American Institute of Mathematical Sciences

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June  2012, 32(6): 2147-2164. doi: 10.3934/dcds.2012.32.2147

On the number of limit cycles in general planar piecewise linear systems

Song-Mei Huan 1, and  Xiao-Song Yang 2,

1. 

Department of Mathematics, Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China
2. 

Department of Mathematics, Huazhong University of Science and Technology, Wuhan, 430074, China

Received  January 2011 Revised  June 2011 Published  February 2012

Much progress has been made in planar piecewise smooth dynamical systems. However there remain many important problems to be solved even in planar piecewise linear systems. In this paper, we investigate the number of limit cycles of planar piecewise linear systems with two linear regions sharing the same equilibrium. By studying the implicit Poincaré map induced by the discontinuity boundary, some cases when there exist at most 2 limit cycles is completely investigated. Especially, based on these results we provide an example along with numerical simulations to illustrate the existence of 3 limit cycles thus have a negative answer to the conjecture by M. Han and W. Zhang [11](J. Differ.Equations 248 (2010) 2399-2416) that piecewise linear systems with only two regions have at most 2 limit cycles.
Keywords: Limit cycle, implicit Poincaré map, piecewise linear systems..
Mathematics Subject Classification: Primary: 34C99, 34C07; Secondary: 37E9.
Citation: Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147
References:
[1]

A. A. Andronov, A. Vitt and S. Khaikin, "Theroy of Oscillators," Pergamon Press, Oxford-New York-Toronto, Ont., 1966.  Google Scholar

[2]

B. Coll, A. 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.  Google Scholar

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V. Carmona, E. Freire, E. Ponce and F. Torres, On simplifying and classifying piecewise-linear systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 49 (2002), 609-620.  Google Scholar

[4]

V. Carmona, E. Freire, E. Ponce and F. Torres, The continuous matching of two stable linear systems can be unstable, Discrete Contin. Dyn. Syst., 16 (2006), 689-703. doi: 10.3934/dcds.2006.16.689.  Google Scholar

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M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems. Theory and Application," Applied Mathematical Sciences, 163, Springer-Verlag London Ltd., London, 2008. Google Scholar

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Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Anal., 69 (2008), 3610-3628. doi: 10.1016/j.na.2007.09.045.  Google Scholar

[7]

E. Freire, E. Ponce and F. Torres, Hopf-like bifurcation in planar piecewise linear systems, Publ. Mat., 41 (1997), 135-148.  Google Scholar

[8]

E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of symmetrical continuous piecewise liinear systems with three zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 1675-1702. doi: 10.1142/S0218127402005509.  Google Scholar

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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.  Google Scholar

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F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632. doi: 10.1088/0951-7715/14/6/311.  Google Scholar

[11]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002.  Google Scholar

[12]

T. Küpper and S. Moritz, Generalized Hopf bifurcation for non-smooth planar systems. Non-smooth mechanics, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2483-2496.  Google Scholar

[13]

Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874.  Google Scholar

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J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106. doi: 10.1142/S0218127403006352.  Google Scholar

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X. Liu and M. Han, Hopf bifurcation for nonsmooth Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 2401-2415. doi: 10.1142/S0218127409024177.  Google Scholar

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J. Llibre and E. Ponce, Hopf bifurcation from infinity for planar control systems, Publicacions Matemàtiques, 41 (1997), 181-198.  Google Scholar

[17]

J. Llibre and E. Ponce, Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields, Nonlinear Anal., 36 (1999), 623-653. doi: 10.1016/S0362-546X(98)00175-8.  Google Scholar

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J. Llibre and E. Ponce, Piecewise linear feedback systems with arbitrary number of limit cycles, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 13 (2003), 895-904. doi: 10.1142/S0218127403007047.  Google Scholar

[19]

J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.  Google Scholar

[20]

D. J. W. Simpson and J. D. Meiss, Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows, Phys. Lett. A, 371 (2007), 213-220. doi: 10.1016/j.physleta.2007.06.046.  Google Scholar

[21]

A. Tonnelier, On the number of limit cycles in piecewise-linear Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1417-1422. doi: 10.1142/S0218127405012624.  Google Scholar

[22]

V. A. Gaiko and W. T. van Horssen, A piecewise linear dynamical system with two dropping sections, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 1367-1372. doi: 10.1142/S021812740902369X.  Google Scholar

[23]

Y. Zou and T. Küpper, Generalized Hopf bifurcation emanated from a corner for piecewise smooth planar systems, Nonlinear Anal., 62 (2005), 1-17. doi: 10.1016/j.na.2004.06.004.  Google Scholar

[24]

Y. Zou, T. Küpper and W.-J. Beyn, Generalized Hopf bifurcations for planar Filippov systems continuous at the origin, J. Nonlinear Sci., 16 (2006), 159-177. doi: 10.1007/s00332-005-0606-8.  Google Scholar

show all references

References:
[1]

A. A. Andronov, A. Vitt and S. Khaikin, "Theroy of Oscillators," Pergamon Press, Oxford-New York-Toronto, Ont., 1966.  Google Scholar

[2]

B. Coll, A. 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.  Google Scholar

[3]

V. Carmona, E. Freire, E. Ponce and F. Torres, On simplifying and classifying piecewise-linear systems, IEEE Trans. Circuits Systems I Fund. Theory Appl., 49 (2002), 609-620.  Google Scholar

[4]

V. Carmona, E. Freire, E. Ponce and F. Torres, The continuous matching of two stable linear systems can be unstable, Discrete Contin. Dyn. Syst., 16 (2006), 689-703. doi: 10.3934/dcds.2006.16.689.  Google Scholar

[5]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, "Piecewise-Smooth Dynamical Systems. Theory and Application," Applied Mathematical Sciences, 163, Springer-Verlag London Ltd., London, 2008. Google Scholar

[6]

Z. Du, Y. Li and W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlinear Anal., 69 (2008), 3610-3628. doi: 10.1016/j.na.2007.09.045.  Google Scholar

[7]

E. Freire, E. Ponce and F. Torres, Hopf-like bifurcation in planar piecewise linear systems, Publ. Mat., 41 (1997), 135-148.  Google Scholar

[8]

E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of symmetrical continuous piecewise liinear systems with three zones, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 1675-1702. doi: 10.1142/S0218127402005509.  Google Scholar

[9]

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.  Google Scholar

[10]

F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity, 14 (2001), 1611-1632. doi: 10.1088/0951-7715/14/6/311.  Google Scholar

[11]

M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equations, 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002.  Google Scholar

[12]

T. Küpper and S. Moritz, Generalized Hopf bifurcation for non-smooth planar systems. Non-smooth mechanics, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 2483-2496.  Google Scholar

[13]

Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874.  Google Scholar

[14]

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106. doi: 10.1142/S0218127403006352.  Google Scholar

[15]

X. Liu and M. Han, Hopf bifurcation for nonsmooth Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 2401-2415. doi: 10.1142/S0218127409024177.  Google Scholar

[16]

J. Llibre and E. Ponce, Hopf bifurcation from infinity for planar control systems, Publicacions Matemàtiques, 41 (1997), 181-198.  Google Scholar

[17]

J. Llibre and E. Ponce, Bifurcation of a periodic orbit from infinity in planar piecewise linear vector fields, Nonlinear Anal., 36 (1999), 623-653. doi: 10.1016/S0362-546X(98)00175-8.  Google Scholar

[18]

J. Llibre and E. Ponce, Piecewise linear feedback systems with arbitrary number of limit cycles, Int. J. Bifurc. Chaos Appl. Sci. Engrg., 13 (2003), 895-904. doi: 10.1142/S0218127403007047.  Google Scholar

[19]

J. Llibre, E. Ponce and F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity, 21 (2008), 2121-2142. doi: 10.1088/0951-7715/21/9/013.  Google Scholar

[20]

D. J. W. Simpson and J. D. Meiss, Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows, Phys. Lett. A, 371 (2007), 213-220. doi: 10.1016/j.physleta.2007.06.046.  Google Scholar

[21]

A. Tonnelier, On the number of limit cycles in piecewise-linear Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1417-1422. doi: 10.1142/S0218127405012624.  Google Scholar

[22]

V. A. Gaiko and W. T. van Horssen, A piecewise linear dynamical system with two dropping sections, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 1367-1372. doi: 10.1142/S021812740902369X.  Google Scholar

[23]

Y. Zou and T. Küpper, Generalized Hopf bifurcation emanated from a corner for piecewise smooth planar systems, Nonlinear Anal., 62 (2005), 1-17. doi: 10.1016/j.na.2004.06.004.  Google Scholar

[24]

Y. Zou, T. Küpper and W.-J. Beyn, Generalized Hopf bifurcations for planar Filippov systems continuous at the origin, J. Nonlinear Sci., 16 (2006), 159-177. doi: 10.1007/s00332-005-0606-8.  Google Scholar

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