Singularity And Dynamics On Discontinuous Vector Fields

This book discussed fundamental problems in dynamics, which extensively exist in engineering, natural and social sciences. The book presented a basic theory for the interactions among many dynamical systems and for a system whose motions are constrained naturally or artificially. The methodology and techniques presented in this book are applicable to discontinuous dynamical systems in physics, engineering and control. In addition, they may provide useful tools to solve non-traditional dynamics in biology, stock market and internet network et al, which cannot be easily solved by the traditional Newton mechanics. The new ideas and concepts will stimulate ones’ thought and creativities in corresponding subjects. The author also used the simple, mathematical language to write this book. Therefore, this book is very readable, which can be either a textbook for senior undergraduate and graduate students or a reference book for researches in dynamics. Challenging continuous Newton's dynamics Original theory and seeds of new researches in the field Wide spectrum of applications in science and engineering Systematic presentation and clear illustrations

Singularities arise naturally in a huge number of different areas of mathematics and science. As a consequence, singularity theory lies at the crossroads of paths that connect many of the most important areas of applications of mathematics with some of its most abstract regions. The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis, geometry, physics, or other science (functions, varieties, mappings, vector or tensor fields, differential equations, models, etc.) on parameters. The articles collected here can be grouped under three headings. (A) Singularities of real maps; (B) Singular complex variables; and (C) Singularities of homomorphic maps.

A workshop on Singularities, Bifuraction and Dynamics was held at Warwick in July 1989, as part of a year-long symposium on Singularity Theory and its applications. The proceedings fall into two halves: Volume I mainly on connections with algebraic geometry and volume II on connections with dynamical systems theory, bifurcation theory and applications in the sciences. The papers are original research, stimulated by the symposium and workshop: All have been refereed and none will appear elsewhere. The main topic of volume II is new methods for the study of bifurcations in nonlinear dynamical systems, and applications of these.

"Discontinuous Dynamical Systems on Time-varying Domains" is the first monograph focusing on this topic. While in the classic theory of dynamical systems the focus is on dynamical systems on time-invariant domains, this book presents discontinuous dynamical systems on time-varying domains where the corresponding switchability of a flow to the time-varying boundary in discontinuous dynamical systems is discussed. From such a theory, principles of dynamical system interactions without any physical connections are presented. Several discontinuous systems on time-varying domains are analyzed in detail to show how to apply the theory to practical problems. The book can serve as a reference book for researchers, advanced undergraduate and graduate students in mathematics, physics and mechanics. Dr. Albert C. J. Luo is a professor at Southern Illinois University Edwardsville, USA. His research is involved in the nonlinear theory of dynamical systems. His main contributions are in the following aspects: a stochastic and resonant layer theory in nonlinear Hamiltonian systems, singularity on discontinuous dynamical systems, and approximate nonlinear theories for a deformable-body.

The dream of mathematical modeling is of systems evolving in a continuous, deterministic, predictable way. Unfortunately continuity is lost whenever the `rules of the game' change, whether a change of behavioural regime, or a change of physical properties. From biological mitosis to seizures. From rattling machine parts to earthquakes. From individual decisions to economic crashes. Where discontinuities occur, determinacy is inevitably lost. Typically the physical laws of such change are poorly understood, and too ill-defined for standard mathematics. Discontinuities offer a way to make the bounds of scientific knowledge a part of the model, to analyse a system with detail and rigour, yet still leave room for uncertainty. This is done without recourse to stochastic modeling, instead retaining determinacy as far as possible, and focussing on the geometry of the many outcomes that become possible when it breaks down. In this book the foundations of `piecewise-smooth dynamics' theory are rejuvenated, given new life through the lens of modern nonlinear dynamics and asymptotics. Numerous examples and exercises lead the reader through from basic to advanced analytical methods, particularly new tools for studying stability and bifurcations. The book is aimed at scientists and engineers from any background with a basic grounding in calculus and linear algebra. It seeks to provide an invaluable resource for modeling discontinuous systems, but also to empower the reader to develop their own novel models and discover as yet unknown phenomena.

This volume contains extended abstracts outlining selected talks and other selected presentations given by participants throughout the "Intensive Research Program on Advances in Nonsmooth Dynamics 2016", held at the Centre de Recerca Matemàtica (CRM) in Barcelona from February 1st to April 29th, 2016. They include brief research articles reporting new results, descriptions of preliminary work or open problems, and outlines of prominent discussion sessions. The articles are all the result of direct collaborations initiated during the research program. The topic is the theory and applications of Nonsmooth Dynamics. This includes systems involving elements of: impacting, switching, on/off control, hybrid discrete-continuous dynamics, jumps in physical properties, and many others. Applications include: electronics, climate modeling, life sciences, mechanics, ecology, and more. Numerous new results are reported concerning the dimensionality and robustness of nonsmooth models, shadowing variables, numbers of limit cycles, discontinuity-induced bifurcations and chaos, determinacy-breaking, stability criteria, and the classification of attractors and other singularities. This material offers a variety of new exciting problems to mathematicians, but also a diverse range of new tools and insights for scientists and engineers making use of mathematical modeling and analysis. The book is intended for established researchers, as well as for PhD and postdoctoral students who want to learn more about the latest advances in these highly active areas of research.