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Attractors Under Discretisation

  • Book
  • © 2017

Overview

Authors:
  1. Xiaoying Han

    1. Department of Mathematics and Statistics, Auburn University, Auburn, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Peter Kloeden

    1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Combines a highly readable style, clear proofs, and many examples
  • Reviews essential earlier work on the discretisation of attractors and saddle points in autonomous systems
  • Introduces cutting-edge work on the discretisation of attractors in nonautonomous systems
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xi
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  2. Dynamical Systems and Numerical Schemes

    1. Front Matter

      Pages 1-1
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    2. Lyapunov Stability and Dynamical Systems

      • Xiaoying Han, Peter Kloeden
      Pages 3-10

    3. One Step Numerical Schemes

      • Xiaoying Han, Peter Kloeden
      Pages 11-31

  3. Steady States Under Discretisation

    1. Front Matter

      Pages 33-33
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    2. Linear Systems

      • Xiaoying Han, Peter Kloeden
      Pages 35-39

    3. Lyapunov Functions

      • Xiaoying Han, Peter Kloeden
      Pages 41-48

    4. Dissipative Systems with Steady States

      • Xiaoying Han, Peter Kloeden
      Pages 49-53

    5. Saddle Points Under Discretisation

      • Xiaoying Han, Peter Kloeden
      Pages 55-66

  4. Autonomous Attractors Under Discretisation

    1. Front Matter

      Pages 67-67
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    2. Dissipative Systems with Attractors

      • Xiaoying Han, Peter Kloeden
      Pages 69-75

    3. Lyapunov Functions for Attractors

      • Xiaoying Han, Peter Kloeden
      Pages 77-81

    4. Discretisation of an Attractor: General Case

      • Xiaoying Han, Peter Kloeden
      Pages 83-88

  5. Nonautonomous Limit Sets Under Discretisation

    1. Front Matter

      Pages 89-89
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    2. Dissipative Nonautonomous Systems

      • Xiaoying Han, Peter Kloeden
      Pages 91-97

    3. Discretisation of Nonautonomous Limit Sets

      • Xiaoying Han, Peter Kloeden
      Pages 99-104

    4. Variable Step Size Discretisation of Autonomous Attractors

      • Xiaoying Han, Peter Kloeden
      Pages 105-109

    5. Discretisation of a Uniform Pullback Attractor

      • Xiaoying Han, Peter Kloeden
      Pages 111-117

  6. Back Matter

    Pages 119-122
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Keywords

About this book

This work focuses on the preservation of attractors and saddle points of ordinary differential equations under discretisation. In the 1980s, key results for autonomous ordinary differential equations were obtained – by Beyn for saddle points and by Kloeden & Lorenz for attractors. One-step numerical schemes with a constant step size were considered, so the resulting discrete time dynamical system was also autonomous. One of the aims of this book is to present new findings on the discretisation of dissipative nonautonomous dynamical systems that have been obtained in recent years, and in particular to examine the properties of nonautonomous omega limit sets and their approximations by numerical schemes – results that are also of importance for autonomous systems approximated by a numerical scheme with variable time steps, thus by a discrete time nonautonomous dynamical system.

Reviews

“The book is based on lecture notes while the material is based on papers, partially by the authors. … But the lecture style is quite accessible and the book has a logical structure with theorems and proofs, and clear examples. The proofs are polished and the text reads easily. A target audience would be advanced students and researchers in dynamical systems with a more theoretical focus, in which the lecturers could provide any missing details.” (Hil G. E. Meijer, Mathematical reviews, September, 2018)​

Authors and Affiliations

  • Department of Mathematics and Statistics, Auburn University, Auburn, USA

    Xiaoying Han

  • School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, China

    Peter Kloeden

About the authors

​Xiaoying Han obtained her PhD from the University at Buffalo, USA in 2007 and is currently a professor at Auburn University, USA.  Her main research interests are in random and nonautonomous dynamical systems and their applications.  In addition to mathematical analysis of dynamical systems, she is also interested in the modelling and simulation of applied dynamical systems in biology, chemical engineering, ecology, material sciences, etc.  Professor Han is a co-author of the books “Applied Nonautonomous and Random Dynamical Systems” (with T. Caraballo) published in the SpringerBrief series and “Random Ordinary Differential Equations and Their Numerical Solutions” (with P. E. Kloeden) published by Springer. 

Peter E. Kloeden completed his PhD and DSc at the University of Queensland, Australia in 1975 and 1995.  He is currently a Professor of Mathematics at Huazhong University of Science & Technology in China, and an affiliated professor at Auburn University, USA. He has wide interests in the applications of mathematical analysis, numerical analysis, stochastic analysis and dynamical systems.  Professor Kloeden is a co-author of several influential books on nonautonomous dynamical systems, metric spaces of fuzzy sets, and in particular “Numerical Solutions of Stochastic Differential equations” (with E. Platen) published by Springer in 1992.  He is a Fellow of the Society of Industrial and Applied Mathematics and was awarded the W.T. & Idalia Reid Prize in 2006.  His current interests focus on nonautonomous and random dynamical systems and their applications in the biological sciences.

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