BROWN UNIV PROVIDENCE R I LEFSCHETZ CENTER FOR DYNAMICAL SYSTEMS
These lectures are concerned only with some aspects of bifurcation theory in the local theory of nonlinear oscillations in equations with delays that is, behavior of solutions near an equilibrium. In particular, how the qualitative behavior of solutions change is shown as parameters vary. A detailed study of the local theory is important in order to know the types of solutions to expect in a global problem. Of course, there is no reason to only study local theory near an equilibrium. One should study how the qualitative behavior changes near any invariant set - for example, behavior near a periodic orbit, behavior near an orbit which connects a saddle point to itself, etc. More complicated behavior is expected near these large invariant sets. One can obtain invariant torii, homoclines points which exhibit a chaotic behavior, etc.