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On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling
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 |
$\left\{ \begin{array}{lcl} \dot x&=&-y+\delta p(x,y)+\varepsilon P_n(x,y),\\ \dot y&=&x+\delta q(x,y)+\varepsilon Q_n(x,y), \end{array} \right.$ |
$δ, \varepsilon$ |
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. |

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 |
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 |
Value of n | Degree of |
1 | 2 |
2 | 2 |
3 | 2 |
4 | 4 |
5 | 6 |
6 | 8 |
7 | 10 |
Value of n | Degree of |
1 | 2 |
2 | 2 |
3 | 2 |
4 | 4 |
5 | 6 |
6 | 8 |
7 | 10 |
Value of n | Degree of |
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 |
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 |
1 | 1 |
2 | 2 |
3 | 3 |
4 | 5 |
5 | 7 |
6 | 9 |
Value of n | Degree of |
1 | 1 |
2 | 2 |
3 | 3 |
4 | 5 |
5 | 7 |
6 | 9 |
Value of n | Degree of |
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 |
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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