epsilon) in the alpha-delta plane and these give rise to disconnected (or isolated) branches of solution in the alpha-lambda plane that appear to be torn from the main solution branch. The integer n acts as a count of the number of oscillations on these solution branches. As epsilon(2)lambda(-1) -> 0 we see layer behaviour occurring about the top and bottom outer solutions and as epsilon(2)lambda(-1) -> infinity we find convergence to the middle outer solution. (C) 2021 Published by Elsevier Inc.