Overview
- Introduces new framework for nonautonomous dynamical systems
- Develops theoretical foundations of impulsive functional differential equations, including linear and nonlinear systems, stability, and invariant manifold theory
- Spotlights recent advances in stability and bifurcation
- Contains detailed calculations to support application-driven approach
- Delivers material in self-contained, three-part structure
Part of the book series: IFSR International Series in Systems Science and Systems Engineering (IFSR, volume 34)
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Table of contents (20 chapters)
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Front Matter
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Impulsive Functional Differential Equations
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Front Matter
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Finite-Dimensional Ordinary Impulsive Differential Equations
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Front Matter
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Singular and Nonsmooth Phenomena
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Front Matter
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Applications
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Front Matter
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Keywords
About this book
This monograph presents the most recent progress in bifurcation theory of impulsive dynamical systems with time delays and other functional dependence. It covers not only smooth local bifurcations, but also some non-smooth bifurcation phenomena that are unique to impulsive dynamical systems. The monograph is split into four distinct parts, independently addressing both finite and infinite-dimensional dynamical systems before discussing their applications. The primary contributions are a rigorous nonautonomous dynamical systems framework and analysis of nonlinear systems, stability, and invariant manifold theory. Special attention is paid to the centre manifold and associated reduction principle, as these are essential to the local bifurcation theory. Specifying to periodic systems, the Floquet theory is extended to impulsive functional differential equations, and this permits an exploration of the impulsive analogues of saddle-node, transcritical, pitchfork and Hopf bifurcations.
Readers will learn how techniques of classical bifurcation theory extend to impulsive functional differential equations and, as a special case, impulsive differential equations without delays. They will learn about stability for fixed points, periodic orbits and complete bounded trajectories, and how the linearization of the dynamical system allows for a suitable definition of hyperbolicity. They will see how to complete a centre manifold reduction and analyze a bifurcation at a nonhyperbolic steady state.
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Authors and Affiliations
Bibliographic Information
Book Title: Bifurcation Theory of Impulsive Dynamical Systems
Authors: Kevin E.M. Church, Xinzhi Liu
Series Title: IFSR International Series in Systems Science and Systems Engineering
DOI: https://doi.org/10.1007/978-3-030-64533-5
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Hardcover ISBN: 978-3-030-64532-8Published: 25 March 2021
Softcover ISBN: 978-3-030-64535-9Published: 29 March 2022
eBook ISBN: 978-3-030-64533-5Published: 24 March 2021
Series ISSN: 1574-0463
Series E-ISSN: 2698-5497
Edition Number: 1
Number of Pages: XVII, 388
Number of Illustrations: 17 b/w illustrations, 12 illustrations in colour
Topics: Dynamical Systems and Ergodic Theory, Vibration, Dynamical Systems, Control, Analysis