Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Phase Resetting in an Asymptotically Phaseless System: On the Phase
  Response of Limit Cycles Verging on a Heteroclinic Orbit   Rhythmic behaviors in neural systems often combine features of limit cycle
dynamics (stability and periodicity) with features of near heteroclinic or near
homoclinic cycle dynamics (extended dwell times in localized regions of phase
space). Proximity of a limit cycle to one or more saddle equilibria can have a
profound effect on the timing of trajectory components and response to both
fast and slow perturbations, providing a possible mechanism for adaptive
control of rhythmic motions. Reyn showed that for a planar dynamical system
with a stable heteroclinic cycle (or separatrix polygon), small perturbations
satisfying a net inflow condition will generically give rise to a stable limit
cycle (Reyn, 1980; Guckenheimer and Holmes, 1983). Here we consider the
asymptotic behavior of the infinitesimal phase response curve (iPRC) for
examples of two systems satisfying Reyn's inflow criterion, (i) a smooth system
with a chain of four hyperbolic saddle points and (ii) a piecewise linear
system corresponding to local linearization of the smooth system about its
saddle points. For system (ii), we obtain exact expressions for the limit cycle
and the iPRC as a function of a parameter, mu>0, representing the distance from
a heteroclinic bifurcation point. In the limit, as mu approaches zero, we find
that perturbations parallel to the unstable eigenvector direction in a
piecewise linear region lead to divergent phase response, as previously
observed (Brown, Moehlis and Holmes (2004), Neural Computation). In contrast to
previous work, we find that perturbations parallel to the stable eigenvector
direction can lead to either divergent or convergent phase response, depending
on the phase at which the perturbation occurs. In the smooth system (i), we
show numerical evidence of qualitatively similar phase specific sensitivity to
perturbation.

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The categories of this paper are: hep-ph
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th