| Homoclinic solutions are of great interest in several applications involving ordinary differential equations. In nucleation models, they correspond to a transition state between the homogeneous equilibrium and more complicated phase separated states. When studying traveling wave equations, they often appear as solitary waves, which are important in water wave theory and optical wave propagation. In the context of dynamical systems, homoclinic orbits are associated with bifurcations of periodic orbits or they act as organizing centers for chaotic behavior. In this thesis we present a new spectral method for the computation of homoclinic solutions to ordinary differential equations. The method is based on Hermite-Fourier expansions of homoclinic solutions. It demonstrates exponential convergence of a computed solution to the actual one. This method can also be used to approximate values of nonlinear functionals along homoclinic orbits. It is demonstrated on the examples from phase separation, chemical reaction dynamics, and metastability. |
