Complex nonlinearity

More about the book

This book explores the prediction and control of nonlinear and chaotic dynamics in high-dimensional complex systems, addressing both physical and non-physical phenomena and their underlying geometro-topological changes. It begins with a foundational overview of nonlinear dynamics, attractors, and chaos, including contemporary chaos-control techniques. The second chapter examines the edge of chaos through various phase transitions—equilibrium, non-equilibrium, oscillatory, fractal, and noise-induced—while also touching on synergetics. Unlike linear dynamics, which operates in flat, Euclidean spaces, nonlinear dynamics is grounded in curved, Riemannian geometry, necessitating tools from nonlinear tensor algebra and analysis. The extreme nonlinearity of chaos is linked to changes in the topology of the configuration manifold. The third chapter delves into the relationship between geometry, topology change, and complex nonlinearity. The fourth chapter introduces general nonlinear dynamics through path integrals and their action-amplitude formalism, building on Feynman’s sum over histories and extending it to sums over geometries and topologies. The final chapter synthesizes these concepts, presenting a unified framework that integrates chaos, phase transitions, geometrical dynamics, and topology change via path integrals. A comprehensive bibliography and detailed index support researchers and students across various fields, includ