Heteroclinic bifurcation and singularly perturbed boundary value problems
A singularly perturbed boundary value problem \(\dot x=f(x,y,\epsilon),\epsilon \dot y=g(x,y,\epsilon)\) with boundary conditions \[ B_ 1(x(\omega_ 0),y(\omega_ 0),\epsilon)=0,\quad B_ 2(x(\omega_ 2+\omega),y(\omega_ 0+\omega),\epsilon)=0 \] is studied. Given a candidate for the 0th order approximation which exhibits boundary and interior layers, a complete procedure is presented to compute higher order expansions. Error estimates are given and the results are extended to periodic boundary conditions. Several ideas from dynamical systems are employed, e.g. exponential dichotomies, Fredholm's alternative, and heteroclinic bifurcations. The geometric interpretation of the solution structure is very helpful to understand the prepositions and conclusions.
singularly perturbed boundary value problem
scientific article; zbMATH DE number 4163281
scientific article