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This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the H�non map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward H�non mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincare map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincare mapping in addition to other analytical methods.

Forms, Quadratic. --- Differential equations, Linear. --- Bifurcation theory. --- Differentiable dynamical systems. --- Proof theory.

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This book covers the central role that bifurcations play in nonlinear phenomena, explaining mechanisms of how stability is gained or lost. It emphasizes practical and computational methods for analyzing dynamical systems. A wide range of phenomena between equilibrium and chaos is explained and illustrated by examples from science and engineering. The book is a practical guide for performing parameter studies and includes exercises. Combining an introduction on the textbook level with an exposition of computational methods, this book addresses the mathematical needs of scientists and engineers. It should be of interest to those in a wide variety of disciplines, including physics, mechanical engineering, electrical engineering, chemistry and chemical engineering, biology, and medicine. Both graduate students (in courses on dynamical systems, stability analysis, differential equations, and chaos) and professionals will be able to use the book equally well. The introduction avoids mathematical formalism, and the only required background is calculus. In the third edition there is a chapter on applications and extensions of standard ODE approaches, for example, to delay equations, to differential-algebraic equations, and to reaction-diffusion problems. Additional material is inserted, including the topics deterministic risk, pattern formation, and control of chaos, and many further references. Review of Earlier Edition: "The outcome is impressive. The book is beautifully written in a style that seeks not only to develop the subject matter but also to expose the thought processes behind the mathematics." Proceedings of the Edinburgh Mathematical Society

Ergodic theory. Information theory --- Numerical analysis --- Mathematics --- Mathematical physics --- Classical mechanics. Field theory --- Physics --- Applied physical engineering --- Engineering sciences. Technology --- Computer. Automation --- ICT (informatie- en communicatietechnieken) --- toegepaste wiskunde --- economie --- wiskunde --- ingenieurswetenschappen --- fysica --- numerieke analyse --- dynamica --- informatietheorie --- Bifurcation theory. --- Stability. --- Bifurkation --- Stabilität --- (Math.) --- EQUATIONS DIFFERENTIELLES ORDINAIRES --- SYSTEMES DIFFERENTIELS DYNAMIQUES --- BIFURCATIONS --- THEORIE QUALITATIVE --- STABILITE

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This book provides a modern investigation into the bifurcation phenomena of physical and engineering problems. Systematic methods - based on asymptotic, probabilistic, and group-theoretic standpoints - are used to examine experimental and computational data from numerous examples (soil, sand, kaolin, concrete, domes). For mathematicians, static bifurcation theory for finite-dimensional systems, as well as its implications for practical problems, is illuminated by the numerous examples. Engineers may find this book, with its minimized mathematical formalism, to be a useful introduction to modern bifurcation theory. This second edition strengthens the theoretical backgrounds of group representation theory and its application, uses of block-diagonalization in bifurcation analysis, and includes up-to-date topics of the bifurcation analysis of diverse materials from rectangular parallelepiped sand specimens to honeycomb cellular solids. Reviews of first edition: "The present book gives a wide and deep description of imperfect bifurcation behaviour in engineering problems. … the book offers a number of systematic methods based on contemporary mathematics. … On balance, the reviewed book is very useful as it develops a modern static imperfect bifurcation theory and fills the gap between mathematical theory and engineering practice." (Zentralblatt MATH, 2003) "The current book is a graduate-level text that presents an overview of imperfections and the prediction of the initial post-buckling response of a system. ... Imperfect Bifurcation in Structures and Materials provides an extensive range of material on the role of imperfections in stability theory. It would be suitable for a graduate-level course on the subject or as a reference to research workers in the field." ( Applied Mechanics Reviews, 2003) "This book is a comprehensive treatment of the static bifurcation problems found in (mainly civil/structural) engineering applications.... The text is well written and regularly interspersed with illustrative examples. The mathematical formalism is kept to a minimum and the 194 figures break up the text and make this a highly readable and informative book. ... In summary a comprehensive treatment of the subject which is very well put together and of interest to all researchers working in this area: recommended." (UK Nonlinear News, 2002).

Bifurcation theory. --- Differential equations, Nonlinear -- Numerical solutions. --- Electronic books. -- local. --- Engineering mathematics --- Bifurcation theory --- Structural analysis (Engineering) --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Mathematical Theory --- Mathematical models --- Differential equations, Nonlinear --- Numerical solutions. --- Engineering. --- Dynamics. --- Ergodic theory. --- System theory. --- Applied mathematics. --- Engineering mathematics. --- Structural mechanics. --- Control engineering. --- Control. --- Appl.Mathematics/Computational Methods of Engineering. --- Systems Theory, Control. --- Dynamical Systems and Ergodic Theory. --- Structural Mechanics. --- Control engineering --- Control equipment --- Control theory --- Engineering instruments --- Automation --- Programmable controllers --- Architectural engineering --- Engineering, Architectural --- Structural mechanics --- Structures, Theory of --- Structural engineering --- Engineering --- Engineering analysis --- Mathematical analysis --- Systems, Theory of --- Systems science --- Science --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Construction --- Industrial arts --- Technology --- Philosophy --- Numerical analysis --- Stability --- Numerical solutions --- Systems theory. --- Differentiable dynamical systems. --- Mechanics. --- Mechanics, Applied. --- Control and Systems Theory. --- Mathematical and Computational Engineering. --- Solid Mechanics. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Applied mechanics --- Engineering, Mechanical --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Mathematical models.