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This book provides a self-contained presentation of the physical and mathematical laws governing complex systems. Complex systems arising in natural, engineering, environmental, life and social sciences are approached from a unifying point of view using an array of methodologies such as microscopic and macroscopic level formulations, deterministic and probabilistic tools, modeling and simulation. The book can be used as a textbook by graduate students, researchers and teachers in science, as well as non-experts who wish to have an overview of one of the most open, markedly interdisciplinary and fast-growing branches of present-day science. Table of Contents The Phenomenology of Complex Systems: Complexity, a New Paradigm Signatures of Complexity Onset of Complexity Four Case Studies Summing Up Deterministic View: Dynamical Systems, Phase Space, Stability Levels of Description Normal Forms The Limit of Universality Deterministic Chaos Emergence Coupling-Induced Complexity Modeling Complexity Beyond Physical Science Probabilistic Description: Need for a Probabilistic Approach Probability Distributions and Their Evolution Laws The Retrieval of Universality Complexity in the Probabilistic Description Emergence Revisited Transitions Between States Simulating Complex Systems Disorder-Generated Complexity Complexity, Entropy and Information: Information Entropy Dynamical Entropies Information Entropy Production Large Deviations, Fluctuation Theorems and the Probabilistic Properties of Time Sequences Algorithmic Complexity and Computation Dynamical Systems as Information Sources: Scaling Rules and Selection Further Information Measures Summing Up Prediction: Communicating with a Complex System Classical Approaches and Their Limitations Nonlinear Data Analysis The Monitoring of Complex Fields The Predictability Horizon Recurrence Extreme Events Selected Topics: The Arrow of Time Nanosystems Atmospheric Dynamics Climate Dynamics Networks Perspectives on Biological Complexity Equilibrium Versus Nonequilibrium in Complexity and Self-Organization Epistemological Insights from Complex Systems Outlook. The Future of Complexity