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Unusual bifurcation of a Neumann boundary value problem. (English) Zbl 07249289
Summary: 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 \(\mu, \delta\). Using asymptotic and numerical methods it is shown that by perturbing \(\delta\) the problem in the limit \(\mu \to 0\) 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 \(\mu\) and \(\delta\) vary it is shown that the main branch of solutions will ‘tear’ for \(\mu = n^{-2}\) where \(n\) is an even integer. Blow-up regions, isolated islands of solution, and for \(\mu \ll 1\) “exponentially small” regions, all occur: a structure that is difficult for conventional path following algorithms to find.
MSC:
34D15 Singular perturbations of ordinary differential equations
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