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

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

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

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

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.

Keywords: Limit cycles, non-linear centers, averaging method.

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

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

References:

[1]	L. Ahlfors, Complex Analysis International Series in Pure and Applied Mathematics 7, McGraw-Hill, 1978. Google Scholar
[2]	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. Google Scholar
[3]	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. Google Scholar
[4]	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. Google Scholar
[5]	J. Chavarriga and M. Sabatini, A Survey of Isochronous Centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70. doi: 10.1007/BF02969404. Google Scholar
[6]	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. Google Scholar
[7]	C. Colin and L. Chengzhi, Limit Cycles of Differential Equations Birkhäuser Verlag, 2007. Google Scholar
[8]	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. Google Scholar
[9]	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. Google Scholar
[10]	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. Google Scholar
[11]	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. Google Scholar
[12]	Y. Ilyashenko and J. Llibre, A restricted version of Hilbert's 16th problem for quadratic vector fields, Mosc. Math. J., 10 (2010), 317-335. Google Scholar
[13]	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. Google Scholar
[14]	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. Google Scholar
[15]	J. Llibre, Periodic Solutions Via Averaging Theory, Notes of the Advanced Course RTNS2014 held in Bellaterra (CRM), January 27–31, 2014. Google Scholar
[16]	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. Google Scholar
[17]	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. Google Scholar
[18]	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. Google Scholar
[19]	W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964), 21-36. Google Scholar
[20]	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. Google Scholar
[21]	J. Murdock, A. Sanders and F. Verhultst, Averaging methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci, 59, Springer, 2007. Google Scholar
[22]	I. A. Pleshkan, A new method of investigating the isochronocity of a system of two differential equations, Differential Equations, 5 (1969), 796-802. Google Scholar
[23]	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. Google Scholar
[24]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second edition. Universitext. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8. Google Scholar
[25]	E. T. Whittaker and G. N. Watson, A Course on Modern Analysis, second edition. Cambridge University Press, 1915. doi: 10.1017/CBO9780511608759. Google Scholar
[26]	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. Google Scholar

show all references

References:

[1]	L. Ahlfors, Complex Analysis International Series in Pure and Applied Mathematics 7, McGraw-Hill, 1978. Google Scholar
[2]	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. Google Scholar
[3]	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. Google Scholar
[4]	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. Google Scholar
[5]	J. Chavarriga and M. Sabatini, A Survey of Isochronous Centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70. doi: 10.1007/BF02969404. Google Scholar
[6]	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. Google Scholar
[7]	C. Colin and L. Chengzhi, Limit Cycles of Differential Equations Birkhäuser Verlag, 2007. Google Scholar
[8]	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. Google Scholar
[9]	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. Google Scholar
[10]	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. Google Scholar
[11]	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. Google Scholar
[12]	Y. Ilyashenko and J. Llibre, A restricted version of Hilbert's 16th problem for quadratic vector fields, Mosc. Math. J., 10 (2010), 317-335. Google Scholar
[13]	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. Google Scholar
[14]	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. Google Scholar
[15]	J. Llibre, Periodic Solutions Via Averaging Theory, Notes of the Advanced Course RTNS2014 held in Bellaterra (CRM), January 27–31, 2014. Google Scholar
[16]	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. Google Scholar
[17]	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. Google Scholar
[18]	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. Google Scholar
[19]	W. S. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations, 3 (1964), 21-36. Google Scholar
[20]	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. Google Scholar
[21]	J. Murdock, A. Sanders and F. Verhultst, Averaging methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci, 59, Springer, 2007. Google Scholar
[22]	I. A. Pleshkan, A new method of investigating the isochronocity of a system of two differential equations, Differential Equations, 5 (1969), 796-802. Google Scholar
[23]	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. Google Scholar
[24]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second edition. Universitext. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8. Google Scholar
[25]	E. T. Whittaker and G. N. Watson, A Course on Modern Analysis, second edition. Cambridge University Press, 1915. doi: 10.1017/CBO9780511608759. Google Scholar
[26]	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. Google Scholar

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

Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download as excel

*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 Options

Download as excel

*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 Options

Download as excel

*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 Options

Download as excel

*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 Options

Download as excel

*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

Table Options

Download as excel

*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

[1]	Olusola Kolebaje, Ebenezer Bonyah, Lateef Mustapha. The first integral method for two fractional non-linear biological models. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 487-502. doi: 10.3934/dcdss.2019032
[2]	Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136
[3]	Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133
[4]	Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : i-iv. doi: 10.3934/dcdss.201702i
[5]	Dmitry Dolgopyat. Bouncing balls in non-linear potentials. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 165-182. doi: 10.3934/dcds.2008.22.165
[6]	Dorin Ervin Dutkay and Palle E. T. Jorgensen. Wavelet constructions in non-linear dynamics. Electronic Research Announcements, 2005, 11: 21-33.
[7]	Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
[8]	Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[9]	Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[10]	Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004
[11]	Jaume Llibre, Lucyjane de A. S. Menezes. On the limit cycles of a class of discontinuous piecewise linear differential systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1835-1858. doi: 10.3934/dcdsb.2020005
[12]	Victoriano Carmona, Soledad Fernández-García, Antonio E. Teruel. Saddle-node of limit cycles in planar piecewise linear systems and applications. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5275-5299. doi: 10.3934/dcds.2019215
[13]	Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
[14]	Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236
[15]	Shimin Li, Jaume Llibre. On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5885-5901. doi: 10.3934/dcdsb.2019111
[16]	José-Luis Bravo, Manuel Fernández. Limit cycles of non-autonomous scalar ODEs with two summands. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1091-1102. doi: 10.3934/cpaa.2013.12.1091
[17]	Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[18]	Pablo Ochoa. Approximation schemes for non-linear second order equations on the Heisenberg group. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1841-1863. doi: 10.3934/cpaa.2015.14.1841
[19]	Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193
[20]	Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302

2019 Impact Factor: 1.338

Tools

Metrics

Tools

Article outline

Figures and Tables

2019 Impact Factor: 1.338

Tools

Figures and Tables

2019 Impact Factor: 1.338

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors