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.

中文翻译：

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Neumann 边值问题的异常分岔

摘要 分析了流体动力学中由拟微分方程降阶引起的具有Neumann边界条件的一族二阶非线性微分方程的边界层行为。该问题通过两个扰动参数 μ , δ 来考虑。使用渐近和数值方法表明，通过扰动 δ，极限 μ → 0 中的问题从具有简化方程的对称解（包括外解）的齐次问题变为非齐次问题，一般来说，没有非零外极限解（带边界层）。通过研究随着 μ 和 δ 变化的分岔图，表明对于 μ = n − 2，其中 n 是偶数，解的主要分支将“撕裂”。爆炸区域，孤立的解决方案，