Deterministic chaotic properties

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

According to Stuart A. Kauffman (1991) there is no generally accepted definition for the term complexity . However, there is consensus on certain properties of complex systems. One of these is deterministic chaos, which we have already mentioned. An ordered, non-linear dynamic system can undergo conversion to a chaotic state when slight, hardly noticeable perturbations act on it. Even very small differences in the initial conditions of complex systems can lead to great differences in the development of the system. Thus, the theory of complex systems no longer uses the well-known cause and effect principle. 

In the absence of deterministic chaos in the time evolution of the wave functions of bounded systems, the focus of quantum chaos research shifted towards the identification of the fingerprints of classical chaos in the properties of -0- The usual procedure is to start with a classically chaotic system, quantize it canonically, and then try to identify those characteristics of V in the semiclassical limit (ft -) 0) that give away the chaoticity of the underlying classically chaotic system. 

That is, the system is completely deterministic there is no randomness at all. However, the evolution of the values of the variables is exquisitely sensitive to their exact initial values. That is, very slightly different initial values of the variables will soon evolve in very different ways. Because these initial conditions can only be specified with finite accuracy, the exact long-term behavior of the system is unpredictable, although its statistical properties can often be determined. Thus, chaotic systems are deterministic, yet unpredictable in the long run. 

Gaspard, R, Klages, R. Chaotic and fractal properties of deterministic diffusion-reaction processes. Chaos 8(2), 409—423 (1998). http //link.aip.Org/link/7CHA/8/409/l... 

Gaspard, P. Klages, R. Chaotic and Fractal Properties of Deterministic Diffusion-Reaction Processes. Cftaor 1998,8 (2), 409-423. 

TURBULENCE is chaotic fluid flow characterized by the appearance of three-dimensional, irregular swirls. These swirls are called eddies, and usually turbulence consists of many different sizes of eddies superimposed on each other. In the presence of turbulence, fluid masses with different properties are mixed rapidly. Atmospheric turbulence usually refers to the small-scale chaotic flow of air in the Earth s atmosphere. This type of turbulence results from vertical wind shear and convection and usually occurs in the atmospheric boundary layer and in clouds. On a horizontal scale of order 1000 km, the disturbances by synoptic weather systems are sometimes referred to as two-dimensional turbulence. Deterministic description of turbulence is difficult because of the chaotic character of turbulence and the large range of scales involved. Consequently, turbulence is treated in terms of statistical quantities. Insight in the physics of atmospheric turbulence is important, for instance, for the construction of buildings and structures, the mixing of air properties, and the dispersion of air pollution. Turbulence also plays an... 

Although there are many definitions of chaos (Gleick, 1987), for our purposes a chaotic system may be defined as one having three properties deterministic dynamics, aperiodicity, and sensitivity to initial conditions. Our first requirement implies that there exists a set of laws, in the case of homogeneous chemical reactions, rate laws, that is, first-order ordinary differential equations, that govern the time evolution of the system. It is not necessary that we be able to write down these laws, but they must be specifiable, at least in principle, and they must be complete, that is, the system cannot be subject to hidden and/or random influences. The requirement of aperiodicity means that the behavior of a chaotic system in time never repeats. A truly chaotic system neither reaches a stationary state nor behaves periodically in its phase space, it traverses an infinite path, never passing more than once through the same point. 

Lorenz, E. N. (1%3) Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130 Manneville, P. (1981) Statistical properties of chaotic solutions of a unidimensional model for phase turbulence. Phys. Lett. 84A, 129... 

The results summarized above are consistent with the general view developed at the beginning of this section on the robustness of the chaotic attractor and p icularly of the statistical properties of the dynamics toward fluctuations. This is rather remarkable, since the Willamowski-Rossler attractor is highly inhomogeneous and certainly not everywhere hyperbolic. It suggests that deterministic chaos in real-world systems where attractor inhomogeneity is ubiquitous retains fully its relevance in that it determines, up to small corrections, the most probable states of the system. 

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