Figure 1.1.
Phase portrait of $ (1.3)|_{\varepsilon = \lambda = 0} $
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January
2021, 20(1): 55-75.
doi: 10.3934/cpaa.2020257
Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters
| 1. | Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China |
| 2. | Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China |
In this paper, we concern with the problem of limit cycle bifurcation for a class of piecewise smooth cubic systems. Using the first order Melnikov function we prove that at least thirteen limit cycles can be bifurcated from periodic solutions surrounding the center.
Keywords:
Limit cycle bifurcation,
piecewise smooth system,
multiple parameters,
near-Hamiltonian system,
Melnikov function.
Mathematics Subject Classification:
Primary: 34C05, 34C07; Secondary: 37G15.
Citation:
Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis,
2021, 20
(1)
: 55-75.
doi: 10.3934/cpaa.2020257
![]()
References:
| [1] |
L. P. da Cruz, D. D. Novaes and J. Torregrosa,
New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 226 (2019), 4170-4203.
doi: 10.1016/j.jde.2018.09.032. |
| [2] |
J. Gin$\acute{e}$ and J. Llibre,
Limit cycles of cubic polynomial vector fields via the averaging theory, Nonlinear Anal., 66 (2007), 1707-1721.
doi: 10.1016/j.na.2006.02.016. |
| [3] |
M. Grau, F. Ma$\tilde{n}$osas and J. Villadelprat,
A Chebyshew criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.
doi: 10.1090/S0002-9947-2010-05007-X. |
| [4] |
L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 6pp.
doi: 10.1016/j.aml.2019.106189. |
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M. Han, G. Chen and C. Sun,
On the number of limit cycles in near-Hamiltonian polynomial systems, Int. J. Bifur. Chaos, 17 (2007), 2033-2047.
doi: 10.1142/S0218127407018208. |
| [6] |
M. Han and L. Sheng,
Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.
doi: 10.11948/2015061. |
| [7] |
M. Han and Y. Xiong,
Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters, Chaos Solitons Fractals, 68 (2014), 20-29.
doi: 10.1016/j.chaos.2014.07.005. |
| [8] |
S. Huan and X. Yang,
On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.
doi: 10.3934/dcds.2012.32.2147. |
| [9] |
S. Karlin and W. J. Studden, Tchebycheff systems: With Applications in Analysis and Statistics, Pure Appa. math., Interscience Publishers, New York, London, Sydney, 1966. |
| [10] |
F. Liang and M. Han,
Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Solitons Fractals, 45 (2012), 454-464.
doi: 10.1016/j.chaos.2011.09.013. |
| [11] |
S. Li and T. Huang,
Limit cycles for piecewise smooth perturbations of a cubic polynomial differential center, J. Differ. Equ., 2015 (2015), 1-17.
|
| [12] |
X. Liu and M. Han,
Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Int. J. Bifur. Chaos, 5 (2010), 1379-1390.
doi: 10.1142/S021812741002654X. |
| [13] |
J. Llibre and J. Itikawa,
Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math., 277 (2015), 171-191.
doi: 10.1016/j.cam.2014.09.007. |
| [14] |
C. Li, W. Li, J. Llibre and Z. Zhang,
Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differ. Equ., 180 (2002), 307-333.
doi: 10.1006/jdeq.2001.4064. |
| [15] |
J. Llibre, C. Mereu and D. D. Novaes,
Averaging theory for discontinuous piecewise differential systems, J. Differ. Equ., 258 (2015), 4007-4032.
doi: 10.1016/j.jde.2015.01.022. |
| [16] |
J. Llibre, D. D. Novaes and M. A. Teixeira,
On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244.
doi: 10.1016/j.bulsci.2014.08.011. |
| [17] |
J. Llibre and G. $\acute{S}$wirszcz,
On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Ser. A. Math. Anal., 18 (2011), 203-314.
|
| [18] |
S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Int. J. Bifur. Chaos, 24 (2014), 8pp.
doi: 10.1142/S0218127414500357. |
| [19] |
J. N. Mather,
Stability of $C^{\infty}$ Mappings: I. The Division Theorem, Ann. Math., 87 (1968), 89-104.
doi: 10.2307/1970595. |
| [20] |
D. D. Novaes and J. Torregrose,
On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl., 448 (2017), 171-186.
doi: 10.1016/j.jmaa.2016.10.076. |
| [21] |
J. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer New York, 2007. |
| [22] |
Y. Xiong, Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters, J. Math. Anal. Appl., 421 (2015), 260-275.
doi: 10.1016/j.jmaa.2014.07.013. |
| [23] |
Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Int. J. Bifur. Chaos, 27 (2017), 14pp.
doi: 10.1142/S0218127417501267. |
show all references
References:
| [1] |
L. P. da Cruz, D. D. Novaes and J. Torregrosa,
New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 226 (2019), 4170-4203.
doi: 10.1016/j.jde.2018.09.032. |
| [2] |
J. Gin$\acute{e}$ and J. Llibre,
Limit cycles of cubic polynomial vector fields via the averaging theory, Nonlinear Anal., 66 (2007), 1707-1721.
doi: 10.1016/j.na.2006.02.016. |
| [3] |
M. Grau, F. Ma$\tilde{n}$osas and J. Villadelprat,
A Chebyshew criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.
doi: 10.1090/S0002-9947-2010-05007-X. |
| [4] |
L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 6pp.
doi: 10.1016/j.aml.2019.106189. |
| [5] |
M. Han, G. Chen and C. Sun,
On the number of limit cycles in near-Hamiltonian polynomial systems, Int. J. Bifur. Chaos, 17 (2007), 2033-2047.
doi: 10.1142/S0218127407018208. |
| [6] |
M. Han and L. Sheng,
Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput., 5 (2015), 809-815.
doi: 10.11948/2015061. |
| [7] |
M. Han and Y. Xiong,
Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters, Chaos Solitons Fractals, 68 (2014), 20-29.
doi: 10.1016/j.chaos.2014.07.005. |
| [8] |
S. Huan and X. Yang,
On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147-2164.
doi: 10.3934/dcds.2012.32.2147. |
| [9] |
S. Karlin and W. J. Studden, Tchebycheff systems: With Applications in Analysis and Statistics, Pure Appa. math., Interscience Publishers, New York, London, Sydney, 1966. |
| [10] |
F. Liang and M. Han,
Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems, Chaos Solitons Fractals, 45 (2012), 454-464.
doi: 10.1016/j.chaos.2011.09.013. |
| [11] |
S. Li and T. Huang,
Limit cycles for piecewise smooth perturbations of a cubic polynomial differential center, J. Differ. Equ., 2015 (2015), 1-17.
|
| [12] |
X. Liu and M. Han,
Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Int. J. Bifur. Chaos, 5 (2010), 1379-1390.
doi: 10.1142/S021812741002654X. |
| [13] |
J. Llibre and J. Itikawa,
Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math., 277 (2015), 171-191.
doi: 10.1016/j.cam.2014.09.007. |
| [14] |
C. Li, W. Li, J. Llibre and Z. Zhang,
Linear estimation of the number of zeros of Abelian integrals for some cubic isochronous centers, J. Differ. Equ., 180 (2002), 307-333.
doi: 10.1006/jdeq.2001.4064. |
| [15] |
J. Llibre, C. Mereu and D. D. Novaes,
Averaging theory for discontinuous piecewise differential systems, J. Differ. Equ., 258 (2015), 4007-4032.
doi: 10.1016/j.jde.2015.01.022. |
| [16] |
J. Llibre, D. D. Novaes and M. A. Teixeira,
On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math., 139 (2015), 229-244.
doi: 10.1016/j.bulsci.2014.08.011. |
| [17] |
J. Llibre and G. $\acute{S}$wirszcz,
On the limit cycles of polynomial vector fields, Dyn. Contin. Discrete Impuls. Ser. A. Math. Anal., 18 (2011), 203-314.
|
| [18] |
S. Li and Y. Zhao, Limit cycles of perturbed cubic isochronous center via the second order averaging method, Int. J. Bifur. Chaos, 24 (2014), 8pp.
doi: 10.1142/S0218127414500357. |
| [19] |
J. N. Mather,
Stability of $C^{\infty}$ Mappings: I. The Division Theorem, Ann. Math., 87 (1968), 89-104.
doi: 10.2307/1970595. |
| [20] |
D. D. Novaes and J. Torregrose,
On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl., 448 (2017), 171-186.
doi: 10.1016/j.jmaa.2016.10.076. |
| [21] |
J. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2$^{nd}$ edition, Applied Mathematical Sciences, Springer New York, 2007. |
| [22] |
Y. Xiong, Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters, J. Math. Anal. Appl., 421 (2015), 260-275.
doi: 10.1016/j.jmaa.2014.07.013. |
| [23] |
Y. Xiong, M. Han and V. G. Romanovski, The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with two saddles, Int. J. Bifur. Chaos, 27 (2017), 14pp.
doi: 10.1142/S0218127417501267. |
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