May  2022, 21(5): 1833-1855. doi: 10.3934/cpaa.2022049

The number of limit cycles by perturbing a piecewise linear system with three zones

1. 

School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

2. 

School of Mathematical Sciences, Zhejiang University, Hangzhou 310058, China

* Corresponding author

Received  August 2021 Revised  January 2022 Published  May 2022 Early access  March 2022

Fund Project: The authors are supported by National Natural Science Foundation of China (11701289)

First, this paper provides a new proof for the expression of the generalized first order Melnikov function on piecewise smooth differential systems with multiply straight lines. Then, by using the Melnikov function, we consider the limit cycle bifurcation problem of a 3-piecewise near Hamiltonian system with two switching lines, obtaining $ 2n+3[\frac{n+1}{2}] $ limit cycles near the double generalized homoclinic loop.

Citation: Xiaolei Zhang, Yanqin Xiong, Yi Zhang. The number of limit cycles by perturbing a piecewise linear system with three zones. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1833-1855. doi: 10.3934/cpaa.2022049
References:
[1]

K. J. Åström, Oscillations in systems with relay feedback, in Adaptive Control, Filtering, and Signal Processing, Springer, New York, (1995), 1–25. doi: 10.1007/978-1-4419-8568-2_1.

[2] S. Banerjee and G. C. Verghese, Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 1999. 
[3]

M. Cai and M. Han, Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters, Commun. Pure Appl. Anal., 20 (2021), 55-75.  doi: 10.3934/cpaa.2020257.

[4]

H. ChenD. LiJ. Xie and Y. Yue, Limit cycles in planar continuous piecewise linear systems, Commun. Nonlinear Sci., 47 (2017), 438-454.  doi: 10.1016/j.cnsns.2016.12.006.

[5]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[6]

E. FreireE. PonceF. Rodrigo and F. Torres, Bifurcation sets of symmetrical continuous piecewise linear systems with three zones, Int. J. Bifurcat. Chaos, 12 (2002), 1675-1702.  doi: 10.1142/S0218127402005509.

[7] M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013. 
[8]

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.

[9]

D. Hilbert, Mathematical problems, M. Newton, Trans. Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.

[10]

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

[11]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Cont. Dyn. S., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[12]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (2014), 340-353.  doi: 10.1016/j.jmaa.2013.08.064.

[13]

S. LiH. ChenT. Chen and K. Wu, A note on the Bendixson-Dulac Theorem for Refractedsystems with multiple zones, J. Non. Mod. Anal., 3 (2021), 79-85. 

[14]

M. F. S. LimaC. Pessoa and W. F. Pereira, Limit cycles bifurcating from a period annulus in continuous piecewise linear differential systems with three zones, Int. J. Bifurcat. Chaos, 27 (2017), 1750022.  doi: 10.1142/S0218127417500225.

[15]

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.

[16]

J. LlibreM. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real, 14 (2013), 2002-2012.  doi: 10.1016/j.nonrwa.2013.02.004.

[17]

J. LlibreR. RamírezV. Ramírez and N. Sadovskaia, The 16th Hilbert problem restricted to circular algebraic limit cycles, J. Differ. Equ., 260 (2016), 5726-5760.  doi: 10.1016/j.jde.2015.12.019.

[18]

H. P. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223.  doi: 10.1016/0001-8708(70)90023-X.

[19]

J. S. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[20]

A. Tonnelier, The McKean's caricature of the Fitzhugh–Nagumo model Ⅰ. The space-clamped system, SIAM J. Appl. Math., 63 (2003), 459-484.  doi: 10.1137/S0036139901393500.

[21]

A. Tonnelier, On the number of limit cycles in piecewise-linear Liénard systems, Int. J. Bifurcat. Chaos, 15 (2005), 1417-1422.  doi: 10.1142/S0218127405012624.

[22]

Y. Xiong and M. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Appl. Math. Comput., 242 (2014), 47-64.  doi: 10.1016/j.amc.2014.05.035.

[23]

Y. Xiong and M. Han, New lower bounds for the Hilbert number of polynomial systems of Liénard type, J. Differ. Equ., 257 (2014), 2565-2590.  doi: 10.1016/j.jde.2014.05.058.

[24]

Y. Xiong and M. Han, Limit cycle bifurcations in discontinuous planar systems with multiple lines, J. Appl. Anal. Comput., 10 (2020), 361-377.  doi: 10.11948/20190274.

[25]

Y. Xiong and C. Wang, Limit cycle bifurcations of planar piecewise differential systems with three zones, Nonlinear Anal. Real, 61 (2021), 18 pp. doi: 10.1016/j.nonrwa.2021.103333.

show all references

References:
[1]

K. J. Åström, Oscillations in systems with relay feedback, in Adaptive Control, Filtering, and Signal Processing, Springer, New York, (1995), 1–25. doi: 10.1007/978-1-4419-8568-2_1.

[2] S. Banerjee and G. C. Verghese, Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 1999. 
[3]

M. Cai and M. Han, Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters, Commun. Pure Appl. Anal., 20 (2021), 55-75.  doi: 10.3934/cpaa.2020257.

[4]

H. ChenD. LiJ. Xie and Y. Yue, Limit cycles in planar continuous piecewise linear systems, Commun. Nonlinear Sci., 47 (2017), 438-454.  doi: 10.1016/j.cnsns.2016.12.006.

[5]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[6]

E. FreireE. PonceF. Rodrigo and F. Torres, Bifurcation sets of symmetrical continuous piecewise linear systems with three zones, Int. J. Bifurcat. Chaos, 12 (2002), 1675-1702.  doi: 10.1142/S0218127402005509.

[7] M. Han, Bifurcation Theory of Limit Cycles, Science Press, Beijing, 2013. 
[8]

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.

[9]

D. Hilbert, Mathematical problems, M. Newton, Trans. Bull. Amer. Math. Soc., 8 (1902), 437-479.  doi: 10.1090/S0002-9904-1902-00923-3.

[10]

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

[11]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Cont. Dyn. S., 32 (2012), 2147-2164.  doi: 10.3934/dcds.2012.32.2147.

[12]

S. Huan and X. Yang, On the number of limit cycles in general planar piecewise linear systems of node-node types, J. Math. Anal. Appl., 411 (2014), 340-353.  doi: 10.1016/j.jmaa.2013.08.064.

[13]

S. LiH. ChenT. Chen and K. Wu, A note on the Bendixson-Dulac Theorem for Refractedsystems with multiple zones, J. Non. Mod. Anal., 3 (2021), 79-85. 

[14]

M. F. S. LimaC. Pessoa and W. F. Pereira, Limit cycles bifurcating from a period annulus in continuous piecewise linear differential systems with three zones, Int. J. Bifurcat. Chaos, 27 (2017), 1750022.  doi: 10.1142/S0218127417500225.

[15]

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.

[16]

J. LlibreM. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real, 14 (2013), 2002-2012.  doi: 10.1016/j.nonrwa.2013.02.004.

[17]

J. LlibreR. RamírezV. Ramírez and N. Sadovskaia, The 16th Hilbert problem restricted to circular algebraic limit cycles, J. Differ. Equ., 260 (2016), 5726-5760.  doi: 10.1016/j.jde.2015.12.019.

[18]

H. P. McKean, Nagumo's equation, Adv. Math., 4 (1970), 209-223.  doi: 10.1016/0001-8708(70)90023-X.

[19]

J. S. NagumoS. Arimoto and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1962), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[20]

A. Tonnelier, The McKean's caricature of the Fitzhugh–Nagumo model Ⅰ. The space-clamped system, SIAM J. Appl. Math., 63 (2003), 459-484.  doi: 10.1137/S0036139901393500.

[21]

A. Tonnelier, On the number of limit cycles in piecewise-linear Liénard systems, Int. J. Bifurcat. Chaos, 15 (2005), 1417-1422.  doi: 10.1142/S0218127405012624.

[22]

Y. Xiong and M. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Appl. Math. Comput., 242 (2014), 47-64.  doi: 10.1016/j.amc.2014.05.035.

[23]

Y. Xiong and M. Han, New lower bounds for the Hilbert number of polynomial systems of Liénard type, J. Differ. Equ., 257 (2014), 2565-2590.  doi: 10.1016/j.jde.2014.05.058.

[24]

Y. Xiong and M. Han, Limit cycle bifurcations in discontinuous planar systems with multiple lines, J. Appl. Anal. Comput., 10 (2020), 361-377.  doi: 10.11948/20190274.

[25]

Y. Xiong and C. Wang, Limit cycle bifurcations of planar piecewise differential systems with three zones, Nonlinear Anal. Real, 61 (2021), 18 pp. doi: 10.1016/j.nonrwa.2021.103333.

Figure 1.  The curve of $ \Gamma_h $
Figure 2.  The phase portrait of system $ (1.8){|}_{\varepsilon = 0} $
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