Chaos global

It is a recent discovery, presented in Chapter 6, that behind the apparent chaos of Table 4.2 there are some simple regularities which enable one to predict the r vs local and global behaviour on the basis of the open-circuit... 

WIGGINS, S. Global Bifurcations and Chaos. Springer-Verlag, New York, Berlin, Heidelberg, 1988... 

The reduction of chaos for 9 = 1.45 is presented in the intensity portraits of Fig. 39. However, as is seen in Fig. 38a, there is a small region (1.68 < 9 < 1.80) where the system behaves orderly in the classical case but the quantum correction leads to chaos. By way of an example for 9=1.75, the classical system, after quantum correction, loses its orderly features and the limit cycle settles into a chaotic trajectory. Generally, Lyapunov analysis shows that the transition from classical chaos to quantum order is very common. For example, this kind of transition appears for 9 = 3.5 where chaos is reduced to periodic motion on a limit cycle. Therefore a global reduction of chaos can be said to take place in the whole region of the parameter 0 < 9 < 7. 

Natsch, A., Keel, C., Pfirter, H. A., Haas, D. Defago, G. (1994). Contribution of the global regulator gene gacA to persistence and dissemination of Pseudomonas fluorescens biocontrol strain CHAO introduced into soil microcosms. Applied and Environmental Microbiology, 60, 2553-60. 

Li, B. and Siegel, R., Global analysis of a model pulsing drug delivery oscillator based on chemomechanical feedback with hysteresis, Chaos, Vol. 10, No. 3, 2000, pp. 682-690. 

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

With these properties, if locally separates phase space, as illustrated in the scheme (Figure 1). It is very important to note that even if if has codimension 1 and is locally a separatrix, it does mean in n DOFs that if neither has a simple geometry, because it is subject to stretching and folding because of chaos [24-26], nor separates globally (see Ref. 27). Let us now make a summary of the... 

S. Wiggins, Global Bifurcation and Chaos—Analytical Methods, Springer, New York, 1988. 

The answer to this question is that, although the constants Ck undoubtedly exist, their analytical properties may be so complicated that they do not impose any restrictions on the motion of the system. This is immediately clear since the process of finding the constants involves the inversion of a system of 2/ nonlinear coupled equations. Theorems in mathematics assure us of the local existence of 2/ explicit functions Cfc, but globally, i.e. for all values of p,q and t, they may only be defined with the help of infinitely many branches. Therefore, we can divide the constants of the motion into two classes, useful and useless . The useful constants of the motion possess a simple analytical structure, a finite number of branches, and are valid for all time t. Such constants actually restrict the motion of the system to a sub-manifold of phase space. Thus, the presence of a useful constant of the motion results in a simplification of the mechanical system at hand. The analytical properties of the useless constants are so unbehevably intricate and complex that they do not result in a reduction of the dimensionahty of phase space. Their presence is no obstacle for chaos. 

Before we conclude this section we have to ask an important question Is it possible to observe experimentally the onset of global chaos predicted to occur at K = Kc On the classical level the answer to this... 

In the concluding remark of Section 5.2 we asked the question whether the transition from confined chaos to global chaos K = Kc can be seen in an experiment with diatomic molecules. The technical feasibility of such an experiment is discussed in Section 5.4. Here we ask the more modest question whether, and if so, how, the transition to global chaos manifests itself within the framework of the quantum kicked rotor. Since the transition to global chaos is primarily a classical phenomenon, we expect that we have the best chance of seeing any manifestation of this transition in the quantum kicked rotor the more classical we prepare its initial state and control parameters. Thus, we choose a small value... 

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