-
Previous Article
Eigenvalues for a nonlocal pseudo $p-$Laplacian
- DCDS Home
- This Issue
-
Next Article
Calderón-Zygmund estimate for homogenization of parabolic systems
Normal forms of planar switching systems
1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China, China |
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. |
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. |
[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. |
[1] |
Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2621-2634. doi: 10.3934/dcdsb.2021151 |
[2] |
Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 |
[3] |
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 |
[4] |
Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 |
[5] |
Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 |
[6] |
John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 |
[7] |
Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 |
[8] |
Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 |
[9] |
Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 |
[10] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[11] |
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 |
[12] |
Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 |
[13] |
Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129 |
[14] |
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 |
[15] |
Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6185-6205. doi: 10.3934/dcdsb.2021013 |
[16] |
Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 |
[17] |
Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 |
[18] |
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 |
[19] |
Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 |
[20] |
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 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]