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doi: 10.3934/dcdsb.2022132
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Limit cycles in a switching Liénard system

1. 

School of Mathematical Sciences, Beihang University, Beijing 100191, China

2. 

School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (MOE), Beijing Normal University, Beijing 100875, China

* Corresponding author: Laigang Guo

Received  February 2022 Revised  May 2022 Early access July 2022

Fund Project: The second author is supported by the Fundamental Research Funds for the Central Universities 2021NTST32

In this paper, we consider a class of quadratic switching Liénard systems with three switching lines. We give an algorithm for computing the Lyapunov constants of this system. Based on this method, we obtain a center condition and three limit cycles bifurcating from the focus $ (0,0) $. Further, an example of quadratic switching systems is constructed to show the existence of six limit cycles bifurcating from the center. This is a new low bound on the maximal number of small-amplitude limit cycles obtained in such quadratic switching systems.

Citation: Xiangyu Wang, Laigang Guo. Limit cycles in a switching Liénard system. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022132
References:
[1]

A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators: International Series in Physics, Elsevier, London, 2013.

[2]

L. Chen and M. Wang, The relative position and the number of limit cycles of a quadratic differential systems, Acta Math. Sinica, 22 (1979), 751-758. 

[3]

X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848.  doi: 10.1016/j.camwa.2010.04.019.

[4]

B. CollA. Gasull and R. Prohens, Degenerate Hopf bifurcation in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.  doi: 10.1006/jmaa.2000.7188.

[5]

B. CollR. Prohens and A. Gasull, The center problem for discontinuous Liénard differential equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1751-1761.  doi: 10.1142/S0218127499001231.

[6]

L. P. C. da CruzD. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 266 (2019), 4170-4203.  doi: 10.1016/j.jde.2018.09.032.

[7]

A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.

[8]

A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 13 (2003), 1755-1765.  doi: 10.1142/S0218127403007618.

[9]

L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 106189, 6 pp. doi: 10.1016/j.aml.2019.106189.

[10]

L. GuoP. Yu and Y. Chen, Bifurcation analysis on a class of $Z_2$-equivariant cubic switching systems showing eighteen limit cycles, J. Diff. Eqns., 266 (2019), 1221-1244.  doi: 10.1016/j.jde.2018.07.071.

[11]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar system, J. Differ. Equ., 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[12]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.

[13]

F. Li and P. Yu, A note on the paper "Center and isochronous center conditions for switching systems associated with elementary singular points", Commun. Nonlinear Sci. Numer. Simul., 90 (2020), 105405, 9 pp. doi: 10.1016/j.cnsns.2020.105405.

[14]

S. Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica, 23 (1980), 153-158. 

[15]

Y. Tian and P. Yu, Center conditions in a switching Bantin system, J. Diff. Eqns., 259 (2015), 1203-1226.  doi: 10.1016/j.jde.2015.02.044.

[16]

Y. WangM. Han and D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos Solitons Fractals, 83 (2016), 158-177.  doi: 10.1016/j.chaos.2015.11.041.

[17]

Y. Wu, L. Guo and Y. Chen, Hopf bifurcation of $Z_2$-equivariant generalized Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850069, 12 pp. doi: 10.1142/S0218127418500694.

[18]

P. YuM. Han and X. Zhang, Eighteen limit cycles around two symmetric foci in a cubic planar switching polynomial system, J. Differ. Equ., 275 (2021), 939-959.  doi: 10.1016/j.jde.2020.11.001.

show all references

References:
[1]

A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators: International Series in Physics, Elsevier, London, 2013.

[2]

L. Chen and M. Wang, The relative position and the number of limit cycles of a quadratic differential systems, Acta Math. Sinica, 22 (1979), 751-758. 

[3]

X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848.  doi: 10.1016/j.camwa.2010.04.019.

[4]

B. CollA. Gasull and R. Prohens, Degenerate Hopf bifurcation in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690.  doi: 10.1006/jmaa.2000.7188.

[5]

B. CollR. Prohens and A. Gasull, The center problem for discontinuous Liénard differential equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1751-1761.  doi: 10.1142/S0218127499001231.

[6]

L. P. C. da CruzD. D. Novaes and J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differ. Equ., 266 (2019), 4170-4203.  doi: 10.1016/j.jde.2018.09.032.

[7]

A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.

[8]

A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 13 (2003), 1755-1765.  doi: 10.1142/S0218127403007618.

[9]

L. F. S. Gouveia and J. Torregrosa, 24 crossing limit cycles in only one nest for piecewise cubic systems, Appl. Math. Lett., 103 (2020), 106189, 6 pp. doi: 10.1016/j.aml.2019.106189.

[10]

L. GuoP. Yu and Y. Chen, Bifurcation analysis on a class of $Z_2$-equivariant cubic switching systems showing eighteen limit cycles, J. Diff. Eqns., 266 (2019), 1221-1244.  doi: 10.1016/j.jde.2018.07.071.

[11]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar system, J. Differ. Equ., 248 (2010), 2399-2416.  doi: 10.1016/j.jde.2009.10.002.

[12]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0103843.

[13]

F. Li and P. Yu, A note on the paper "Center and isochronous center conditions for switching systems associated with elementary singular points", Commun. Nonlinear Sci. Numer. Simul., 90 (2020), 105405, 9 pp. doi: 10.1016/j.cnsns.2020.105405.

[14]

S. Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica, 23 (1980), 153-158. 

[15]

Y. Tian and P. Yu, Center conditions in a switching Bantin system, J. Diff. Eqns., 259 (2015), 1203-1226.  doi: 10.1016/j.jde.2015.02.044.

[16]

Y. WangM. Han and D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos Solitons Fractals, 83 (2016), 158-177.  doi: 10.1016/j.chaos.2015.11.041.

[17]

Y. Wu, L. Guo and Y. Chen, Hopf bifurcation of $Z_2$-equivariant generalized Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850069, 12 pp. doi: 10.1142/S0218127418500694.

[18]

P. YuM. Han and X. Zhang, Eighteen limit cycles around two symmetric foci in a cubic planar switching polynomial system, J. Differ. Equ., 275 (2021), 939-959.  doi: 10.1016/j.jde.2020.11.001.

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