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

Normal forms of planar switching systems

Xingwu Chen 1, and  Weinian Zhang 1,

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.
Keywords: normal form, Switching system, monodromy, Hopf bifurcation..
Mathematics Subject Classification: Primary: 37G05, 37G10; Secondary: 34C0.
Citation: Xingwu Chen, Weinian Zhang. Normal forms of planar switching systems. Discrete & 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer, London, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. F. Filippov, Differential Equation with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[13]

I. Flügge-Lutz, Discontinuous Automatic Control, Princeton University Press, Princeton, 1953.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[17]

J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969.  Google Scholar

[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.  Google Scholar

[19]

M. Kunze, Non-Smooth Dynamical Systems, Springer, Berlin, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[20]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Ya. Z. Tsypkin, Relay Control Systems, Cambridge University Press, Cambridge, 1984.  Google Scholar

[27]

V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automatic Control, 22 (1977), 212-222.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer, London, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. F. Filippov, Differential Equation with Discontinuous Righthand Sides, Kluwer Academic, Amsterdam, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[13]

I. Flügge-Lutz, Discontinuous Automatic Control, Princeton University Press, Princeton, 1953.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. K. Hale, Ordinary Differential Equations, Wiley, New York, 1969.  Google Scholar

[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.  Google Scholar

[19]

M. Kunze, Non-Smooth Dynamical Systems, Springer, Berlin, 2000. doi: 10.1007/BFb0103843.  Google Scholar

[20]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Ya. Z. Tsypkin, Relay Control Systems, Cambridge University Press, Cambridge, 1984.  Google Scholar

[27]

V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automatic Control, 22 (1977), 212-222.  Google Scholar

[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.  Google Scholar

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