By Vladimir G. Ivancevic
Complex Nonlinearity: Chaos, section Transitions, Topology switch and course Integrals is a e-book approximately prediction & keep watch over of basic nonlinear and chaotic dynamics of high-dimensional advanced platforms of varied actual and non-physical nature and their underpinning geometro-topological switch.
The publication begins with a textbook-like reveal on nonlinear dynamics, attractors and chaos, either temporal and spatio-temporal, together with glossy options of chaos–control. bankruptcy 2 turns to the sting of chaos, within the type of section transitions (equilibrium and non-equilibrium, oscillatory, fractal and noise-induced), in addition to the similar box of synergetics. whereas the typical degree for linear dynamics includes of flat, Euclidean geometry (with the corresponding calculation instruments from linear algebra and analysis), the normal level for nonlinear dynamics is curved, Riemannian geometry (with the corresponding instruments from nonlinear, tensor algebra and analysis). the intense nonlinearity – chaos – corresponds to the topology switch of this curved geometrical level, often known as configuration manifold. bankruptcy three elaborates on geometry and topology switch in relation with advanced nonlinearity and chaos. bankruptcy four develops common nonlinear dynamics, non-stop and discrete, deterministic and stochastic, within the designated type of direction integrals and their action-amplitude formalism. This so much usual framework for representing either part transitions and topology swap begins with Feynman’s sum over histories, to be speedy generalized into the sum over geometries and topologies. The final bankruptcy places the entire formerly built recommendations jointly and provides the unified kind of advanced nonlinearity. right here we've got chaos, part transitions, geometrical dynamics and topology swap, all operating jointly within the kind of direction integrals.
The aim of this e-book is to supply a significant reader with a major medical device that would let them to truly practice a aggressive study in smooth advanced nonlinearity. It features a entire bibliography at the topic and a close index. goal readership contains all researchers and scholars of complicated nonlinear structures (in physics, arithmetic, engineering, chemistry, biology, psychology, sociology, economics, drugs, etc.), operating either in industry/clinics and academia.
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Additional info for Complex Nonlinearity - Chaos, Phase Transition, Topology Change and Path Integrals
Soon afterwards George Birkhoﬀ published a much stronger version of the theorem (see [CAM05]). Recall that Henri Poincar´e (April 29, 1854–July 17, 1912), was one of France’s greatest mathematicians and theoretical physicists, and a philosopher of science. Poincar´e is often described as the last ‘universalist’ (after Gauss), capable of understanding and contributing in virtually all parts of mathematics. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.
It prompted many experiments and some theoretical development by B. Van der Pol , G. Duﬃng, and D. Hayashi . They found other systems in which the nonlinear oscillator played a role and classiﬁed the possible motions 10 Recall from [II06b] that in diﬀerential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying diﬀerentiable functions on that manifold. According to the basic insights of Marston Morse, a diﬀerentiable function on a manifold will, in a typical case, reﬂect the topology quite directly.
Krylov , who understood that a physical billiard was a dynamical system on a surface of negative curvature, but with the curvature concentrated along the lines of collision. Sinai, who was the ﬁrst to show that a physical billiard can be ergodic, knew Krylov’s work well. On the other hand, the work of Lord Rayleigh also received vigorous development. It prompted many experiments and some theoretical development by B. Van der Pol , G. Duﬃng, and D. Hayashi . They found other systems in which the nonlinear oscillator played a role and classiﬁed the possible motions 10 Recall from [II06b] that in diﬀerential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying diﬀerentiable functions on that manifold.