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Author: David John Warwick Simpson
Publisher: World Scientific
ISBN: 9814293849
Category : Science
Languages : en
Pages : 255
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Book Description
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: David John Warwick Simpson
Publisher: World Scientific
ISBN: 9814293849
Category : Science
Languages : en
Pages : 255
Get Book Here
Book Description
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
ISBN: 1846287081
Category : Mathematics
Languages : en
Pages : 497
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Book Description
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: Viktor Avrutin
Publisher: World Scientific
ISBN: 9811204713
Category : Mathematics
Languages : en
Pages : 649
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Book Description
The investigation of dynamics of piecewise-smooth maps is both intriguing from the mathematical point of view and important for applications in various fields, ranging from mechanical and electrical engineering up to financial markets. In this book, we review the attracting and repelling invariant sets of continuous and discontinuous one-dimensional piecewise-smooth maps. We describe the bifurcations occurring in these maps (border collision and degenerate bifurcations, as well as homoclinic bifurcations and the related transformations of chaotic attractors) and survey the basic scenarios and structures involving these bifurcations. In particular, the bifurcation structures in the skew tent map and its application as a border collision normal form are discussed. We describe the period adding and incrementing bifurcation structures in the domain of regular dynamics of a discontinuous piecewise-linear map, and the related bandcount adding and incrementing structures in the domain of robust chaos. Also, we explain how these structures originate from particular codimension-two bifurcation points which act as organizing centers. In addition, we present the map replacement technique which provides a powerful tool for the description of bifurcation structures in piecewise-linear and other form of invariant maps to a much further extent than the other approaches.
Author: David John Warwick Simpson
Publisher: World Scientific
ISBN: 9814293857
Category : Mathematics
Languages : en
Pages : 255
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Book Description
1. Fundamentals of piecewise-smooth, continuous systems. 1.1. Applications. 1.2. A framework for local behavior. 1.3. Existence of equilibria and fixed points. 1.4. The observer canonical form. 1.5. Discontinuous bifurcations. 1.6. Border-collision bifurcations. 1.7. Poincaré maps and discontinuity maps. 1.8. Period adding. 1.9. Smooth approximations -- 2. Discontinuous bifurcations in planar systems. 2.1. Periodic orbits. 2.2. The focus-focus case in detail. 2.3. Summary and classification -- 3. Codimension-two, discontinuous bifurcations. 3.1. A nonsmooth, saddle-node bifurcation. 3.2. A nonsmooth, Hopf bifurcation. 3.3. A codimension-two, discontinuous Hopf bifurcation -- 4. The growth of Saccharomyces cerevisiae. 4.1. Mathematical model. 4.2. Basic mathematical observations. 4.3. Bifurcation structure. 4.4. Simple and complicated stable oscillations -- 5. Codimension-two, border-collision bifurcations. 5.1. A nonsmooth, saddle-node bifurcation. 5.2. A nonsmooth, period-doubling bifurcation -- 6. Periodic solutions and resonance tongues. 6.1. Symbolic dynamics. 6.2. Describing and locating periodic solutions. 6.3. Resonance tongue boundaries. 6.4. Rotational symbol sequences. 6.5. Cardinality of symbol sequences. 6.6. Shrinking points. 6.7. Unfolding shrinking points -- 7. Neimark-Sacker-like bifurcations. 7.1. A two-dimensional map. 7.2. Basic dynamics. 7.3. Limiting parameter values. 7.4. Resonance tongues. 7.5. Complex phenomena relating to resonance tongues. 7.6. More complex phenomena
Author: Yuri Kuznetsov
Publisher: Springer Science & Business Media
ISBN: 1475739788
Category : Mathematics
Languages : en
Pages : 648
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Book Description
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: Zhanybai T Zhusubaliyev
Publisher: World Scientific
ISBN: 9814485632
Category : Science
Languages : en
Pages : 377
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Book Description
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: Zhanybai T. Zhusubaliyev
Publisher: World Scientific
ISBN: 9812384200
Category : Mathematics
Languages : en
Pages : 377
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Book Description
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: Stephen Wiggins
Publisher: Springer Science & Business Media
ISBN: 1461210429
Category : Mathematics
Languages : en
Pages : 505
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Book Description
Global Bifurcations and Chaos: Analytical Methods is unique in the literature of chaos in that it not only defines the concept of chaos in deterministic systems, but it describes the mechanisms which give rise to chaos (i.e., homoclinic and heteroclinic motions) and derives explicit techniques whereby these mechanisms can be detected in specific systems. These techniques can be viewed as generalizations of Melnikov's method to multi-degree of freedom systems subject to slowly varying parameters and quasiperiodic excitations. A unique feature of the book is that each theorem is illustrated with drawings that enable the reader to build visual pictures of global dynamcis of the systems being described. This approach leads to an enhanced intuitive understanding of the theory.
Author: Grigorios Dimitriadis
Publisher: John Wiley & Sons
ISBN: 1118756460
Category : Technology & Engineering
Languages : en
Pages : 944
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Book Description
Introduction to Nonlinear Aeroelasticity Introduces the latest developments and technologies in the area of nonlinear aeroelasticity Nonlinear aeroelasticity has become an increasingly popular research area in recent years. There have been many driving forces behind this development, increasingly flexible structures, nonlinear control laws, materials with nonlinear characteristics and so on. Introduction to Nonlinear Aeroelasticity covers the theoretical basics in nonlinear aeroelasticity and applies the theory to practical problems. As nonlinear aeroelasticity is a combined topic, necessitating expertise from different areas, the book introduces methodologies from a variety of disciplines such as nonlinear dynamics, bifurcation analysis, unsteady aerodynamics, non-smooth systems and others. The emphasis throughout is on the practical application of the theories and methods, so as to enable the reader to apply their newly acquired knowledge Key features: Covers the major topics in nonlinear aeroelasticity, from the galloping of cables to supersonic panel flutter Discusses nonlinear dynamics, bifurcation analysis, numerical continuation, unsteady aerodynamics and non-smooth systems Considers the practical application of the theories and methods Covers nonlinear dynamics, bifurcation analysis and numerical methods Accompanied by a website hosting Matlab code Introduction to Nonlinear Aeroelasticity is a comprehensive reference for researchers and workers in industry and is also a useful introduction to the subject for graduate and undergraduate students across engineering disciplines.
Author: Robert A. Meyers
Publisher: Springer Science & Business Media
ISBN: 1461418054
Category : Mathematics
Languages : en
Pages : 1885
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Book Description
Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
