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

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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.
[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.
[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.
[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.
[5]	J. Chavarriga and M. Sabatini, A Survey of Isochronous Centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70. doi: 10.1007/BF02969404.
[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.
[7]	C. Colin and L. Chengzhi, Limit Cycles of Differential Equations Birkhäuser Verlag, 2007.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	J. Llibre, Periodic Solutions Via Averaging Theory, Notes of the Advanced Course RTNS2014 held in Bellaterra (CRM), January 27–31, 2014.
[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.
[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.
[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.
[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.
[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.
[21]	J. Murdock, A. Sanders and F. Verhultst, Averaging methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci, 59, Springer, 2007.
[22]	I. A. Pleshkan, A new method of investigating the isochronocity of a system of two differential equations, Differential Equations, 5 (1969), 796-802.
[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.
[24]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second edition. Universitext. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.
[25]	E. T. Whittaker and G. N. Watson, A Course on Modern Analysis, second edition. Cambridge University Press, 1915. doi: 10.1017/CBO9780511608759.
[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.

show all references

References:

[1]	L. Ahlfors, Complex Analysis International Series in Pure and Applied Mathematics 7, McGraw-Hill, 1978.
[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.
[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.
[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.
[5]	J. Chavarriga and M. Sabatini, A Survey of Isochronous Centers, Qualitative Theory of Dynamical Systems, 1 (1999), 1-70. doi: 10.1007/BF02969404.
[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.
[7]	C. Colin and L. Chengzhi, Limit Cycles of Differential Equations Birkhäuser Verlag, 2007.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	J. Llibre, Periodic Solutions Via Averaging Theory, Notes of the Advanced Course RTNS2014 held in Bellaterra (CRM), January 27–31, 2014.
[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.
[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.
[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.
[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.
[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.
[21]	J. Murdock, A. Sanders and F. Verhultst, Averaging methods in Nonlinear Dynamical Systems 2nd edition, Appl. Math. Sci, 59, Springer, 2007.
[22]	I. A. Pleshkan, A new method of investigating the isochronocity of a system of two differential equations, Differential Equations, 5 (1969), 796-802.
[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.
[24]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second edition. Universitext. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-61453-8.
[25]	E. T. Whittaker and G. N. Watson, A Course on Modern Analysis, second edition. Cambridge University Press, 1915. doi: 10.1017/CBO9780511608759.
[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.

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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*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

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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

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*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

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