Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Piecewise smooth systems near a co-dimension 2 discontinuity manifold:
  can one say what should happen?   We consider a piecewise smooth system in the neighborhood of a co-dimension 2
discontinuity manifold $\Sigma$. Within the class of Filippov solutions, if
$\Sigma$ is attractive, one should expect solution trajectories to slide on
$\Sigma$. It is well known, however, that the classical Filippov
convexification methodology is ambiguous on $\Sigma$. The situation is further
complicated by the possibility that, regardless of how sliding on $\Sigma$ is
taking place, during sliding motion a trajectory encounters so-called generic
first order exit points, where $\Sigma$ ceases to be attractive.
  In this work, we attempt to understand what behavior one should expect of a
solution trajectory near $\Sigma$ when $\Sigma$ is attractive, what to expect
when $\Sigma$ ceases to be attractive (at least, at generic exit points), and
finally we also contrast and compare the behavior of some regularizations
proposed in the literature.
  Through analysis and experiments we will confirm some known facts, and
provide some important insight: (i) when $\Sigma$ is attractive, a solution
trajectory indeed does remain near $\Sigma$, viz. sliding on $\Sigma$ is an
appropriate idealization (of course, in general, one cannot predict which
sliding vector field should be selected); (ii) when $\Sigma$ loses attractivity
(at first order exit conditions), a typical solution trajectory leaves a
neighborhood of $\Sigma$; (iii) there is no obvious way to regularize the
system so that the regularized trajectory will remain near $\Sigma$ as long as
$\Sigma$ is attractive, and so that it will be leaving (a neighborhood of)
$\Sigma$ when $\Sigma$ looses attractivity.
  We reach the above conclusions by considering exclusively the given piecewise
smooth system, without superimposing any assumption on what kind of dynamics
near $\Sigma$ (or sliding motion on $\Sigma$) should have been taking place.

23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom