Complex Nonlinearity: Chaos, Phase Transitions, Topology Change
and Path Integrals is a book about prediction & control of
general nonlinear and chaotic dynamics of high-dimensional complex
systems of various physical and non-physical nature and their
underpinning geometro-topological change.
The book starts with a textbook-like expose on nonlinear
dynamics, attractors and chaos, both temporal and spatio-temporal,
including modern techniques of chaos-control. Chapter 2 turns to
the edge of chaos, in the form of phase transitions (equilibrium
and non-equilibrium, oscillatory, fractal and noise-induced), as
well as the related field of synergetics. While the natural stage
for linear dynamics comprises of flat, Euclidean geometry (with the
corresponding calculation tools from linear algebra and analysis),
the natural stage for nonlinear dynamics is curved, Riemannian
geometry (with the corresponding tools from nonlinear, tensor
algebra and analysis). The extreme nonlinearity - chaos -
corresponds to the topology change of this curved geometrical
stage, usually called configuration manifold. Chapter 3 elaborates
on geometry and topology change in relation with complex
nonlinearity and chaos. Chapter 4 develops general nonlinear
dynamics, continuous and discrete, deterministic and stochastic, in
the unique form of path integrals and their action-amplitude
formalism. This most natural framework for representing both phase
transitions and topology change starts with Feynman's sum over
histories, to be quickly generalized into the sum over geometries
and topologies. The last Chapter puts all the previously developed
techniques together and presents the unified form of complex
nonlinearity. Here we have chaos, phase transitions, geometrical
dynamics and topology change, all working together in the form of
path integrals.
The objective of this book is to provide a serious reader with a
serious scientific tool that will enable them to actually perform a
competitive research in modern complex nonlinearity. It includes a
comprehensive bibliography on the subject and a detailed index.
Target readership includes all researchers and students of complex
nonlinear systems (in physics, mathematics, engineering, chemistry,
biology, psychology, sociology, economics, medicine, etc.), working
both in industry/clinics and academia.
General
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