American Institute of Mathematical Sciences

Advanced
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 - A, 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]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435
[2]

Noufel Frikha, Valentin Konakov, Stéphane Menozzi. Well-posedness of some non-linear stable driven SDEs. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 849-898. doi: 10.3934/dcds.2020302
[3]

Jaume Llibre, Claudia Valls. Rational limit cycles of abel equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021007
[4]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405
[5]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145
[6]

Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021001
[7]

Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2020  doi: 10.3934/mfc.2021001
[8]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[9]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124
[10]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003
[11]

Hideki Murakawa. Fast reaction limit of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1047-1062. doi: 10.3934/dcdss.2020405
[12]

Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014
[13]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[14]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001
[15]

Theresa Lange, Wilhelm Stannat. Mean field limit of ensemble square root filters - discrete and continuous time. Foundations of Data Science, 2021  doi: 10.3934/fods.2021003
[16]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013
[17]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020180
[18]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095
[19]

Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129
[20]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (97)
  • HTML views (57)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences