DISSIPATIVE LATTICE DYNAMICAL SYSTEMS

There is an extensive literature in the form of papers (but no books) on lattice dynamical systems. The book focuses on dissipative lattice dynamical systems and their attractors of various forms such as autonomous, nonautonomous and random. The existence of such attractors is established by showing that the corresponding dynamical system has an appropriate kind of absorbing set and is asymptotically compact in some way.

There is now a very large literature on lattice dynamical systems, especially on attractors of all kinds in such systems. We cannot hope to do justice to all of them here. Instead, we have focused on key areas of representative types of lattice systems and various types of attractors. Our selection is biased by our own interests, in particular to those dealing with biological applications. One of the important results is the approximation of Heaviside switching functions in LDS by sigmoidal functions.

Nevertheless, we believe that this book will provide the reader with a solid introduction to the field, its main results and the methods that are used to obtain them.

Contents:

• Background:
  • Lattice Dynamical Systems: A Preview
  • Dynamical Systems
• Laplacian LDS:
  • Lattice Laplacian Models
  • Approximation of Attractors of LDS
  • Non-Autonomous Laplacian Lattice Systems in Weighted Sequence Spaces
• A Selection of Lattice Models:
  • Lattice Dynamical Systems with Delays
  • Set-Valued Lattice Models
  • Second Order Lattice Dynamical Systems
  • Discrete Time Lattice Systems
  • Three Topics in Brief
• Stochastic and Random LDS:
  • Random Dynamical Systems
  • Stochastic LDS with Additive Noise
  • Stochastic LDS with Multiplicative Noise
  • Stochastic Lattice Models with Fractional Brownian Motions
• Hopfield Lattice Models:
  • Hopfield Neural Network Lattice Model
  • The Hopfield Lattice Model in Weighted Spaces
  • A Random Hopfield Lattice Model
• LDS in Biology:
  • FitzHugh-Nagumo Lattice Model
  • The Amari Lattice Neural Field Model
  • Stochastic Neural Field Models with Nonlinear Noise
  • Lattice Systems with Switching Effects and Delayed Recovery

Readership: Researchers ranging from doctoral students to professors in mathematics and related areas. Can also be used in lecture for a graduate course.

Key Features:

• Readable introduction leading to actually research results
• Only book specifically on the topic
• An area of considerable current research activity
• Experienced authors