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On the number of limit cycles in general planar piecewise linear systems
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 |
References:
[1] |
A. A. Andronov, A. Vitt and S. Khaikin, "Theroy of Oscillators," Pergamon Press, Oxford-New York-Toronto, Ont., 1966. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
E. Freire, E. Ponce and F. Torres, Hopf-like bifurcation in planar piecewise linear systems, Publ. Mat., 41 (1997), 135-148. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
J. Llibre and E. Ponce, Hopf bifurcation from infinity for planar control systems, Publicacions Matemàtiques, 41 (1997), 181-198. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
A. A. Andronov, A. Vitt and S. Khaikin, "Theroy of Oscillators," Pergamon Press, Oxford-New York-Toronto, Ont., 1966. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
E. Freire, E. Ponce and F. Torres, Hopf-like bifurcation in planar piecewise linear systems, Publ. Mat., 41 (1997), 135-148. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
J. Llibre and E. Ponce, Hopf bifurcation from infinity for planar control systems, Publicacions Matemàtiques, 41 (1997), 181-198. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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