Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamics of planar vector fields near a non-smooth equilibrium   In this paper we contribute to qualitative and geometric analysis of planar
piecewise smooth vector fields, which consist of two smooth vector fields
separated by the straight line $y=0$ and sharing the origin as a non-degenerate
equilibrium. In the sense of $\Sigma$-equivalence, we provide a sufficient
condition for linearization and give phase portraits and normal forms for these
linearizable vector fields. This condition is hard to be weakened because there
exist vector fields which are not linearizable when this condition is not
satisfied. Regarding perturbations, a necessary and sufficient condition for
local $\Sigma$-structural stability is established when the origin is still an
equilibrium of both smooth vector fields under perturbations. In the opposition
to this case, we prove that for any piecewise smooth vector field studied in
this paper there is a limit cycle bifurcating from the origin, and there are
some piecewise smooth vector fields such that for any positive integer $m$
there is a perturbation having exactly $m$ limit cycles bifurcating from the
origin. Here $m$ maybe infinity.

28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex