Limit cycles in an $ m $-piecewise discontinuous polynomial differential system

Ziguo Jiang; Ziguo Jiang

doi:10.3934/math.2024177

AIMS Mathematics

2024, Volume 9, Issue 2: 3613-3629. doi: 10.3934/math.2024177

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Limit cycles in an $ m $-piecewise discontinuous polynomial differential system

Ziguo Jiang ^{1,2
,
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1.
School of Mathematics, Aba Teachers University, Wenchuan, Sichuan, 623002, China
2.
Computational Mathematics and Applied Statistics Laboratory, Aba Teachers University, Wenchuan, Sichuan, 623002, China

Received: 16 November 2023 Revised: 24 December 2023 Accepted: 02 January 2024 Published: 09 January 2024
MSC : 34C05, 34C25, 34C29

In this paper, I study a planar $ m $-piecewise discontinuous polynomial differential system $ \dot{x} = y, \dot{y} = -x-\varepsilon(f(x, y)+g_m(x, y)h(x)) $, which has a linear center in each zone partitioned by those switching lines, where $ f(x, y) = \sum_{i+j = 0}^na_{ij}x^iy^j $, $ h(x) = \sum_{j = 0}^lb_jx^j, a_{ij}, b_j\in\mathbb{R}, n, l\in\mathbb{N} $, and $ g_m(x, y) $ with the positive even number $ m $ as the union of $ m/2 $ different straight lines passing through the origin of coordinates dividing the plane into sectors of angle $ 2\pi/m $. Using the averaging theory, I provide the lower bound $ L_m(n, l) $ for the maximun number of limit cycles, which bifurcates which bifurcating from the annulus of the origin of this system.
- limit cycle,
- piecewise differential equation,
- averaging method
Citation: Ziguo Jiang. Limit cycles in an $ m $-piecewise discontinuous polynomial differential system[J]. AIMS Mathematics, 2024, 9(2): 3613-3629. doi: 10.3934/math.2024177

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Abstract

In this paper, I study a planar $ m $-piecewise discontinuous polynomial differential system $ \dot{x} = y, \dot{y} = -x-\varepsilon(f(x, y)+g_m(x, y)h(x)) $, which has a linear center in each zone partitioned by those switching lines, where $ f(x, y) = \sum_{i+j = 0}^na_{ij}x^iy^j $, $ h(x) = \sum_{j = 0}^lb_jx^j, a_{ij}, b_j\in\mathbb{R}, n, l\in\mathbb{N} $, and $ g_m(x, y) $ with the positive even number $ m $ as the union of $ m/2 $ different straight lines passing through the origin of coordinates dividing the plane into sectors of angle $ 2\pi/m $. Using the averaging theory, I provide the lower bound $ L_m(n, l) $ for the maximun number of limit cycles, which bifurcates which bifurcating from the annulus of the origin of this system.

References

[1]	F. Dumortier, A. Guzmán, C. Rousseau, Finite cyclicity of elementary graphics surrounding a focus or center in quadratic systems, Qual. Th. Dyn. Syst., 3 (2002), 123–154. https://doi.org/10.1007/BF02969336 doi: 10.1007/BF02969336
[2]	C. Li, C. Liu, J. Yang, A cubic system with thirteen limit cycles, J. Differ. Equ., 246 (2009), 3609–3619. https://doi.org/10.1016/j.jde.2009.01.038 doi: 10.1016/j.jde.2009.01.038
[3]	Y. Liu, J. Li, $Z_2$-equivariant cubic systems which yield 13 limit cycles, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 781–800. https://doi.org/10.1007/s10255-014-0420-x doi: 10.1007/s10255-014-0420-x
[4]	M. di Bernardo Laurea, A. R. Champneys, C. J. Budd, P. Kowalczyk, Piecewise-smooth dynamical systems: Theory and applications, London: Springer, 2008. https://doi.org/10.1007/978-1-84628-708-4
[5]	B. Coll, A. Gasull, R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671–690. https://doi.org/10.1006/jmaa.2000.7188 doi: 10.1006/jmaa.2000.7188
[6]	A. Gasull, J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurcat. Chaos, 13 (2003), 1755–1765. https://doi.org/10.1142/S0218127403007618 doi: 10.1142/S0218127403007618
[7]	D. D. Novaes, L. A. Silva, Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields, J. Differ. Equ., 300 (2021), 565–596. https://doi.org/10.1016/j.jde.2021.08.008 doi: 10.1016/j.jde.2021.08.008
[8]	M. Han, W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differ. Equ., 248 (2010), 2399–2416. https://doi.org/10.1016/j.jde.2009.10.002 doi: 10.1016/j.jde.2009.10.002
[9]	S. M. Huan, X. S. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin. Dyn. Syst., 32 (2012), 2147–2164. https://doi.org/10.3934/dcds.2012.32.2147 doi: 10.3934/dcds.2012.32.2147
[10]	J. Llibre, E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), 325–335.
[11]	C. Buzzi, C. Pessoa, J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst., 33 (2013), 3915–3936. https://doi.org/10.3934/DCDS.2013.33.3915 doi: 10.3934/DCDS.2013.33.3915
[12]	J. L. Cardoso, J. Llibre, D. D. Novaes, D. J. Tonon, Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields, Dyn. Syst., 35 (2020), 490–514. https://doi.org/10.1080/14689367.2020.1722064 doi: 10.1080/14689367.2020.1722064
[13]	E. Freire, E. Ponce, F. Torres, The discontinuous matching of two planar linear foci can have three nested crossing limit cycles, Publ. Math., 2014,221–253. https://doi.org/10.5565/PUBLMAT_Extra14_13
[14]	J. Llibre, M. Ordóñez, E. Ponce, On the exisentence and uniquness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real, 14 (2013), 2002–2012. https://doi.org/10.1016/j.nonrwa.2013.02.004 doi: 10.1016/j.nonrwa.2013.02.004
[15]	X. Chen, V. G. Romanovski, W. Zhang, Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl., 432 (2015), 1058–1076. https://doi.org/10.1016/j.jmaa.2015.07.036 doi: 10.1016/j.jmaa.2015.07.036
[16]	J. K. Hale, Ordinary differential equations, New York: Robert E. Krieger, 1980.
[17]	J. Llibre, D. D. Novaes, M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. des Sci. Math., 139 (2015), 229–244. https://doi.org/10.1016/j.bulsci.2014.08.011 doi: 10.1016/j.bulsci.2014.08.011
[18]	J. Llibre, D. D. Novaes, C. A. B. Rodrigues, Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones, Physica D, 353-354 (2017), 1–10. https://doi.org/10.1016/j.physd.2017.05.003 doi: 10.1016/j.physd.2017.05.003
[19]	X. Chen, J. Llibre, W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Contin. Dyn. Syst. B, 22 (2017), 3953–3965. https://doi.org/10.3934/dcdsb.2017203 doi: 10.3934/dcdsb.2017203
[20]	R. M. Martins, A. C. Mereu, Limit cycles in discontinuous classical Liénard equations, Nonlinear Anal. Real, 20 (2014), 67–73. https://doi.org/10.1016/j.nonrwa.2014.04.003 doi: 10.1016/j.nonrwa.2014.04.003
[21]	T. M. P. De Abreu, R. M. Martins, Sharp estimates for the number of limit cycles in discontinuous generalized Liénard equations, 2023, arXiv: 2307.09599v1. https://doi.org/10.48550/arXiv.2307.09599
[22]	C. Henry, Differential equations with discontinuous right-hand side for planning procedures. J. Econ. Theory, 4 (1972), 545–551. https://doi.org/10.1016/0022-0531(72)90138-X doi: 10.1016/0022-0531(72)90138-X
[23]	A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of ocillators, New York: Dover, 1966. https://doi.org/10.1016/C2013-0-06631-5
[24]	M. Kunze, T. Kupper, Qualitative bifurcation analysis of a non-smooth friction-oscillator model, Z. Angew. Math. Phys., 48 (1997), 87–101. https://doi.org/10.1007/PL00001471 doi: 10.1007/PL00001471
[25]	J. Llibre, M. A. Teixeira, Limit cycles for $m$-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys., 66 (2015), 51–66. https://doi.org/10.1007/s00033-013-0393-2 doi: 10.1007/s00033-013-0393-2
[26]	G. Dong, C. Liu, Note on limit cycles for $m$-piecewise discontinuous polynomial Liénard differential equtions. Z. Angew. Math. Phys., 68 (2017), 97. https://doi.org/10.1007/s00033-017-0844-2 doi: 10.1007/s00033-017-0844-2
[27]	D. D. Novaes, On nonsmooth perturbations of nondegenerate planar centers, Publ. Math., 2014,395–420. https://doi.org/10.5565/PUBLMAT_Extra14_20
[28]	V. Carmona, F. Fernández-Sánchez, D. D. Novaes, Uniform upper bound for the number of limit cycles of planar piecewise linear differential systems with two zones separated by a straight line, Appl. Math. Lett., 137 (2023), 108501. https://doi.org/10.1016/j.aml.2022.108501 doi: 10.1016/j.aml.2022.108501
[29]	I. S. Berezin, N. P. Zhidkov, Computing methods, Oxford: Pergamon Press, 1965. https://doi.org/10.1016/C2013-0-01726-4
[30]	B. Coll, A. Gasull, R. Prohens, Bifurcation of limit cycles from two families of centers, Dyn. Contin. Discrete Implus. Syst. Ser. A Math. Anal., 12 (2005), 275–287.

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