Abstract
Discontinuous dynamical systems exist everywhere in the real world. One used to adopt continuous models for approximate descriptions of discontinuous dynamical systems. However, such continuous modeling cannot provide adequate predictions of discontinuous dynamical systems, and also makes the problems solving be more complicated and inaccurate. In the real world, the discontinuous modeling of dynamical systems is absolute, but the continuous modeling is relative. In other words, the continuous description of dynamical systems is an approximation of the discontinuous problems. To better describe the real world, one should realize that discontinuous models can provide adequate and real predications of engineering systems. For any discontinuous dynamical system, there are many continuous subsystems in different domains or different time intervals, and the dynamical properties of any continuous subsystems are different from that of the adjacent continuous subsystems. Thus, we have two types of discontinuous dynamical systems. (i) In two different adjacent time intervals, dynamical systems are different. When a dynamical system reaches the switching time, this system will be switched to another different dynamical system. With such switching, the discontinuous dynamical system is called the switching system. (ii) In phase space, there are many different domains. On any two adjacent domains, distinct dynamical systems are defined. Thus, once a flow of a sub-system arrives to its boundary, the switchability and/or transport laws on the boundary should be addressed because of such a difference between two adjacent subsystems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aizerman, M.A., Pyatnitskii, E.S., 1974, Foundation of a theory of discontinuous systems. 1, Automatic and Remote Control, 35, 1066–1079.
Aizerman, M.A., Pyatnitskii, E.S., 1974, Foundation of a theory of discontinuous systems. 2, Automatic and Remote Control, 35, 1241–1262.
Birkhoff, C.D., 1927, On the periodic motions of dynamical systems, Acta Mathematica, 50, 359–379.
den Hartog, J.P., 1930, Forced vibration with combined viscous and Coulomb damping, Phil, Magazine, VII (9), 801–817.
den Hartog, J.P., 1931, Forced vibrations with Coulomb and viscous damping, Transactions of the American Society of Mechanical Engineers, 53, 107–115.
DeCarlo, R.A., Zak, S.H. and Matthews, G.P., 1988, Variable structure control of nonlinear multivariable systems: A tutorial, Proceedings of the IEEE, 76, 212–232.
Filippov, A.F., 1964, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, Series 2, 42, 199–231.
Filippov, A.F., 1988, Differential Equations with Discontinuous Righthand Sides, Dordrecht: Kluwer Academic Publishers.
Leine, R.I., van Campen, D.H. and van de Vrande, B.L., 2000, Bifurcations in nonlinear discontinuous systems, Nonlinear Dynamics, 23, 105–164.
Levinson, N., 1949, A second order differential equation with singular solutions, Annals of Mathematics, 50, 127–153.
Levitan, E.S., 1960, Forced oscillation of a spring-mass system having combined coulomb and viscous damping, Journal of the Acoustical Society of America, 32, 1265–1269.
Lu, C., 2007, Existence of slip and stick periodic motions in a non-smooth dynamical system, Chaos, Solitons and Fractals, 35, 949–959.
Luo, A.C.J., 2005a, A theory for non-smooth dynamical systems on connectable domains, Communication in Nonlinear Science and Numerical Simulation, 10, 1–55.
Luo, A.C.J., 2005b, Imaginary, sink and source flows in the vicinity of the separatrix of nonsmooth dynamic system, Journal of Sound and Vibration, 285, 443–456.
Luo, A.C.J., 2006, Singularity and Dynamics on Discontinuous Vector Fields, Amsterdam: Elsevier.
Luo, A.C.J., 2008a, Global Transversality, Resonance and Chaotic Dynamics, Singapore: World Scientific.
Luo, A.C.J., 2008b, A theory for flow swtichability in discontinuous dynamical systems, Nonlinear Analysis: Hybrid Systems, 2, 1030–1061.
Luo, A.C.J. and Gegg, B.C., 2006a, On the mechanism of stick and non-stick periodic motion in a forced oscillator including dry-friction, ASME Journal of Vibration and Acoustics, 128, 97–105.
Luo, A.C.J. and Gegg, B.C., 2006b, Stick and non-stick periodic motions in a periodically forced, linear oscillator with dry friction, Journal of Sound and Vibration, 291, 132–168.
Luo, A.C.J. and Gegg, B.C., 2006c, Periodic motions in a periodically forced oscillator moving on an oscillating belt with dry friction, ASME Journal of Computational and Nonlinear Dynamics, 1, 212–220.
Luo, A.C.J. and Gegg, B.C., 2006d, Dynamics of a periodically excited oscillator with dry friction on a sinusoidally time-varying, traveling surface, International Journal of Bifurcation and Chaos, 16, 3539–3566.
Luo, A.C.J. and Thapa, S., 2008, Periodic motions in a simplified brake dynamical system with a periodic excitation, Communication in Nonlinear Science and Numerical Simulation, 14, 2389–2412.
Poincaré, H., 1892, Les Methods Nouvelles de la Mecanique Celeste, Vol. 1, Paris: Gauthier-Villars.
Utkin, V. I., 1976, Variable structure systems with sliding modes, IEEE Transactions on Automatic Control, AC-22, 212–222.
Utkin, V.I., 1978, Sliding Modes and Their Application in Variable Structure Systems, Moscow: Mir.
Utkin, V.I., 1981, Sliding Regimes in Optimization and Control Problem, Moscow: Nauka.
Zhusubaliyev, Z. and Mosekilde, E., 2003, Bifurcations and Chaos in Piecewise-smooth Dynamical Systems, Singapore: World Scientific.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Luo, A.C.J. (2012). Introduction. In: Discontinuous Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22461-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-22461-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22460-7
Online ISBN: 978-3-642-22461-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)