Author : Piotr Sebastian Kowalczyk
Publisher :
Page : 0 pages
File Size : 22,70 MB
Release : 2003
Category :
ISBN :
Author : Piotr Kowalczyk
Publisher :
Page : pages
File Size : 19,17 MB
Release : 2003
Category :
ISBN :
Author : Zhanybai T. Zhusubaliyev
Publisher : World Scientific
Page : 377 pages
File Size : 25,60 MB
Release : 2003
Category : Mathematics
ISBN : 9812384200
Technical problems often lead to differential equations with piecewise-smooth right-hand sides. Problems in mechanical engineering, for instance, violate the requirements of smoothness if they involve collisions, finite clearances, or stick-slip phenomena. Systems of this type can display a large variety of complicated bifurcation scenarios that still lack a detailed description.This book presents some of the fascinating new phenomena that one can observe in piecewise-smooth dynamical systems. The practical significance of these phenomena is demonstrated through a series of well-documented and realistic applications to switching power converters, relay systems, and different types of pulse-width modulated control systems. Other examples are derived from mechanical engineering, digital electronics, and economic business-cycle theory.The topics considered in the book include abrupt transitions associated with modified period-doubling, saddle-node and Hopf bifurcations, the interplay between classical bifurcations and border-collision bifurcations, truncated bifurcation scenarios, period-tripling and -quadrupling bifurcations, multiple-choice bifurcations, new types of direct transitions to chaos, and torus destruction in nonsmooth systems.In spite of its orientation towards engineering problems, the book addresses theoretical and numerical problems in sufficient detail to be of interest to nonlinear scientists in general.
Author : David John Warwick Simpson
Publisher : World Scientific
Page : 255 pages
File Size : 12,19 MB
Release : 2010
Category : Science
ISBN : 9814293849
Real-world systems that involve some non-smooth change are often well-modeled by piecewise-smooth systems. However there still remain many gaps in the mathematical theory of such systems. This doctoral thesis presents new results regarding bifurcations of piecewise-smooth, continuous, autonomous systems of ordinary differential equations and maps. Various codimension-two, discontinuity induced bifurcations are unfolded in a rigorous manner. Several of these unfoldings are applied to a mathematical model of the growth of Saccharomyces cerevisiae (a common yeast). The nature of resonance near border-collision bifurcations is described; in particular, the curious geometry of resonance tongues in piecewise-smooth continuous maps is explained in detail. NeimarkSacker-like border-collision bifurcations are both numerically and theoretically investigated. A comprehensive background section is conveniently provided for those with little or no experience in piecewise-smooth systems.
Author : Mario Bernardo
Publisher : Springer Science & Business Media
Page : 497 pages
File Size : 35,75 MB
Release : 2008-01-01
Category : Mathematics
ISBN : 1846287081
This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction expounds the ubiquity of such models via numerous. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level. Almost no mathematical background is assumed other than basic calculus and algebra.
Author : Quentin Brandon
Publisher :
Page : pages
File Size : 27,3 MB
Release : 2009
Category :
ISBN :
In the field of dynamical system analysis, piecewise-smooth models have grown in popularity due to there greater flexibility and accuracy in representing some hybrid systems in applications such as electronics or mechanics. Hybrid dynamical systems have two sets of variables, one which evolve in a continuous space, and the other in a discrete one. Most analytical methods require the orbit to be smooth during objective intervals, so that some special treatments are inevitable to study the existence and stability of solutions in hybrid dynamical systems. Based on a piecewise-smooth model, where the orbit of the system is broken down into locally smooth pieces, and a hybrid bifurcation analysis method, using a Poincare map with sections ruled by the switching conditions of the system, we review the analysis process in details. Then we apply it to various extensions of the Alpazur oscillator, originally a nonsmooth 2-dimension switching oscillator. The original Alpazur oscillator, as a simple nonlinear switching system, was a perfect candidate to prove the efficiency of the approach. Each of its extensions shows a new scenario and how it can be handled, in order to illustrate the generality of the model. Finally, and in order to show more of the implementation we used for our own computer-based analysis tool, some of the most relevant numerical methods we used are introduced. It is noteworthy that the emphasis has been put on autonomous systems because the treatment of non-autonomous ones only requires a simplification (no time variation). This study brings a strong and general framework for the bifurcation analysis of nonlinear hybrid dynamical systems, illustrated by some results. Among them, some interesting local and global properties of the Alpazur Oscillator are revealed, such as the presence of a cascade of cusps in the bifurcation diagram. Our work resulted in the implementation of an analysis tool, implemented in C++, using the numerical methods that we chose for this particular purpose, such as the numerical approximation of the second derivative elements in the Jacobian matrix.
Author : Michal Fečkan
Publisher : Springer Science & Business Media
Page : 387 pages
File Size : 28,53 MB
Release : 2011-05-30
Category : Science
ISBN : 3642182690
"Bifurcation and Chaos in Discontinuous and Continuous Systems" provides rigorous mathematical functional-analytical tools for handling chaotic bifurcations along with precise and complete proofs together with concrete applications presented by many stimulating and illustrating examples. A broad variety of nonlinear problems are studied involving difference equations, ordinary and partial differential equations, differential equations with impulses, piecewise smooth differential equations, differential and difference inclusions, and differential equations on infinite lattices as well. This book is intended for mathematicians, physicists, theoretically inclined engineers and postgraduate students either studying oscillations of nonlinear mechanical systems or investigating vibrations of strings and beams, and electrical circuits by applying the modern theory of bifurcation methods in dynamical systems. Dr. Michal Fečkan is a Professor at the Department of Mathematical Analysis and Numerical Mathematics on the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, Slovakia. He is working on nonlinear functional analysis, bifurcation theory and dynamical systems with applications to mechanics and vibrations.
Author : Yuri Kuznetsov
Publisher : Springer Science & Business Media
Page : 610 pages
File Size : 15,44 MB
Release : 1998-09-18
Category : Mathematics
ISBN : 0387983821
Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
Author : Albert C. J. Luo
Publisher : Springer Science & Business Media
Page : 234 pages
File Size : 46,3 MB
Release : 2009-11-06
Category : Science
ISBN : 3642002536
"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.
Author : Yuri Kuznetsov
Publisher : Springer Science & Business Media
Page : 648 pages
File Size : 45,97 MB
Release : 2013-03-09
Category : Mathematics
ISBN : 1475739788
Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
