The investigators will continue development of new analytic approaches to homoclinic and heteroclinic bifurcation in several important areas of applied dynamical systems. In particular, they will continue development of a new bifurcation function approach to periodic and aperiodic solutions near homoclinic orbits and heteroclinic cycles. They will to continue development of a unified theory of interior layers of singular perturbation problems using heteroclinic bifurcation techniques and a version of the shadowing lemma. They will continue working with shock wave solutions of nonstrictly hyperbolic PDE's and PDE's that change type using heteroclinic bifurcation techniques. They will extend a new rigorous algorithm for numerical continuation of homoclinic and heteroclinic orbits to include continuation to the point where an equilibrium becomes nonhyperbolic. Finally, they will investigate nonhomogeneous solutions of reaction-diffusion equations near spatially homogeneous solutions represented by homoclinic orbits or heteroclinic cycles of an ODE.
