# American Institute of Mathematical Sciences

June  2017, 37(6): 3353-3386. doi: 10.3934/dcds.2017142

## Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree

 Department of Mathematics, IMECC/Unicamp, Campinas/SP, 13083-970, Brazil

* Corresponding author: R. M. Martins

Received  June 2016 Revised  January 2017 Published  February 2017

Fund Project: R. M. Martins is partially supported by Fapesp grant 2015/06903-8. O. M. L. Gomide is supported by Fapesp grant 2013/18168-5.

In this paper we consider planar systems of differential equations of the form
 $\left\{ \begin{array}{lcl} \dot x&=&-y+\delta p(x,y)+\varepsilon P_n(x,y),\\ \dot y&=&x+\delta q(x,y)+\varepsilon Q_n(x,y), \end{array} \right.$
where
 $δ, \varepsilon$
are small parameters, $(p, q)$ are quadratic or cubic homogeneous polynomials such that the unperturbed system ($\varepsilon=0$) has an isochronous center at the origin and $P_n, Q_n$ are arbitrary perturbations. Estimates for the maximum number of limit cycles are provided and these estimatives are sharp for $n≤q 6$ (when $p, q$ are quadratic). When $p, q$ are cubic polynomials and $P_n, Q_n$ are linear, the problem is addressed from a numerical viewpoint and we also study the existence of limit cycles.
Citation: Ricardo M. Martins, Otávio M. L. Gomide. Limit cycles for quadratic and cubic planar differential equations under polynomial perturbations of small degree. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3353-3386. doi: 10.3934/dcds.2017142
##### References:

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##### References:
Projections of $\Omega$ and $\partial\mathcal H$ in the $xy$-plane for different values of $z,w$
Maximum number of limit cycles bifurcating from polynomial perturbations of a given degree of system (S1)
 Perturbation Degree Maximum Number of Bifurcating Limit Cycles 1 1 2 1 3 1 4 2 5 3 6 4 7 5
 Perturbation Degree Maximum Number of Bifurcating Limit Cycles 1 1 2 1 3 1 4 2 5 3 6 4 7 5
Maximum number of limit cycles bifurcating from polynomial perturbations of a given degree of system (S2).
 Perturbation Degree Maximum Number of Bifurcating Limit Cycles 1 0 2 4 3 3 4 4 5 8 6 6
 Perturbation Degree Maximum Number of Bifurcating Limit Cycles 1 0 2 4 3 3 4 4 5 8 6 6
Degree of the polynomial part of $F_1^{[n]}(Z)$ according to the value of $n$.
 Value of n Degree of $f_1^{[n]}$ 1 2 2 2 3 2 4 4 5 6 6 8 7 10
 Value of n Degree of $f_1^{[n]}$ 1 2 2 2 3 2 4 4 5 6 6 8 7 10
Degree of the perturbation in (19), degree of the polynomial part of $F_1^{[n]}(Z)$ according to the value of $n$ and maximum number of limit cycles
 Value of n Degree of $f_1^{[n]}$ Maximum number of limit cycles 1 2 1 2 2 1 3 2 1 4 4 2 5 6 3 6 8 4 7 10 5
 Value of n Degree of $f_1^{[n]}$ Maximum number of limit cycles 1 2 1 2 2 1 3 2 1 4 4 2 5 6 3 6 8 4 7 10 5
Degree of the polynomial part of $F_2^{[n]}(Z)$ according to the value of $n$
 Value of n Degree of $f_1^{[n]}$ 1 1 2 2 3 3 4 5 5 7 6 9
 Value of n Degree of $f_1^{[n]}$ 1 1 2 2 3 3 4 5 5 7 6 9
Degree of the perturbation in (25), degree of the polynomial part of $F_2^{[n]}(Z)$ according to the value of $n$ and maximum number of limit cycles.
 Value of n Degree of $f_1^{[n]}$ Maximum number of limit cycles 1 1 0 2 2 2 3 3 3 4 5 4 5 7 5 6 9 6
 Value of n Degree of $f_1^{[n]}$ Maximum number of limit cycles 1 1 0 2 2 2 3 3 3 4 5 4 5 7 5 6 9 6
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