Complex Nonlinearity

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This book explores the prediction and control of nonlinear and chaotic dynamics in high-dimensional complex systems, focusing on their geometro-topological changes. It begins with an introduction to nonlinear dynamics, attractors, and chaos, including modern chaos-control techniques. The second chapter addresses the edge of chaos through phase transitions—equilibrium, non-equilibrium, oscillatory, fractal, and noise-induced—as well as synergetics. While linear dynamics operate in flat, Euclidean geometry, nonlinear dynamics occur in curved, Riemannian geometry, utilizing tools from nonlinear tensor algebra and analysis. Extreme nonlinearity, or chaos, corresponds to changes in the topology of the curved geometric stage, known as the configuration manifold. The third chapter delves into geometry and topology change in relation to complex nonlinearity and chaos. Chapter four presents general nonlinear dynamics in the form of path integrals and action-amplitude formalism, starting with Feynman’s sum over histories and extending to sums over geometries and topologies. The final chapter integrates these concepts, presenting a unified framework of complex nonlinearity, encompassing chaos, phase transitions, geometrical dynamics, and topology change through path integrals. The book aims to equip serious readers with the tools for competitive research in modern complex nonlinearity, featuring a comprehensive bibliography and detailed