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Synchronization of chaotic systems, a patently nonlinear phenomenon, has emerged as a highly active interdisciplinary research topic at the interface of physics, biology, applied mathematics and engineering sciences. In this connection, time-delay systems described by delay differential equations have developed as particularly suitable tools for modeling specific dynamical systems. Indeed, time-delay is ubiquitous in many physical systems, for example due to finite switching speeds of amplifiers in electronic circuits, finite lengths of vehicles in traffic flows, finite signal propagation times in biological networks and circuits, and quite generally whenever memory effects are relevant. This monograph presents the basics of chaotic time-delay systems and their synchronization with an emphasis on the effects of time-delay feedback which give rise to new collective dynamics. Special attention is devoted to scalar chaotic/hyperchaotic time-delay systems, and some higher order models, occurring in different branches of science and technology as well as to the synchronization of their coupled versions. Last but not least, the presentation as a whole strives for a balance between the necessary mathematical description of the basics and the detailed presentation of real-world applications.

Nonlinear systems. --- Delay lines. --- Artificial delay lines --- Delay circuits --- Delay devices --- Time-delay networks --- Systems, Nonlinear --- Physics. --- System theory. --- Statistical physics. --- Applied mathematics. --- Engineering mathematics. --- Vibration. --- Dynamical systems. --- Dynamics. --- Electronic circuits. --- Nonlinear Dynamics. --- Vibration, Dynamical Systems, Control. --- Systems Theory, Control. --- Appl.Mathematics/Computational Methods of Engineering. --- Complex Networks. --- Circuits and Systems. --- Automatic control --- Automatic timers --- Electric waves --- Electronics --- Waves --- System theory --- Systems theory. --- Systems engineering. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Mathematical and Computational Engineering. --- Applications of Graph Theory and Complex Networks. --- Engineering systems --- System engineering --- Engineering --- Industrial engineering --- System analysis --- Engineering analysis --- Mathematical analysis --- Cycles --- Mechanics --- Sound --- Design and construction --- Mathematics --- Electron-tube circuits --- Electric circuits --- Electron tubes --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Systems, Theory of --- Systems science --- Science --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Physics --- Statics --- Mathematical statistics --- Philosophy --- Statistical methods