Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Discrete Geometric Singular Perturbation Theory   We propose a mathematical formalism for discrete multi-scale dynamical
systems induced by maps which parallels the established geometric singular
perturbation theory for continuous-time fast-slow systems. We identify limiting
maps corresponding to both 'fast' and 'slow' iteration under the map. A notion
of normal hyperbolicity is defined by a spectral gap requirement for the
multipliers of the fast limiting map along a critical fixed-point manifold $S$.
We provide a set of Fenichel-like perturbation theorems by reformulating
pre-existing results so that they apply near compact, normally hyperbolic
submanifolds of $S$. The persistence of the critical manifold $S$, local
stable/unstable manifolds $W^{s/u}_{loc}(S)$ and foliations of
$W^{s/u}_{loc}(S)$ by stable/unstable fibers is described in detail. The
practical utility of the resulting discrete geometric singular perturbation
theory (DGSPT) is demonstrated in applications. First, we use DGSPT to identify
singular geometry corresponding to excitability, relaxation, chaotic and
non-chaotic bursting in a map-based neural model. Second, we derive results
which relate the geometry and dynamics of fast-slow ODEs with non-trivial
time-scale separation and their Euler-discretized counterpart. Finally, we show
that fast-slow ODE systems with fast rotation give rise to fast-slow Poincar\'e
maps, the geometry and dynamics of which can be described in detail using
DGSPT.

29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph