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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
##### Software:
Matlab; MATLAB ODE suite; ode113; Ode15s; ode23; ode23s; ode45
Full Text:
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