  Fast-slow systems are notoriously difficult to analyze with rigorous
numerics, since the qualitative properties of the solution space change
fundamentally when the so-called small parameter $\epsilon$ is varied from 0 to
small non-zero values. In this dissertation I develop a computer-assisted
rigorous method which can be used in combination with topological tools for
proving the existence of period and connecting orbits in the near-zero
parameter regime. As an application, I prove the existence of periodic and
homoclinic orbits in the FitzHugh-Nagumo system, for $\epsilon \in
(0,\epsilon_0]$ with an explicit $\epsilon_0$.
  This dissertation was prepared under supervision of prof. Piotr Zgliczy\'nski
and submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy, awarded at the Jagiellonian University, Department of
Mathematics and Computer Science in June 2016. Some of the results of this
dissertation have been previously published in A. Czechowski and P.
Zgliczy\'nski. Existence of periodic solutions of the FitzHugh-Nagumo equations
for an explicit range of the small parameter. arXiv:1502.02451, SIAM J. Appl.
Dyn. Syst., 15(3), 1615-1655 and text overlap may occur. The dissertation also
contains some (yet) unpublished results, in particular concerning the existence
of homoclinic orbits for explicit ranges of $\epsilon$.
