# American Institute of Mathematical Sciences

December  2016, 36(12): 6715-6736. doi: 10.3934/dcds.2016092

## Normal forms of planar switching systems

 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China, China

Received  November 2014 Revised  August 2016 Published  October 2016

In this paper we study normal forms of planar differential systems with a non-degenerate equilibrium on a single switching line, i.e., the equilibrium is a non-degenerate equilibrium of both the upper system and the lower one. In the sense of $C^0$ conjugation we find all normal forms for linear switching systems and use them together with switching near-identity transformations to normalize second order terms, showing the reduction of normal forms. We prove that only one of those 19 types of linear normal form decides if the system is monodromic. With the monodromic linear normal form, we compute the second order monodromic normal form, which gives a condition under which exactly one limit cycle arises from a Hopf bifurcation.
Citation: Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6715-6736. doi: 10.3934/dcds.2016092
##### References:
 [1] D. V. Anosov and V. I. Arnold, Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems, Encyclopedia of Mathematical Science, Vol. 1, Springer, Berlin, 1988. doi: 10.1007/978-3-642-61551-1. [2] S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 2001. doi: 10.1109/9780470545393. [3] M. di Bernardo, C. J. Budd and A. R. Champneys, Grazing, skipping and sliding: Analysis of the nonsmooth dynamics of the dc/dc buck converter, Nonlinearity, 11 (1998), 859-890. doi: 10.1088/0951-7715/11/4/007. [4] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer, London, 2008. [5] B. Brogliato, Nonsmooth Mechanics, Models, dynamics and control. Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016. doi: 10.1007/978-3-319-28664-8. [6] C. J. Budd, Non-smooth dynamical systems and the grazing bifurcation, in Nonlinear Mathematics and Its Applications (Guildford, 1995), Cambridge University Press, Cambridge, 1996, 219-235. [7] X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848. doi: 10.1016/j.camwa.2010.04.019. [8] X. Chen and W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differential Equa., 252 (2012), 2877-2899. doi: 10.1016/j.jde.2011.10.013. [9] S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639. [10] B. Coll, A. Gasull and R. Prohens, Center-focus and isochronous center problems for discontinuous differential equations, Discrete Contin. Dyn. Syst., 6 (2000), 609-624. doi: 10.3934/dcds.2000.6.609. [11] B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690. doi: 10.1006/jmaa.2000.7188. [12] A. F. Filippov, Differential Equation with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9. [13] I. Flügge-Lutz, Discontinuous Automatic Control, Princeton University Press, Princeton, 1953. [14] 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. [15] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765. doi: 10.1142/S0218127403007618. [16] M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equa., 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016. [17] J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969. [18] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equa., 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002. [19] M. Kunze, Non-Smooth Dynamical Systems, Springer, Berlin, 2000. doi: 10.1007/BFb0103843. [20] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998. [21] Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurc. Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874. [22] F. Mañosas and P. J. Torres, Isochronicity of a class of piecewise continuous oscillators, Proc. Amer. Math. Soc., 133 (2005), 3027-3035. doi: 10.1090/S0002-9939-05-07873-1. [23] I. I. Pleskan and K. S. Sibirskii, On the problem of the center for systems with discontinuous right-hand sides, (Russian) Differencial'nye Uravnenija, 9 (1973), 1817-1825. [24] 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. [25] D. Shevitz and B. Paden, Lyapunov stability theory of nonsmooth systems, IEEE Trans. Automatic Control, 39 (1994), 1910-1914. doi: 10.1109/9.317122. [26] Ya. Z. Tsypkin, Relay Control Systems, Cambridge University Press, Cambridge, 1984. [27] V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automatic Control, 22 (1977), 212-222. [28] Y. Zou, T. Küpper and W.-J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlin. Sci., 16 (2006), 159-177. doi: 10.1007/s00332-005-0606-8.

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##### References:
 [1] D. V. Anosov and V. I. Arnold, Dynamical Systems I: Ordinary Differential Equations and Smooth Dynamical Systems, Encyclopedia of Mathematical Science, Vol. 1, Springer, Berlin, 1988. doi: 10.1007/978-3-642-61551-1. [2] S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York, 2001. doi: 10.1109/9780470545393. [3] M. di Bernardo, C. J. Budd and A. R. Champneys, Grazing, skipping and sliding: Analysis of the nonsmooth dynamics of the dc/dc buck converter, Nonlinearity, 11 (1998), 859-890. doi: 10.1088/0951-7715/11/4/007. [4] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer, London, 2008. [5] B. Brogliato, Nonsmooth Mechanics, Models, dynamics and control. Third edition. Communications and Control Engineering Series. Springer, [Cham], 2016. doi: 10.1007/978-3-319-28664-8. [6] C. J. Budd, Non-smooth dynamical systems and the grazing bifurcation, in Nonlinear Mathematics and Its Applications (Guildford, 1995), Cambridge University Press, Cambridge, 1996, 219-235. [7] X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl., 59 (2010), 3836-3848. doi: 10.1016/j.camwa.2010.04.019. [8] X. Chen and W. Zhang, Isochronicity of centers in a switching Bautin system, J. Differential Equa., 252 (2012), 2877-2899. doi: 10.1016/j.jde.2011.10.013. [9] S.-N. Chow, C. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639. [10] B. Coll, A. Gasull and R. Prohens, Center-focus and isochronous center problems for discontinuous differential equations, Discrete Contin. Dyn. Syst., 6 (2000), 609-624. doi: 10.3934/dcds.2000.6.609. [11] B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl., 253 (2001), 671-690. doi: 10.1006/jmaa.2000.7188. [12] A. F. Filippov, Differential Equation with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9. [13] I. Flügge-Lutz, Discontinuous Automatic Control, Princeton University Press, Princeton, 1953. [14] 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. [15] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equations, Int. J. Bifurc. Chaos, 13 (2003), 1755-1765. doi: 10.1142/S0218127403007618. [16] M. Guardia, T. M. Seara and M. A. Teixeira, Generic bifurcations of low codimension of planar Filippov systems, J. Differential Equa., 250 (2011), 1967-2023. doi: 10.1016/j.jde.2010.11.016. [17] J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969. [18] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differential Equa., 248 (2010), 2399-2416. doi: 10.1016/j.jde.2009.10.002. [19] M. Kunze, Non-Smooth Dynamical Systems, Springer, Berlin, 2000. doi: 10.1007/BFb0103843. [20] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998. [21] Yu. A. Kuznetsov, S. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J. Bifurc. Chaos, 13 (2003), 2157-2188. doi: 10.1142/S0218127403007874. [22] F. Mañosas and P. J. Torres, Isochronicity of a class of piecewise continuous oscillators, Proc. Amer. Math. Soc., 133 (2005), 3027-3035. doi: 10.1090/S0002-9939-05-07873-1. [23] I. I. Pleskan and K. S. Sibirskii, On the problem of the center for systems with discontinuous right-hand sides, (Russian) Differencial'nye Uravnenija, 9 (1973), 1817-1825. [24] 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. [25] D. Shevitz and B. Paden, Lyapunov stability theory of nonsmooth systems, IEEE Trans. Automatic Control, 39 (1994), 1910-1914. doi: 10.1109/9.317122. [26] Ya. Z. Tsypkin, Relay Control Systems, Cambridge University Press, Cambridge, 1984. [27] V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automatic Control, 22 (1977), 212-222. [28] Y. Zou, T. Küpper and W.-J. Beyn, Generalized Hopf bifurcation for planar Filippov systems continuous at the origin, J. Nonlin. Sci., 16 (2006), 159-177. doi: 10.1007/s00332-005-0606-8.
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