American Institute of Mathematical Sciences

Advanced
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 - A, 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, (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, (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.  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, (2008).   Google Scholar

[5]

B. Brogliato, Nonsmooth Mechanics,, Models, (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), 219.   Google Scholar

[7]

X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems,, Comput. Math. Appl., 59 (2010), 3836.  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.  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, (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.  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.  doi: 10.1006/jmaa.2000.7188.  Google Scholar

[12]

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

[13]

I. Flügge-Lutz, Discontinuous Automatic Control,, Princeton University Press, (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.  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.  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.  doi: 10.1016/j.jde.2010.11.016.  Google Scholar

[17]

J. K. Hale, Ordinary Differential Equations,, Wiley, (1969).   Google Scholar

[18]

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

[19]

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

[20]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer, (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.  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.  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.   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.  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.  doi: 10.1109/9.317122.  Google Scholar

[26]

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

[27]

V. I. Utkin, Variable structure systems with sliding modes,, IEEE Trans. Automatic Control, 22 (1977), 212.   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.  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, (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, (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.  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, (2008).   Google Scholar

[5]

B. Brogliato, Nonsmooth Mechanics,, Models, (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), 219.   Google Scholar

[7]

X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems,, Comput. Math. Appl., 59 (2010), 3836.  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.  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, (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.  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.  doi: 10.1006/jmaa.2000.7188.  Google Scholar

[12]

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

[13]

I. Flügge-Lutz, Discontinuous Automatic Control,, Princeton University Press, (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.  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.  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.  doi: 10.1016/j.jde.2010.11.016.  Google Scholar

[17]

J. K. Hale, Ordinary Differential Equations,, Wiley, (1969).   Google Scholar

[18]

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

[19]

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

[20]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory,, Springer, (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.  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.  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.   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.  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.  doi: 10.1109/9.317122.  Google Scholar

[26]

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

[27]

V. I. Utkin, Variable structure systems with sliding modes,, IEEE Trans. Automatic Control, 22 (1977), 212.   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.  doi: 10.1007/s00332-005-0606-8.  Google Scholar

[1]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[2]

Larry M. Bates, Francesco Fassò. No monodromy in the champagne bottle, or singularities of a superintegrable system. Journal of Geometric Mechanics, 2016, 8 (4) : 375-389. doi: 10.3934/jgm.2016012
[3]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[4]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[5]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[6]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[7]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[8]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[9]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979
[10]

Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031
[11]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183
[12]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[13]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[14]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[15]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[16]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[17]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[18]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[19]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[20]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences