Lin's method and homoclinic bifurcations for functional differential equations of mixed type

We extend Lin's method for use in the setting of parameter-dependent nonlinear functional dierential equations of mixed type (MFDEs). We show that the presence of M-homoclinic and M-periodic solutions that bifurcate from a prescribed homoclinic connection can be detected by studying a nite dimensional bifurcation equation. As an application, we describe the codimension two orbit-ip bifurcation in the setting of
MFDEs.
Key words: mixed type functional dierential equation, homoclinic bifurcation, Lin's method, orbit-ip bifurcation, nite dimensional reduction, exponential dichotomies,
advanced and retarded arguments.