Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rapid evolution of complex limit cycles   The current article studies certain problems related to complex cycles of
holomorphic foliations with singularities in the complex plane. We focus on the
case when polynomial differential one-form gives rise to a foliation by Riemann
surfaces. In this setting, a complex cycle is defined as a nontrivial element
of the fundamental group of a leaf from the foliation. Whenever the polynomial
foliation comes from a perturbation of an exact one-form, one can introduce the
notion of a multi-fold cycle. This type of cycle has at least one
representative that determines a free homotopy class of loops in an open
fibered subdomain of the complex plane. The topology of this subdomain is
closely related to the exact one-form mentioned earlier. We introduce and study
the notion of multi-fold cycles of a close-to-integrable polynomial foliation.
We also explore how these cycles correspond to periodic orbits of a certain
Poincar\'e map associated with the foliation. Finally, we discuss the tendency
of a continuous family of multi-fold limit cycles to escape from certain large
open domains in the complex plane as the foliation converges to its integrable
part.

27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ