# American Institute of Mathematical Sciences

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

 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.
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:

show all references

##### References:
Projections of $\Omega$ and $\partial\mathcal H$ in the $xy$-plane for different values of $z,w$
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
 Perturbation Degree Maximum Number of Bifurcating Limit Cycles 1 1 2 1 3 1 4 2 5 3 6 4 7 5
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
 Perturbation Degree Maximum Number of Bifurcating Limit Cycles 1 0 2 4 3 3 4 4 5 8 6 6
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
 Value of n Degree of $f_1^{[n]}$ 1 2 2 2 3 2 4 4 5 6 6 8 7 10
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
 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
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
 Value of n Degree of $f_1^{[n]}$ 1 1 2 2 3 3 4 5 5 7 6 9
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
 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