Bifurcation tearing in a forced Duffing equation

Abstract

The structure of solution branches of the equation ε 2 u x x = u 3 − λ u + δ g ( x ) , for x ∈ R is studied, for real constants, ε , λ and δ , where g ( x ) is periodic in x and u satisfies Neumann boundary conditions. For α : = u ( 0 ) bifurcation structure in the α – δ plane occurs that for small ε leads to families of singular asymptotic solutions with interior transition layers between outer solutions. In contrast, bifurcation in the α – λ plane produces families of oscillatory solutions that for small ε have large numbers, n , of oscillations between envelope outer solutions. The n label of the solution branch is defined by its local bifurcation at ε 2 n 2 = λ > 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 α – δ plane leads to connected branches of solutions in the α – λ – δ space. But when n is odd the presence of g ( x ) leads to cusp catastrophes (for small ε ) in the α – δ plane and these give rise to disconnected (or isolated) branches of solution in the α – λ 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 ε 2 λ − 1 → 0 we see layer behaviour occurring about the top and bottom outer solutions and as ε 2 λ − 1 → ∞ we find convergence to the middle outer solution.