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Discontinuous Dynamical Systems

In: Principles of Discontinuous Dynamical Systems

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  • Marat Akhmet

    (Middle East Technical University, Department of Mathematics)

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.

Suggested Citation

  • Marat Akhmet, 2010. "Discontinuous Dynamical Systems," Springer Books, in: Principles of Discontinuous Dynamical Systems, chapter 0, pages 113-137, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-6581-3_8
    DOI: 10.1007/978-1-4419-6581-3_8
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