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

Ricardo M. Martins; Otávio M. L. Gomide

doi:10.3934/dcds.2017142

Article Contents

2017, Volume 37, Issue 6: 3353-3386. Doi: 10.3934/dcds.2017142

This issue Previous Article On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling Next Article Generalized inhomogeneous Strichartz estimates

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

Ricardo M. Martins^, and
Otávio M. L. Gomide

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

* Corresponding author: R. M. Martins
* Corresponding author: R. M. Martins

Received: June 2016

Revised: January 2017

Published: May 2017

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

Abstract / Introduction Full Text(HTML) Figure(1) / Table(6) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

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

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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

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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

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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

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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

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DownLoad: CSV

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

| Show Table

DownLoad: CSV

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

| Show Table

DownLoad: CSV

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