Unusual bifurcation of a Neumann boundary value problem

Abstract

Boundary layer behaviour of a family of second order nonlinear differential equations with Neumann boundary condition arising from an order-reduction of a pseudo-differential equation in fluid dynamics is analysed. The problem is considered with two perturbation parameters μ,δ. Using asymptotic and numerical methods it is shown that by perturbing δ the problem in the limit μ→0 changes from a homogeneous problem with symmetric solutions (including outer solutions) of a simplified equation, to a non-homogeneous problem which, in general, does not have a non-zero outer limit solution (with boundary layers). By studying the bifurcation diagrams as μ and δ vary it is shown that the main branch of solutions will ‘tear’ for μ=n−2 where n is an even integer. Blow-up regions, isolated islands of solution, and for μ≪1 “exponentially small” regions, all occur: a structure that is difficult for conventional path following algorithms to find.