Jankovic, Sally2023-09-192023-09-192023-05https://hdl.handle.net/11299/257125University of Minnesota Ph.D. dissertation. May 2023. Major: Mathematics. Advisor: Arnd Scheel. 1 computer file (PDF); vi, 80 pages + 1 compressed folder of supplementary files.Coherent structures in pattern-forming systems, such as pulses and spikes, are often mathematically represented as homoclinic orbits. We here present a generalized method for finding such homoclinic solutions to 2nd-order ODEs using a posteriori data derived from approximate solutions to boundary value problems on truncated intervals. We then show in the opposite direction that the a priori existence of a non-degenerate homoclinic implies the existence of a family of solutions to a Dirichlet boundary-value problem, with an explicit lower bound on domain size necessary to obtain a solution. In each direction, we also provide explicit error estimates as a function of truncation error. We lastly apply our method to find a family of homoclinics in the Lengyel-Epstein system and compute a minimum domain size for the existence of finite-domain solutions using the a priori proof.ena posteriori errora priori errorcomputer-assisted proofhomoclinicpattern formationThesis or Dissertation