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Introduction to the Qualitative Theory of Differential Systems

Planar, Symmetric and Continuous Piecewise Linear Systems

  • Textbook
  • © 2014

Overview

Authors:
  1. Jaume Llibre

    1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

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  2. Antonio E. Teruel

    1. Dept. de Matemàtiques i Informàtica, Universitat de les Illes Balears, Palma-Illes Balears, Spain

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  • Excellent for learning the use of basic results on qualitative theory of differential systems
  • Illustrates how to use the Poincaré map for studying the periodic orbits of a differential system
  • Shows the importance of compactification of the domain of definition of a differential system for the understanding of the global dynamics of the system
  • Points out the importance of bifurcation diagrams for describing the different dynamics of differential systems depending on parameters
  • Includes supplementary material: sn.pub/extras

Part of the book series: Birkhäuser Advanced Texts Basler Lehrbücher (BAT)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction and Statement of the Main Results

    • Jaume Llibre, Antonio E. Teruel
    Pages 1-18

  3. Basic Elements of the Qualitative Theory of Ordinary Differential Equations

    • Jaume Llibre, Antonio E. Teruel
    Pages 19-59

  4. Fundamental Systems

    • Jaume Llibre, Antonio E. Teruel
    Pages 61-118

  5. Return Maps

    • Jaume Llibre, Antonio E. Teruel
    Pages 119-187

  6. Phase Portraits

    • Jaume Llibre, Antonio E. Teruel
    Pages 189-281

  7. Back Matter

    Pages 283-289
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Keywords

About this book

The book deals with continuous piecewise linear differential systems in the plane with three pieces separated by a pair of parallel straight lines. Moreover, these differential systems are symmetric with respect to the origin of coordinates. This class of systems driven by concrete applications is of interest in engineering, in particular in control theory and the design of electric circuits. By studying these particular differential systems we will introduce the basic tools of the qualitative theory of ordinary differential equations, which allow us to describe the global dynamics of these systems including the infinity. The behavior of their solutions, their parametric stability or instability and their bifurcations are described. The book is very appropriate for a first course in the qualitative theory of differential equations or dynamical systems, mainly for engineers, mathematicians, and physicists.

Reviews

From the book reviews:

“This book is devoted to the study of piecewise linear differential systems (PWLSs), from the point of view of the qualitative theory of differential equations. … this book could serve well both as a textbook for a course in piecewise linear differential equations and as a reference in which important tools of current research are thoroughly explained and their use is illustrated.” (Claudio Pessoa, Mathematical Reviews, November, 2014)

“The monograph presents a detailed study of an important family of continuous piecewise linear differential systems. … The book is of great interest for researches working on planar piecewise linear differential systems. It can also serve as a reference book for those interested in the global phase portraits and the bifurcation sets of fundamental systems. The book is also highly suitable for students and engineers wishing to learn the theory of planar piecewise linear systems … .” (Valery Romanovski, zbMATH, Vol. 1286 (2), 2014)

Authors and Affiliations

  • Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    Jaume Llibre

  • Dept. de Matemàtiques i Informàtica, Universitat de les Illes Balears, Palma-Illes Balears, Spain

    Antonio E. Teruel

About the authors

Jaume Llibre is full professor at the Autonomous University of Barcelona (Spain) and a member of the Royal Academy of Sciences and Arts of Barcelona. He was a long-term visitor at different important universities and research institutes and is the author of many papers and some books. His main results deal with the periodic orbits either of different kinds of self-maps, or of vector fields, Hamiltonian systems and celestial mechanics. Antonio E. Teruel is associated professor at the University of the Balear Islands (Spain). He is the author of some papers about piecewise linear differential systems. His main results deal with piecewise linear models of differential behaviors.

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