Bifurcation from a homoclinic orbit in partial functional differential equations (Q1811376)

The paper deals with the one-parameter family of partial functional-differential equations of the form \[ u^\prime (t) = Au(t) + L(u_t) + g(u_t,\varepsilon), \tag{1} \] where \(A\) is the generator of an analytic semigroup, \(L\) is a linear operator and \(g\) is a smooth nonlinear functional. The authors study the differentiability of the solutions with respect to the initial values and parameters and prove smoothness of the stable and unstable manifolds. A Shil'nikov map is constructed and the authors prove a generalization of the Shil'nikov theorem for (1).