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: May 31, 2016

Revised: December 31, 2016

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 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.

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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

| Show Table

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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References

	L. Ahlfors, Complex Analysis International Series in Pure and Applied Mathematics 7, McGraw-Hill, 1978.
	T. Boni , P. Mardesic and C. Rousseau , Linearization of isochronous centers, Journal of Differential Equations, 121 (1995) , 67-108. doi: 10.1006/jdeq.1995.1122.
	A. Buică and J. Llibre , Averaging methods for finding periodic orbits via Brouwer degree, Bulletin des Sciences Mathématiques, 128 (2004) , 7-22. doi: 10.1016/j.bulsci.2003.09.002.
	L. Cairó and J. Llibre , Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3, J. Math. Anal. Appl., 331 (2007) , 1284-1298. doi: 10.1016/j.jmaa.2006.09.066.
	J. Chavarriga and M. Sabatini , A Survey of Isochronous Centers, Qualitative Theory of Dynamical Systems, 1 (1999) , 1-70. doi: 10.1007/BF02969404.
	C. Chicone and M. Jacobs , Bifurcation of limit cycles from quadratic isochrones, Journal of Differential Equations, 91 (1991) , 268-326. doi: 10.1016/0022-0396(91)90142-V.
	C. Colin and L. Chengzhi, Limit Cycles of Differential Equations Birkhäuser Verlag, 2007.
	F. Dumortier and R. Roussarie , Abelian integrals and limit cycles, Journal of Differential Equations, 227 (2006) , 116-165. doi: 10.1016/j.jde.2005.08.015.
	J. Giné and J. Llibre , Limit cycles of cubic polynomial vector fields via the averaging theory, Nonlinear Analysis: Theory, Methods & Applications, 66 (2007) , 1707-1721. doi: 10.1016/j.na.2006.02.016.
	M. Han , On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, Journal of Applied Analysis and Computation, 7 (2017) , 788-794.
	M. Han and P. Yu, Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles, Applied Mathematical Sciences, 181, Springer, 2012. doi: 10.1007/978-1-4471-2918-9.
	Y. Ilyashenko and J. Llibre , A restricted version of Hilbert's 16th problem for quadratic vector fields, Mosc. Math. J., 10 (2010) , 317-335.
	S. Li , Y. Shao and J. Li , On the number of limit cycles of a perturbed cubic polynomial differential center, Journal of Mathematical Analysis and Applications, 404 (2013) , 212-220. doi: 10.1016/j.jmaa.2013.03.010.
	J. Llibre and J. Itikawa , Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, Journal of Computational and Applied Mathematics, 277 (2015) , 171-191. doi: 10.1016/j.cam.2014.09.007.
	J. Llibre, Periodic Solutions Via Averaging Theory, Notes of the Advanced Course RTNS2014 held in Bellaterra (CRM), January 27–31, 2014.
	J. Llibre and A. C. Mereu , Limit cycles for discontinuous quadratic differential systems with two zones, Journal of Mathematical Analysis and Applications, 413 (2014) , 763-775. doi: 10.1016/j.jmaa.2013.12.031.
	J. Llibre and A. C. Mereu , Limit cycles for generalized Kukles polynomial differential systems, Nonlinear Analysis, 74 (2011) , 1261-1271. doi: 10.1016/j.na.2010.09.064.
	J. Llibre, R. M. Martins and M. A. Teixeira, Periodic orbits, invariant tori and cylinders of Hamiltonian systems near integrable ones having a return map equal to the identity, J. Math. Phys. , 51 (2010), 082704, 11pp. doi: 10.1063/1.3477937.
	W. S. Loud , Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964) , 21-36.
	R. M. Martins , A. C. Mereu and R. Oliveira , An estimation for the number of limit cycles in a Liénard-like perturbation of a quadratic nonlinear center, Nonlinear Dynamics, 79 (2015) , 185-194. doi: 10.1007/s11071-014-1655-z.
	J. Murdock, A. Sanders and F. Verhultst, Averaging methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci, 59, Springer, 2007.
	I. A. Pleshkan , A new method of investigating the isochronocity of a system of two differential equations, Differential Equations, 5 (1969) , 796-802.
	J. Spanier and K. B. Oldham , The complete elliptic integrals K(p) and E(p) and "The incomplete elliptic integrals F(p;phi) and E(p;phi),", Chs. 61-62 in An Atlas of Functions. Washington, DC: Hemisphere, (1987) , 609-633.
	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second edition. Universitext. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.
	E. T. Whittaker and G. N. Watson, A Course on Modern Analysis, second edition. Cambridge University Press, 1915. doi: 10.1017/CBO9780511608759.
	P. Yu and M. Han , Bifurcation of limit cycles in quadratic Hamiltonian systems with various degree polynomial perturbations, Chaos, Solitons & Fractals, 45 (2012) , 772-794. doi: 10.1016/j.chaos.2012.02.010.