Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence of periodic solutions of the FitzHugh-Nagumo equations for an
  explicit range of the small parameter   The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is
given by fast-slow reaction-diffusion equations, with dependence on a parameter
$\epsilon$ representing the ratio of time scales. It is well known that for all
sufficiently small $\epsilon>0$ the system possesses a periodic traveling wave.
With aid of computer-assisted rigorous computations, we prove the existence of
this periodic orbit in the traveling wave equation for an explicit range
$\epsilon \in (0, 0.0015]$. Our approach is based on a novel method of
combination of topological techniques of covering relations and isolating
segments, for which we provide a self-contained theory. We show that the range
of existence is wide enough, so the upper bound can be reached by standard
validated continuation procedures. In particular, for the range $\epsilon \in
[1.5 \times 10^{-4}, 0.0015]$ we perform a rigorous continuation based on
covering relations and not specifically tailored to the fast-slow setting.
Moreover, we confirm that for $\epsilon=0.0015$ the classical interval
Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds.
Techniques described in this paper can be adapted to other fast-slow systems of
similar structure.

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The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
1
The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: eess