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

• * Corresponding author: R. M. Martins

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.

Mathematics Subject Classification: Primary: 34C14, 34C20; Secondary: 37J15, 37J40.

 Citation: • • Figure 1.  Projections of $\Omega$ and $\partial\mathcal H$ in the $xy$-plane for different values of $z,w$

Table 1.  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

Table 2.  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

Table 3.  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

Table 4.  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

Table 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

Table 6.  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
• Figures(1)

Tables(6)

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