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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 and 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.

[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
Table Options

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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

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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

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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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