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For over 300 years, differential equations have served as an essential tool for describing and analyzing problems in many scientific disciplines. This carefully-written textbook provides an introduction to many of the important topics associated with ordinary differential equations. Unlike most textbooks on the subject, this text includes nonstandard topics such as a chapter on perturbation methods and a section in Chapter 3 that shows how to solve differential equations using Mathematica codes. In addition to the nonstandard topics, this text also contains contemporary material in the area as well as its classical topics. This second edition is updated to be compatible with Mathematica, version 7.0, and all Mathematica codes are in the book itself. This new edition also provides 81 additional exercises, a new section in Chapter 1 on the generalized logistic equation, an additional theorem in Chapter 2 concerning fundamental matrices, and many further enhancements to the first edition. This book can be used either for a second course in ordinary differential equations or as an introductory course for well-prepared students. The prerequisites for this book are three semesters of calculus and a course in linear algebra, although the needed concepts from linear algebra are introduced along with examples in the book. An undergraduate course in analysis is needed for the more theoretical subjects covered in the final two chapters.

Mathematics. --- Ordinary Differential Equations. --- Dynamical Systems and Ergodic Theory. --- Differentiable dynamical systems. --- Differential Equations. --- Mathématiques --- Dynamique différentiable --- Differential equations.

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We are living in an ever more complex world, an epoch where human actions can accordingly acquire far-reaching potentialities. Complex and adaptive dynamical systems are ubiquitous in the world surrounding us and require us to adapt to new realities and the way of dealing with them. This primer has been developed with the aim of conveying a wide range of "commons-sense" knowledge in the field of quantitative complex system science at an introductory level, providing an entry point to this both fascinating and vitally important subject. The approach is modular and phenomenology driven. Examples of emerging phenomena of generic importance treated in this book are: -- The small world phenomenon in social and scale-free networks. -- Phase transitions and self-organized criticality in adaptive systems. -- Life at the edge of chaos and coevolutionary avalanches resulting from the unfolding of all living. -- The concept of living dynamical systems and emotional diffusive control within cognitive system theory. Technical course prerequisites are a basic knowledge of ordinary and partial differential equations and of statistics. Each chapter comes with exercises and suggestions for further reading - solutions to the exercises are also provided.

Mathematics --- Statistical physics --- General biophysics --- Computer science --- Information systems --- ICT (informatie- en communicatietechnieken) --- biofysica --- toegepaste wiskunde --- informatica --- statistiek --- informatiesystemen --- fysica --- Adaptive control systems. --- Differentiable dynamical systems.

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This book offers an up-to-date overview of current research on Dynamics of Small Solar System Bodies and Exoplanets. In course-tested extensive chapters the authors cover topics of theoretical celestial mechanics, physics and dynamics of asteroids, comets, stability of exoplanets and numerical integration codes applied in dynamical astronomy.

Physics. --- Astrophysics and Astroparticles. --- Mathematical Methods in Physics. --- Extraterrestrial Physics, Space Sciences. --- Dynamical Systems and Ergodic Theory. --- Differentiable dynamical systems. --- Mathematical physics. --- Astrophysics. --- Physique --- Dynamique différentiable --- Physique mathématique --- Astrophysique --- Extrasolar planets --- Celestial mechanics --- Solar system

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In the analysis and synthesis of contemporary systems, engineers and scientists are frequently confronted with increasingly complex models that may simultaneously include components whose states evolve along continuous time and discrete instants; components whose descriptions may exhibit nonlinearities, time lags, transportation delays, hysteresis effects, and uncertainties in parameters; and components that cannot be described by various classical equations, as in the case of discrete-event systems, logic commands, and Petri nets. The qualitative analysis of such systems requires results for finite-dimensional and infinite-dimensional systems; continuous-time and discrete-time systems; continuous continuous-time and discontinuous continuous-time systems; and hybrid systems involving a mixture of continuous and discrete dynamics. Filling a gap in the literature, this textbook presents the first comprehensive stability analysis of all the major types of system models described above. Throughout the book, the applicability of the developed theory is demonstrated by means of many specific examples and applications to important classes of systems, including digital control systems, nonlinear regulator systems, pulse-width-modulated feedback control systems, artificial neural networks (with and without time delays), digital signal processing, a class of discrete-event systems (with applications to manufacturing and computer load balancing problems) and a multicore nuclear reactor model. The book covers the following four general topics: * Representation and modeling of dynamical systems of the types described above * Presentation of Lyapunov and Lagrange stability theory for dynamical systems defined on general metric spaces * Specialization of this stability theory to finite-dimensional dynamical systems * Specialization of this stability theory to infinite-dimensional dynamical systems Replete with exercises and requiring basic knowledge of linear algebra, analysis, and differential equations, the work may be used as a textbook for graduate courses in stability theory of dynamical systems. The book may also serve as a self-study reference for graduate students, researchers, and practitioners in applied mathematics, engineering, computer science, physics, chemistry, biology, and economics.

Differentiable dynamical systems --- Stability --- Differentiable dynamical systems. --- Stability. --- Dynamique différentiable --- Systèmes dynamiques --- Stabilité --- 517.9 --- Dynamics --- Mechanics --- Motion --- Vibration --- Benjamin-Feir instability --- Equilibrium --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- Dynamique différentiable. --- Systèmes dynamiques. --- Stabilité.

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This book addresses the fundamental question of how to construct mathematical models for the evolution of dynamical systems from experimentally-obtained time series. It places emphasis on chaotic signals and nonlinear modeling and discusses different approaches to the forecast of future system evolution. In particular, it teaches readers how to construct difference and differential model equations depending on the amount of a priori information that is available on the system in addition to the experimental data sets. This book will benefit graduate students and researchers from all natural sciences who seek a self-contained and thorough introduction to this subject.

Physics. --- Statistical Physics, Dynamical Systems and Complexity. --- Geophysics/Geodesy. --- Quantitative Finance. --- Game Theory/Mathematical Methods. --- Environmental Physics. --- Physical geography. --- Finance. --- Economics, Mathematical. --- Physique --- Géographie physique --- Finances --- Mathématiques économiques --- Chaotic behavior in systems --- -Synergetics --- 519.55 --- System theory --- Self-organizing systems --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- Mathematical models