Bücher der Reihe Frontiers in Applied Dynamical Systems: Reviews and Tutorials

Bifurcation theory is a major topic in dynamical systems theory with profound applications. However, in contrast to autonomous dynamical systems, it is not clear what a bifurcation of a nonautonomous dynamical system actually is, and so far, various different approaches to describe qualitative changes have been suggested in the literature. The aim of this book is to provide a concise survey of the area and equip the reader with suitable tools to tackle nonautonomous problems. A review, discussion and comparison of several concepts of bifurcation is provided, and these are formulated in a unified notation and illustrated by means of comprehensible examples. Additionally, certain relevant tools needed in a corresponding analysis are presented.

This monograph uses braids to explore dynamics on surfaces, with an eye towards applications to mixing in fluids.  The text uses the particular example of taffy pulling devices to represent pseudo-Anosov maps in practice.  In addition, its final chapters also briefly discuss current applications in the emerging field of analyzing braids created from trajectory data.  While written with beginning graduate students, advanced undergraduates, or practicing applied mathematicians in mind, the book is also suitable for pure mathematicians seeking real-world examples.  Readers can benefit from some knowledge of homotopy and homology groups, but these concepts are briefly reviewed.  Some familiarity with Matlab is also helpful for the computational examples.

This book contains two review articles on mathematical physiology that deal with closely related topics but were written and can be read independently.The first article reviews the basic theory of calcium oscillations (common to almost all cell types), including spatio-temporal behaviors such as waves. The second article uses, and expands on, much of this basic theory to show how the interaction of cytosolic calcium oscillators with membrane ion channels can result in highly complex patterns of electrical spiking. Through these examples one can see clearly how multiple oscillatory processes interact within a cell, and how mathematical methods can be used to understand such interactions better. The two reviews provide excellent examples of how mathematics and physiology can learn from each other, and work jointly towards a better understanding of complex cellular processes.Review 1: Richard Bertram, Joel Tabak, Wondimu Teka, Theodore Vo, Martin Wechselberger: Geometric Singular Perturbation Analysis of Bursting Oscillations in Pituitary CellsReview 2: Vivien Kirk, James Sneyd: Nonlinear Dynamics of Calcium

The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations.

Mathematics for a nonsmooth world.- 1930-2010: Nonsmooth dynamics'' linear age.- Discontinuities to model missing knowledge.- Three experiments.- Layers and implementations.- Ideal and non-ideal sliding.- The three experiments revisited.- Further curiosities of hidden dynamics.- Closing remarks: open challenges. 

Mathematical models of such multiple-scale systems are considered singular perturbation problems, and this volume focuses on the geometric approach known as Geometric Singular Perturbation Theory (GSPT).It is the first of its kind that introduces the GSPT in a coordinate-independent manner.

This allows one to bridge successful ensemble Kalman filters with fully nonlinear particle filters, and allows a proper introduction of localization in particle filters, which has been lacking up to now.

This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations.