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Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Among other things, this provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos - the appearance of 'chaotic' motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichmüller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schrödinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

Lyapunov exponents --- Lyapunov stability --- Dynamics --- Lyapunov exponents. --- Lyapunov stability. --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Liapunov stability --- Ljapunov stability --- Control theory --- Stability --- Liapunov exponents --- Lyapunov characteristic exponents --- Differential equations

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Modern complex large-scale dynamical systems exist in virtually every aspect of science and engineering, and are associated with a wide variety of physical, technological, environmental, and social phenomena, including aerospace, power, communications, and network systems, to name just a few. This book develops a general stability analysis and control design framework for nonlinear large-scale interconnected dynamical systems, and presents the most complete treatment on vector Lyapunov function methods, vector dissipativity theory, and decentralized control architectures. Large-scale dynamical systems are strongly interconnected and consist of interacting subsystems exchanging matter, energy, or information with the environment. The sheer size, or dimensionality, of these systems necessitates decentralized analysis and control system synthesis methods for their analysis and design. Written in a theorem-proof format with examples to illustrate new concepts, this book addresses continuous-time, discrete-time, and hybrid large-scale systems. It develops finite-time stability and finite-time decentralized stabilization, thermodynamic modeling, maximum entropy control, and energy-based decentralized control. This book will interest applied mathematicians, dynamical systems theorists, control theorists, and engineers, and anyone seeking a fundamental and comprehensive understanding of large-scale interconnected dynamical systems and control.

Lyapunov stability --- Energy dissipation --- Dynamics --- Large scale systems --- Information Technology --- General and Others --- Lyapunov stability. --- Energy dissipation. --- Dynamics. --- Large scale systems. --- Systems, Large scale --- Dynamical systems --- Kinetics --- Liapunov stability --- Ljapunov stability --- Degradation, Energy --- Dissipation (Physics) --- Energy degradation --- Energy losses --- Losses, Energy --- Engineering systems --- System analysis --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Control theory --- Stability --- Clausius-type inequality. --- KalmanЙakubovichАopov conditions. --- KalmanЙakubovichАopov equations. --- KrasovskiiЌaSalle theorem. --- asymptotic stabilizability. --- combustion processes. --- comparison system. --- compartmental dynamical system theory. --- compartmental dynamical system. --- control Lyapunov function. --- control design. --- control signal. --- control vector Lyapunov function. --- convergence. --- coordination control. --- decentralized affine. --- decentralized control. --- decentralized controller. --- decentralized finite-time stabilizer. --- discrete-time dynamical system. --- dissipativity theory. --- dynamical system. --- ectropy. --- energy conservation. --- energy dissipation. --- energy equipartition. --- energy flow. --- entropy. --- feedback control law. --- feedback interconnection stability. --- feedback stabilizer. --- finite-time stability. --- finite-time stabilization. --- gain margin. --- hybrid closed-loop system. --- hybrid decentralized controller. --- hybrid dynamic controller. --- hybrid finite-time stabilizing controller. --- hybrid vector comparison system. --- hybrid vector dissipation inequality. --- impulsive differential equations. --- impulsive dynamical system. --- interconnected dynamical system. --- large-scale dynamical system. --- law of thermodynamics. --- linear energy exchange. --- maximum entropy control. --- multiagent interconnected system. --- multiagent systems. --- multivehicle coordinated motion control. --- nonconservation of ectropy. --- nonconservation of entropy. --- nonlinear dynamical system. --- optimality. --- plant energy. --- scalar Lyapunov function. --- sector margin. --- semistable dissipation matrix. --- stability analysis. --- stability theory. --- stability. --- state space. --- subsystem decomposition. --- subsystem energy. --- thermoacoustic instabilities. --- thermodynamic modeling. --- time-invariant set. --- time-varying set. --- vector Lyapunov function. --- vector available storage. --- vector comparison system. --- vector dissipation inequality. --- vector dissipative system. --- vector dissipativity theory. --- vector dissipativity. --- vector field. --- vector hybrid supply rate. --- vector lossless system. --- vector required supply. --- vector storage function. --- vector supply rate.