Chaos tools and concepts

In Chapter 1 we discussed some concepts of chaos, its manifestations and appUcations on an introductory level from a purely quaUtative point of view. The concepts were introduced ad hoc and in a pictorial manner. We will now turn to a more detailed investigation of chaos in order to prepare the tools and concepts needed for the discussion of chaotic atomic and molecular systems. 

It was Poincare who introduced a major revolution in the analysis of dynamical systems in classical mechanics. Instead of focussing on the time evolution of individual trajectories of a system, he emphasized the global point of view (Poincare (1892, 1993)). Poincare argued that it was much more important to know about the qualitative behaviour of the solutions of a given dynamical system in different parts of the classical phase space, than to know the detailed time evolution of a special solution. Poincare developed this point of view by analysing the age old question of whether the solar system is stable. Clearly the answer to this question is a quaUtative statement about a property of the solar 

Another of Poincare s new methods was the reduction of the continuous phase-space flow of a classical dynamical system to a discrete mapping. This is certainly one of the most useful techniques ever introduced into the theory of dynamical systems. Modem journals on nonlinear dynamics abound with graphical representations of Poincare mappings. A quick glance into any one of these journals will attest to this fact. Because of the usefulness and the formal simplicity of mappings, this topic is introduced and discussed in Section 2.2. 

The intricate structure of the set S of scattering singularities we encountered in connection with the reaction function in Section 1.1 can be characterized using the concept of fractals introduced by Mandelbrot in 1975 (see also Mandelbrot (1977, 1983)). Fractals are discussed in Section 2.3. 

A useful technique is the method of symbolic dynamics. We will encounter this topic in Section 2.4. Symbolic dynamics enables us to establish connections between dynamical systems. Often it is possible to map a given dynamical system onto an older one which has been studied before. If such a mapping is possible on the symbolic level, the two systems are dynamically equivalent. 

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The mathematical basis of chaos is the number continuum. The existence of deterministic randomness, e.g., a key feature of chaos, relies essentially on the properties of the number continuum. This is why we start our discussion of tools and concepts in chaos theory in the following section with a brief review of some elementary properties of the real numbers. 

The prototype potential surface invoked in chemical kinetics is a two-dimensional surface with a saddle equilibrium point and two exit channels at lower energies. The classical and quantal dynamics of such surfaces has been the object of many studies since the pioneering works by Wigner and Polanyi. Recent advances in nonlinear dynamical systems theory have provided powerful tools, such as the concepts of bifurcations and chaos, to investigate the classical dynamics from a new point of view and to perform the semiclassical... 

But chaos is more than a tool. There are as yet unsolved philosophical problems in its wake. While relativity and quantum mechanics necessitated - and in fact originated from - a careful analysis of the concepts of space, time and measurement, chaos, already on the classical level, forces us to re-think the concepts of determinism and predictability. Thus, classical mechanics could not be further removed from the dusty subject it is usually portrayed as. On the contrary it is at the forefront of modern scientific research. Since path integrals provide a link between classical and quantum mechanics, conceptual and philosophical problems with classical mechanics are bound to manifest themselves on the quantum level. We are only at the beginning of a thorough exploration of these questions. But one fact is established already chaos has a profound in-fiuence on the quantum mechanics of atoms and molecules. This book presents some of the most prominent examples. 

Chaos, Fractals, Selforganization and Disorder Concepts and Tools 2nd Edition By D. Sornette... 

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