Chaos definition

In 1814, J.J. Berzelius succeeded for the first time in systematically naming chemical substances by building on the results of quantitative analyses and on the definition of the term "element by Lavoisier. In the 19th century, the number of known chemical compounds increased so rapidly that it became essential to classify them, to avoid a complete chaos of trivial names (see Section 2.2.4). 

Cell growth and metabolic activities are similarly described as a simple chemical reaction. It is also necessary to establish a definite formula for dry cell matter. The elemental composition of certain strains of microorganism is defined by an empirical formula CHaO/3Ns. The general biochemical reaction for biomass production is based on consumption of organic substrate, as shown below. Substrate oxidation is simplified in the following biochemical oxidation ... 

In my opinion Popper s and Kuhn s hypotheses are complementary (in the sense of Niels Bohr) to one another. A combination of the two is not easy to understand but is definitely not a chaos in the sense of everything goes in developing a new... 

Despite his laboratory s outward calm, Carothers was poised on the brink of an almost superhuman outpouring of scientific achievement. Over the next three years, between 1929 and 1931, he would transform the chaos of organic polymer chemistry with a clarity of focus and definition. He would settle the argument between Staudinger and the rest of Europe s chemists. As a leading polymer scientist later commented, Carothers work was a volcanic eruption, the reverberations of which are still being felt. ... 

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. 

There are unfortunately various definitions of magnetic multipole moments in the literature - a recent paper by Raab16 discusses the various definitions and properties, and creates some order from an apparent chaos. 

I lay with my face on the Persian rug. Pain ran round my jaw and into my eyes, but it seemed a welcome and definite feeling compared to the chaos in my head, swirling... 

On that system were exact TS discovered [39], the importance of mass mismatch between atoms A,B,C underlined and chaos in reactive scattering described [3, 29,40-42]. It must be underlined that studies in atomic physics and celestial dynamics were decisive in a definition of a TS, with less obvious Hamiltonians, see the chapter by Jaffe et al. in this book. 

But how do we decide whether tp t) contains, or does not contain, chaos While classical chaos can be defined rigorously in mathematical terms (see Section 2.2 and Devaney (1992)), there is as yet no generally accepted definition of quantum chaos. The best we can do at this point is to develop and apply certain tests for quantum chaos which axe constructed with the intention of revealing comphcated behaviour of ip t) reminiscent of classical chaos. 

We are now ready for a definition of chaos. Summarizing a century of research in chaos, Devaney (1992) gives the following definition of chaos A system is chaotic if (Cl) periodic orbits are dense,... 

In order to allow for the largest possible class of chaotic systems, the degree of sensitivity is not specified in Devaney s definition of chaos. It turns out that many chaotic systems of practical importance are exponentially sensitive to initial conditions. In this case the sensitivity can be characterized quantitatively with the help of Lyapunov exponents. 

No definition of the term chaos is universally accepted yet, but almost everyone would agree on the three ingredients used in the following working definition ... 

We have seen that the logistic map can exhibit aperiodic orbits for certain parameter values, but how do we know that this is really chaos To be called chaotic, a system should also show sensitive dependence on initial conditions, in the sense that neighboring orbits separate exponentially fast, on average. In Section 9.3 we quantified sensitive dependence by defining the Liapunov exponent for a chaotic differental equation. Now we extend the definition to one-dimensional maps. 

We may wish, for example, to know the probability of finding a gas molecule at a definite spot in the box within which we suppose the gas to have been enclosed. If no external forces act on the molecules, we shall be unable to give any reason why a particle of gas should be at one place in the box rather than at another. Similarly, in this case there is no assignable reason why a particle of the gas should move in one direction rather than in another. We therefore introduce the following hypothesis, the principle of molecular chaos For the molecules of gas in a closed box, in the absence of external forces, all positions in the box and all directions of velocity are equally probable. 

For systems with a continuous spectrum, the concept of chaos is not defined in classical mechanics,66 since at least some trajectories will escape from the interaction region, as t -> oo. One could introduce a definition based on the notion of cantori, which act as barriers in phase space67 and which can serve68 to provide an analogue of a transition-state configuration for systems whose time evolution is constrained. Our own preferred interpretation is based on the discussion of Section III E. The fluctuations in rates are with respect to the zero-time limit of the dynamics. When v = 1, it is not necessary to propagate the system for a finite time in order to be able to predict the rates. The variation in rates are just the inherent fluctuations about the conventional, statistical specific features that... 

Subsection A contains a summary of the formal definitions of chaotic behavior, derived from ergodic theory detailed discussions of this topic may be found elsewhere.11 We comment, in this section, on the gap that must be bridged in order to apply these concepts to chemical dynamics. Subsection B discusses some recent developments in computational signatures of chaos. In Subsection C we review a number of studies that have provided some of these links and that, in some instances, have resulted in new useful computational methods for treating the dynamics of reactions displaying chaotic dynamics. In addition, it includes a subsection on connection between formal ergodic theory and statistical behavior in unimolecular decay. 

Different methods have been developed either for a rapid computation of the LCIs (Cincotta and Simo 2000) or for detecting the structure of the phase space (chaotic zones, weak chaos, regular resonant motion, invariant tori). Especially for this last purpose we quote the frequency map analysis (Laskar 1990, Laskar et al. 1992, Laskar 1993, Lega and Froeschle 1996), the sup-map method (Laskar 1994, Froeschle and Lega 1996), and more recently the fast Lyapunov indicator (hereafter FLI, Froeschle et al. 1997, Froeschle et al. 2000) and the Relative Lyapunov Indicator (Sandor et al. 2000). The definitions and comparisons between different methods including a preliminary version of the FLI have been discussed in Froeschle and Lega (1998, 1999). 

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