Abstract
The book [39] edited by D.V. Anosov and V.I. Arnold considers two fundamentally different dynamical systems: flows and cascades. Roughly speaking, flows are dynamical systems with continuous time and cascades are dynamical systems with discrete time. One of the most important theoretical problems is to consider Discontinuous Dynamical Systems (DDS). That is, the systems whose trajectories are piecewise continuous curves. Analyzing the behavior of the trajectories, we can conclude that DDS combine features of vector fields and maps. They cannot be reduced to flows or cascades but are close to flows since time is continuous. That is why we propose to call them also as Discontinuous Flows (DF). One must emphasize that DF are not differential equations with discontinuous right side, which often have been accepted as DDS [68]. One should also agree that nonautonomous impulsive differential equations, which were thoroughly described in previous chapters are not discontinuous flows.
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References
E. Akalin, M.U. Akhmet, The principles of B-smooth discontinuous flows, Comput. Math. Appl., 49 (2005) 981–995.
M.U. Akhmet, On the smoothness of solutions of impulsive autonomous systems, Nonlinear Anal.: TMA, 60 (2005a) 311–324.
D.V. Anosov, V.I. Arnold, Dynamical Systems, Springer, Berlin, 1994.
M. di Bernardo, A. Nordmark, G. Olivar, Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems, Phys. D, 237 (2008b) 119–136.
E.M. Bonotto, M. Federson, Limit sets and the Poincaré-Bendixson theorem in impulsive semidynamical systems, J. Diff. Eqs., 244 (2008) 2334–2349.
D. Chillingwirth, Differential topology with a view to applications, Pitman, London, 1978.
E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
C. Corduneanu, Principles of differential and integral equations, Chelsea Publishing Co., Bronx, NJ, 1977.
A.F. Filippov, Differential equations with discontinuous right-hand sides, Kluwer, Dordrecht, 1988.
V. Gullemin, A. Pollack, Differential topology, Prentice-Hall, New Jersey, 1974.
P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
S. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990) 120–128.
V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of impulsive differential equations, World Scientific, Singapore, NJ, London, Hong Kong, 1989.
A.D. Myshkis, A.M. Samoilenko, Systems with impulses at fixed moments of time (russian), Math. Sb., 74 (1967) 202–208.
A.B. Nordmark, Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14 (2001) 1517–1542.
T. Pavlidis, A new model for simple neural nets and its application in the design of a neural oscillator, Bull. Math. Biophys., 27 (1965) 215–229.
T. Pavlidis, Stability of a class of discontinuous dynamical systems, Inform. Contrl., 9 (1966) 298–322.
V.F. Rozhko, Lyapunov stability in discontinuous dynamic systems, (russian), Diff. Eqs., 11 (1975) 761–766.
V.F. Rozhko, On a class of almost periodic motions in systems with shocks (russian), Diff. Eqs., 11 (1972) 2012–2022.
S. Wiggins, Global Bifurcation and Chaos: Analytical Methods, Springer, New York, 1988.
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Akhmet, M. (2010). Discontinuous Dynamical Systems. In: Principles of Discontinuous Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6581-3_8
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DOI: https://doi.org/10.1007/978-1-4419-6581-3_8
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