Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On local attraction properties and a stability index for heteroclinic
  connections   Some invariant sets may attract a nearby set of initial conditions but
nonetheless repel a complementary nearby set of initial conditions. For a given
invariant set $X\subset\R^n$ with a basin of attraction $N$, we define a
stability index $\sigma(x)$ of a point $x\in X$ that characterizes the local
extent of the basin. Let $B_{\epsilon}$ denote a ball of radius $\epsilon$
about $x$. If $\sigma(x)>0$, then the measure of $B_{\epsilon}\setminus N$
relative the measure of the ball is $O(\epsilon^{|\sigma(x)|})$, while if
$\sigma(x)<0$, then the measure of $B_{\epsilon}\cap N$ relative the measure of
the ball is of the same order. We show that this index is constant along
trajectories, and relate this orbit invariant to other notions of stability
such as Milnor attraction, essential asymptotic stability and asymptotic
stability relative to a positive measure set. We adapt the definition to local
basins of attraction (i.e. where $N$ is defined as the set of initial
conditions that are in the basin and whose trajectories remain local to $X$).
This stability index is particularly useful for discussing the stability of
robust heteroclinic cycles, where several authors have studied the appearance
of cusps of instability near cycles that are Milnor attractors. We study simple
(robust heteroclinic) cycles in $\R^4$ and show that the local stability
indices (and hence local stability properties) can be calculated in terms of
the eigenvalues of the linearization of the vector field at steady states on
the cycle. In doing this, we extend previous results of Krupa and Melbourne
(1995,2004) and give criteria for simple heteroclinic cycles in $\R^4$ to be
Milnor attractors.

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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: cs
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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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: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: econ