Nonlinear Dynamics in Complex Systems

Concise and accessible, this introduction to the mathematical tools of non-linear dynamics and complex systems theory is aimed at researchers in engineering and the life sciences who are unfamiliar with the formulae, but recognize their analytical potency.

With many areas of science reaching across their boundaries and becoming more and more interdisciplinary, students and researchers in these fields are confronted with techniques and tools not covered by their particular education. Especially in the life- and neurosciences quantitative models based on nonlinear dynamics and complex systems are becoming as frequently implemented as traditional statistical analysis. Unfamiliarity with the terminology and rigorous mathematics may discourage many scientists to adopt these methods for their own work, even though such reluctance in most cases is not justified.

This book bridges this gap by introducing the procedures and methods used for analyzing nonlinear dynamical systems. In Part I, the concepts of fixed points, phase space, stability and transitions, among others, are discussed in great detail and implemented on the basis of example elementary systems. Part II is devoted to specific, non-trivial applications: coordination of human limb movement (Haken-Kelso-Bunz model), self-organization and pattern formation in complex systems (Synergetics), and models of dynamical properties of neurons (Hodgkin-Huxley, Fitzhugh-Nagumo and Hindmarsh-Rose). Part III may serve as a refresher and companion of some mathematical basics that have been forgotten or were not covered in basic math courses. Finally, the appendix contains an explicit derivation and basic numerical methods together with some programming examples as well as solutions to the exercises provided at the end of certain chapters. Throughout this book all derivations are as detailed and explicit as possible, and everybody with some knowledge of calculus should be able to extract meaningful guidance follow and apply the methods of nonlinear dynamics to their own work.

This book is a masterful treatment, one might even say a gift, to the interdisciplinary scientist of the future.

With the authoritative voice of a genuine practitioner,Fuchs is a master teacher of how to handle complex dynamical systems.

What I find beautiful in this book is its clarity, the clear definition of terms, every step explained simply and systematically.

(J.A.Scott Kelso, excerpts from the foreword)

Introduces mathematical tools of complexity and shows how these tools work in important applications useful for engineers, physicists, neuroscientists, psychologists, biologists, etc Well structured presentation of the mathematical background of nonlinear systems theory to those who are not using formula and equations on a daily basis from various fields This book not only presents the problem and solution but also provides each step along the way together with the basic rules that have been applied.

Inhalt

Part I Nonlinear Dynamical Systems.- Introduction.- One-dimensional Systems.- Two-Dimensional Systems.- Higher-Dimensional Systems and Chaos.- Discrete Maps and Iterations in Space.- Stochastic Systems.- Part II: Model Systems.- Haken-Kelso-Bunz (HKB) Model.- Self-organization and Synergetics.- Neuronal Models.- Part III: Mathematical Basics.- Mathematical Basics.- The Coupled HKB System.- Numerical Procedures and Computer Simulations.- Solutions.