Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A new uniqueness criterion for the number of periodic orbits of Abel
  equations   A solution of the Abel equation $\dot{x}=A(t)x^3+B(t)x^2$ such that
$x(0)=x(1)$ is called a periodic orbit of the equation. Our main result proves
that if there exist two real numbers $a$ and $b$ such that the function
$aA(t)+bB(t)$ is not identically zero, and does not change sign in $[0,1]$ then
the Abel differential equation has at most one non-zero periodic orbit.
Furthermore, when this periodic orbit exists, it is hyperbolic. This result
extends the known criteria about the Abel equation that only refer to the cases
where either $A(t)\not\equiv0$ or $B(t)\not\equiv0$ does not change sign. We
apply this new criterion to study the number of periodic solutions of two
simple cases of Abel equations: the one where the functions $A(t)$ and $B(t)$
are 1-periodic trigonometric polynomials of degree one and the case where these
two functions are polynomials with three monomials. Finally, we give an upper
bound for the number of isolated periodic orbits of the general Abel equation
$\dot{x}=A(t)x^3+B(t)x^2+C(t)x,$ when $A(t), B(t)$ and $C(t)$ satisfy adequate
conditions.

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The categories of this paper are: stat
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int