Global Bifurcations And Chaos

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.

Although the application of differential equations to economics is a vast and vibrant area, the subject has not been systematically studied; it is often treated as a subsidiary part of mathematical economics textbooks. This book aims to fill that void by providing a unique blend of the theory of differential equations and their exciting applications to dynamic economics. Containing not just a comprehensive introduction to the applications of the theory of linear (and linearized) differential equations to economic analysis, the book also studies nonlinear dynamical systems, which have only been widely applied to economic analysis in recent years. It provides comprehensive coverage of the most important concepts and theorems in the theory of differential equations in a way that can be understood by any reader who has a basic knowledge of calculus and linear algebra. In addition to traditional applications of the theory to economic dynamics, the book includes many recent developments in different fields of economics.

During the last twenty years, a large number of books on nonlinear chaotic dynamics in deterministic dynamical systems have appeared. These academic tomes are intended for graduate students and require a deep knowledge of comprehensive, advanced mathematics. There is a need for a book that is accessible to general readers, a book that makes it possible to get a good deal of knowledge about complex chaotic phenomena in nonlinear oscillators without deep mathematical study.Chaos, Bifurcations and Fractals Around Us: A Brief Introduction fills that gap. It is a very short monograph that, owing to geometric interpretation complete with computer color graphics, makes it easy to understand even very complex advanced concepts of chaotic dynamics. This invaluable publication is also addressed to lecturers in engineering departments who want to include selected nonlinear problems in full time courses on general mechanics, vibrations or physics so as to encourage their students to conduct further study.

During the last twenty years, a large number of books on nonlinear chaotic dynamics in deterministic dynamical systems have appeared. These academic tomes are intended for graduate students and require a deep knowledge of comprehensive, advanced mathematics. There is a need for a book that is accessible to general readers, a book that makes it possible to get a good deal of knowledge about complex chaotic phenomena in nonlinear oscillators without deep mathematical study.Chaos, Bifurcations and Fractals Around Us: A Brief Introduction fills that gap. It is a very short monograph that, owing to geometric interpretation complete with computer color graphics, makes it easy to understand even very complex advanced concepts of chaotic dynamics. This invaluable publication is also addressed to lecturers in engineering departments who want to include selected nonlinear problems in full time courses on general mechanics, vibrations or physics so as to encourage their students to conduct further study.

This book presents the theoretical frame for studying lumped nonsmooth dynamical systems: the mathematical methods are recalled, and adapted numerical methods are introduced (differential inclusions, maximal monotone operators, Filippov theory, Aizerman theory, etc.). Tools available for the analysis of classical smooth nonlinear dynamics (stability analysis, the Melnikov method, bifurcation scenarios, numerical integrators, solvers, etc.) are extended to the nonsmooth frame. Many models and applications arising from mechanical engineering, electrical circuits, material behavior and civil engineering are investigated to illustrate theoretical and computational developments.

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.

Attractors, Bifurcations, & Chaos - now in its second edition - begins with an introduction to mathematical methods in modern nonlinear dynamics and deals with differential equations. Phenomena such as bifurcations and deterministic chaos are given considerable emphasis, both in the methodological part, and in the second part, containing various applications in economics and in regional science. Coexistence of attractors and the multiplicity of development paths in nonlinear systems are central topics. The applications focus on issues such as business cycles, oligopoly, interregional trade dynamics, and economic development theory.

The local and global bifurcation behavior of various structural and mechanical systems have been examined in detail. The analysis is divided into three main parts. In the first part, global bifurcation analysis is performed for externally excited nonlinear systems with initial imperfections and a semisimple linear operator. Explicit restrictions are obtained on parameters where such systems can exhibit chaotic dynamics. The results are demonstrated on a shallow arch system, which is subjected to a spatio-temporal loading, under various resonance conditions. In the second part, local dynamics is investigated for a parametrically excited nonlinear system with a non-semisimple linear operator. The global dynamics associated with such systems is also examined by imposing the reversible symmetry on the original system. The results from the general analysis, which are applicable to various physical applications, are used to study the flexural-torsional motion of a rectangular beam. In the third part, a systematic formulation of nonlinear oscillations of a spinning disc is obtained. These questions of motion include the effects due to inherent bending rigidity, membrane stresses arising from centrifugal forces, non-axisymmetry of the in-plane and transverse displacements, geometric nonlinearities, aerodynamic damping, parametric excitation due to time varying spin rate, etc. For the constant rotation case, the linearized equations of motion are solved by taking both membrane as well as flexural stiffness effects into account. This leads to a power series solution for the radial shape functions and harmonic solutions for the circumferential shape functions. The two-dimensional eigen-functions thus obtained can describe a disc mode with any number of nodal diameters and nodal circles, and the resulting eigen-frequencies match well with the numerical results. The nonlinear and non-axisymmetric in-plane response is also determined. A two-degree-of-freedom system of nonlinear ordinary differential equations, which governs the dynamic evolution of the amplitudes of traveling waves associated with the dominant mode of the transverse motion, is obtained. The local bifurcations are examined in the resulting equations of motion, both in the presence and the absence of imperfections. The existence of chaotic behavior is also proven in the spinning disc system by showing the existence of single and multi-pulse Silnikov type orbits in the presence of dissipation effects. Throughout this research, the relationship between the mathematical results and their physical implications have been interpreted for the engineering applications considered.

An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help readers develop an intuitive feel for the properties involved.