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May  2022, 21(5): 1793-1809. doi: 10.3934/cpaa.2022047

The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines

Ai Ke 1, , Maoan Han 2,, and  Wei Geng 1,

1. 

Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
2. 

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, 321004, China

* Corresponding author

Received  November 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: This work was supported by National Nature Science Foundation of China (11931016 and 11771296)

In this paper, we give an upper bound (for $ n\geq3 $) and the least upper bound (for $ n = 1,2 $) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree $ n $, respectively. The results improve the conclusions in [19].

Keywords: Limit cycles, bifurcation, isochronous system, Melnikov function, piecewise smooth system.
Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 37G15.
Citation: Ai Ke, Maoan Han, Wei Geng. The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1793-1809. doi: 10.3934/cpaa.2022047
References:
[1]

X. Chen and M. Han, A linear estimate of the number of limit cycles for a piecewise smooth near-hamiltonian system, Qual. Theory Dynam. Syst., 19 (2020), 19 pp. doi: 10.1007/s12346-020-00398-x.

[2]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser Verlag, Basel-Boston-Berlin, 2007.

[3]

J. Giné, J. Llibre, K. Wu and X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differ. Equ., 260 (2016), 4130-4156.  doi: 10.1016/j.jde.2015.11.005.

[4]

M. Han, On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, J. Appl. Anal. Comput., 7 (2017), 788-794.  doi: 10.11948/2017049.

[5]

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.

[6]

M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, J. Nonlinear Model. Anal., 3 (2021), 13-34.  doi: 10.12150/jnma.2021.13.

[7]

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

[8]

E. Horozov and I. D. Iliev, Linear estimate for the number of zeros of abelian integrals with cubic hamiltonians, Nonlinearity, 11 (1998), 1521-1537.  doi: 10.1088/0951-7715/11/6/006.

[9]

N. Hu and Z. Du, Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3436-3448.  doi: 10.1016/j.cnsns.2013.05.012.

[10]

S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, New York, 1966.

[11]

A. Ke and M. Han, Limit cycles from perturbing a piecewise smooth system with a center and a homoclinic loop, Int. J. Bifurcat. Chaos, 31 (2021), 15 PP. doi: 10.1142/S0218127421501595.

[12]

F. Liang, M. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.  doi: 10.1016/j.na.2012.03.022.

[13]

S. Liu, M. Han and J. Li, Bifurcation methods of periodic orbits for piecewise smooth systems, J. Differ. Equ., 275 (2021), 204-233.  doi: 10.1016/j.jde.2020.11.040.

[14]

X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise hamiltonian systems, Int. J. Bifurcat. Chaos, 20 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.

[15]

W. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contr. Differ. Equ., 3 (1964), 21-36.  doi: 10.1017/s002555720004852x.

[16]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain abelian integrals, J. Differ. Equ., 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[17]

H. Tian and M. Han, Limit cycle bifurcations of piecewise smooth near-hamiltonian systems with a switching curve, Discret. Contin. Dynam. Syst. Series B, 26 (2021), 5581-5599.  doi: 10.3934/dcdsb.2020368.

[18]

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

[19]

J. Yang, Picard-fuchs equation applied to quadratic isochronous systems with two switching lines, Int. J. Bifurcat. Chaos, 30 (2020), 17 pp. doi: 10.1142/S021812742050042X.

[20]

J. Yang and L. Zhao, Bounding the number of limit cycles of discontinuous differential systems by using picard-fuchs equations, J. Differ. Equ., 264 (2018), 5734-5757.  doi: 10.1016/j.jde.2018.01.017.

show all references

References:
[1]

X. Chen and M. Han, A linear estimate of the number of limit cycles for a piecewise smooth near-hamiltonian system, Qual. Theory Dynam. Syst., 19 (2020), 19 pp. doi: 10.1007/s12346-020-00398-x.

[2]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkhäuser Verlag, Basel-Boston-Berlin, 2007.

[3]

J. Giné, J. Llibre, K. Wu and X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differ. Equ., 260 (2016), 4130-4156.  doi: 10.1016/j.jde.2015.11.005.

[4]

M. Han, On the maximum number of periodic solutions of piecewise smooth periodic equations by average method, J. Appl. Anal. Comput., 7 (2017), 788-794.  doi: 10.11948/2017049.

[5]

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.

[6]

M. Han and J. Yang, The maximum number of zeros of functions with parameters and application to differential equations, J. Nonlinear Model. Anal., 3 (2021), 13-34.  doi: 10.12150/jnma.2021.13.

[7]

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

[8]

E. Horozov and I. D. Iliev, Linear estimate for the number of zeros of abelian integrals with cubic hamiltonians, Nonlinearity, 11 (1998), 1521-1537.  doi: 10.1088/0951-7715/11/6/006.

[9]

N. Hu and Z. Du, Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 3436-3448.  doi: 10.1016/j.cnsns.2013.05.012.

[10]

S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Interscience, New York, 1966.

[11]

A. Ke and M. Han, Limit cycles from perturbing a piecewise smooth system with a center and a homoclinic loop, Int. J. Bifurcat. Chaos, 31 (2021), 15 PP. doi: 10.1142/S0218127421501595.

[12]

F. Liang, M. Han and V. G. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear hamiltonian system with a homoclinic loop, Nonlinear Anal., 75 (2012), 4355-4374.  doi: 10.1016/j.na.2012.03.022.

[13]

S. Liu, M. Han and J. Li, Bifurcation methods of periodic orbits for piecewise smooth systems, J. Differ. Equ., 275 (2021), 204-233.  doi: 10.1016/j.jde.2020.11.040.

[14]

X. Liu and M. Han, Bifurcation of limit cycles by perturbing piecewise hamiltonian systems, Int. J. Bifurcat. Chaos, 20 (2010), 1379-1390.  doi: 10.1142/S021812741002654X.

[15]

W. Loud, Behavior of the period of solutions of certain plane autonomous systems near centers, Contr. Differ. Equ., 3 (1964), 21-36.  doi: 10.1017/s002555720004852x.

[16]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain abelian integrals, J. Differ. Equ., 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[17]

H. Tian and M. Han, Limit cycle bifurcations of piecewise smooth near-hamiltonian systems with a switching curve, Discret. Contin. Dynam. Syst. Series B, 26 (2021), 5581-5599.  doi: 10.3934/dcdsb.2020368.

[18]

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

[19]

J. Yang, Picard-fuchs equation applied to quadratic isochronous systems with two switching lines, Int. J. Bifurcat. Chaos, 30 (2020), 17 pp. doi: 10.1142/S021812742050042X.

[20]

J. Yang and L. Zhao, Bounding the number of limit cycles of discontinuous differential systems by using picard-fuchs equations, J. Differ. Equ., 264 (2018), 5734-5757.  doi: 10.1016/j.jde.2018.01.017.

Figure 1.  Phase portraits of (1.3) as $ \varepsilon = 0. $
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Figure 2.  The graphs of (a).$ W_{71}(h)+W_{72}(h) $ for $ h\in(0,38) $; (b).$ W_{81}(h)+W_{82}(h) $ for $ h\in(0,80) $; (c).$ W_{91}(h)+W_{92}(h)+W_{93}(h) $ for $ h\in(0,70) $
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Figure 3.  The graph of $ W_{41}(h)+W_{42}(h) $ for $ h\in(-4,-1). $
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Table 1.  New and old upper bounds for $ Z_1(n) $ and $ \tilde{Z}_1(n) $
$ n $ 1 2 3 4 5 6 7 8 9 10 11 12
new 7, 4
(exact)		 9, 5
(exact)		 15, 9 20, 11 25, 13 30, 16 35, 19 40, 22 45, 25 50, 28 55, 31 60, 33
old 28, 17 37, 19 46, 21 55, 23 64, 25 73, 27 82, 29 91, 31 100, 33 109, 35 118, 37 127, 39
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$ n $ 1 2 3 4 5 6 7 8 9 10 11 12
new 7, 4
(exact)		 9, 5
(exact)		 15, 9 20, 11 25, 13 30, 16 35, 19 40, 22 45, 25 50, 28 55, 31 60, 33
old 28, 17 37, 19 46, 21 55, 23 64, 25 73, 27 82, 29 91, 31 100, 33 109, 35 118, 37 127, 39
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