# American Institute of Mathematical Sciences

August  2018, 38(8): 4189-4202. doi: 10.3934/dcds.2018182

## Bifurcation of limit cycles for a family of perturbed Kukles differential systems

 Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Ⅷ-región, Chile

* Corresponding author: S. Rebollo-Perdomo

Received  December 2017 Published  May 2018

Fund Project: The authors are supported by Universidad de Bío-Bío grant DIUBB 167208 2/R.

We consider an integrable non-Hamiltonian system, which belongs to the quadratic Kukles differential systems. It has a center surrounded by a bounded period annulus. We study polynomial perturbations of such a Kukles system inside the Kukles family. We apply averaging theory to study the limit cycles that bifurcate from the period annulus and from the center of the unperturbed system. First, we show that the periodic orbits of the period annulus can be parametrized explicitly through the Lambert function. Later, we prove that at most one limit cycle bifurcates from the period annulus, under quadratic perturbations. Moreover, we give conditions for the non-existence, existence, and stability of the bifurcated limit cycles. Finally, by using averaging theory of seventh order, we prove that there are cubic systems, close to the unperturbed system, with 1 and 2 small limit cycles.

Citation: Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182
##### References:
 [1] J. Chavarriga, I. A. García, E. Sáez and I. Szántó, Limit cycles in Kukles systems of arbitrary degree with invariant ellipse, Nonlinear Anal., 67 (2007), 1005-1014.  doi: 10.1016/j.na.2006.06.035. [2] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750. [3] J. Giné, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65.  doi: 10.1016/j.physd.2013.01.015. [4] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0273-0979-00-00881-8. [5] J. M. Hill, N. G. Lloyd and J. M. Pearson, Limit cycles of a predator-prey model with intratrophic predation, J. Math. Anal. Appl., 349 (2009), 544-555.  doi: 10.1016/j.jmaa.2008.09.022. [6] J. Llibre and A. C. Mereu, Limit cycles for generalized Kukles polynomial differential systems, Nonlinear Anal., 74 (2011), 1261-1271.  doi: 10.1016/j.na.2010.09.064. [7] J. Llibre, D. D. Novaes and M. A. Texeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563. [8] J. Llibre, D. D. Novaes and M. A. Texeira, Corrigendum: Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 2417.  doi: 10.1088/0951-7715/27/9/2417. [9] J. Llibre, S. Rebollo-Perdomo and J. Torregrosa, Limit cycles bifurcating from isochronous surfaces of revolution in $\mathbb{R}^3$, J. Math. Anal. Appl., 381 (2011), 414-426.  doi: 10.1016/j.jmaa.2011.04.009. [10] I. Mezo and G. Keady, Some physical applications of generalized Lambert function, European Journal of Physics, 37 (2016), 065802 (10pp).  doi: 10.1088/0143-0807/37/6/065802. [11] O. Osuna, S. Rebollo-Perdomo and G. Villaseñor, On a class of invariant algebraic curves for Kukles systems, Electron. J. Qual. Theory Differ. Equ., 2016, Paper No. 61, 12 pp. [12] A. P. Sadovskiǐ, Cubic systems of nonlinear oscillations with seven limit cycles, Differ. Equ., 39 (2003), 505-516.  doi: 10.1023/A:1026010926840. [13] E. Sáez and I. Szántó, Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse, Appl. Math. Lett., 25 (2012), 1695-1700.  doi: 10.1016/j.aml.2012.01.039. [14] D. Veberi$\check{c}$, Lambert $W$ function for applications in physics, Comput. Phys. Commun., 183 (2012), 2622-2628.  doi: 10.1016/j.cpc.2012.07.008. [15] H. Zang, T. Zhang, Y. C. Tian and M. O. Tadé, Limit cycles for the Kukles system, J. Dyn. Control Syst., 14 (2008), 283-298.  doi: 10.1007/s10883-008-9036-x.

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##### References:
 [1] J. Chavarriga, I. A. García, E. Sáez and I. Szántó, Limit cycles in Kukles systems of arbitrary degree with invariant ellipse, Nonlinear Anal., 67 (2007), 1005-1014.  doi: 10.1016/j.na.2006.06.035. [2] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert $W$ function, Adv. Comput. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750. [3] J. Giné, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Physica D, 250 (2013), 58-65.  doi: 10.1016/j.physd.2013.01.015. [4] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0273-0979-00-00881-8. [5] J. M. Hill, N. G. Lloyd and J. M. Pearson, Limit cycles of a predator-prey model with intratrophic predation, J. Math. Anal. Appl., 349 (2009), 544-555.  doi: 10.1016/j.jmaa.2008.09.022. [6] J. Llibre and A. C. Mereu, Limit cycles for generalized Kukles polynomial differential systems, Nonlinear Anal., 74 (2011), 1261-1271.  doi: 10.1016/j.na.2010.09.064. [7] J. Llibre, D. D. Novaes and M. A. Texeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 563-583.  doi: 10.1088/0951-7715/27/3/563. [8] J. Llibre, D. D. Novaes and M. A. Texeira, Corrigendum: Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity, 27 (2014), 2417.  doi: 10.1088/0951-7715/27/9/2417. [9] J. Llibre, S. Rebollo-Perdomo and J. Torregrosa, Limit cycles bifurcating from isochronous surfaces of revolution in $\mathbb{R}^3$, J. Math. Anal. Appl., 381 (2011), 414-426.  doi: 10.1016/j.jmaa.2011.04.009. [10] I. Mezo and G. Keady, Some physical applications of generalized Lambert function, European Journal of Physics, 37 (2016), 065802 (10pp).  doi: 10.1088/0143-0807/37/6/065802. [11] O. Osuna, S. Rebollo-Perdomo and G. Villaseñor, On a class of invariant algebraic curves for Kukles systems, Electron. J. Qual. Theory Differ. Equ., 2016, Paper No. 61, 12 pp. [12] A. P. Sadovskiǐ, Cubic systems of nonlinear oscillations with seven limit cycles, Differ. Equ., 39 (2003), 505-516.  doi: 10.1023/A:1026010926840. [13] E. Sáez and I. Szántó, Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse, Appl. Math. Lett., 25 (2012), 1695-1700.  doi: 10.1016/j.aml.2012.01.039. [14] D. Veberi$\check{c}$, Lambert $W$ function for applications in physics, Comput. Phys. Commun., 183 (2012), 2622-2628.  doi: 10.1016/j.cpc.2012.07.008. [15] H. Zang, T. Zhang, Y. C. Tian and M. O. Tadé, Limit cycles for the Kukles system, J. Dyn. Control Syst., 14 (2008), 283-298.  doi: 10.1007/s10883-008-9036-x.
a) Graph of $\mathcal{F}_1(r_0)$ for (20). b) Phase portrait of (20) with $\varepsilon = 1/50$
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