The chaotic regime

The data shown in Fig. 6.9 and Fig. 6.10 confirm our suspicion that for weak microwave fields no chaos mechanisms have to be invoked for an adequate physical understanding of microwave ionization data. The situation, however, is quite different in the case of strong microwave fields. In this case the ionization routes are very comphcated, and the multiphoton pictmre loses its attractiveness. It has to be replaced by a picture based on chaos. Chaos provides a simpler description of the ionization process and consequently a better physical insight. The discussion of the chaotic strong-field regime is the topic of the following section. 

With Fig. 6.5 we estabhshed that the phase space of the classical version of the SSE system contains chaotic regions. But since the SSE system is a manifestly quantum mechanical system, the central question is whether the classical chaos in the SSE system is at all relevant for the quantum dynamics, and, if yes, what are the signatures  

The SSE phase-space portrait shown in Fig. 6.5 reminds us of the phase-space portraits of the kicked rotor presented in Chapter 5. In Fig. 6.5 we can identify resonances and sealing invariant curves. In Chapter 5 we saw that resonance overlap in the standard mapping defines a sudden percolation transition when for K Kc the seahng invariant 

In the case of the kicked rotor we were able to predict the critical perturbation strength by applying the Chirikov overlap criterion. This criterion does two things for us. First, it provides us with an excellent physical picture which explains qualitatively the mechanism of the chaos transition secondly, it provides us with an analytical estimate for the critical field. 

The central idea of the overlap criterion is to focus on the most important resonances in the phase space and to calculate their widths as a function of the strength of the apphed field. In a classical picture the critical field strength is reached when the resonances overlap. 

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Effective computation, such as that required by life processes and the maintenance of evolvability and adaptability in complex systems, requires both the storage and transmission of information. If correlations between separated sites (or agents) of a system are too small - as they are in the ordered regime shown in figure 11.3 -the sites evolve essentially independently of one another and little or no transmission takes place. On the other hand, if the correlations are too strong - as they are in the chaotic regime - distant sites may cooperate so strongly so as to effectively mimic each other s behavior, or worse yet, whatever ordered behavior is present may be... 

Heat conductivity has been studied by placing the end particles in contact with two thermal reservoirs at different temperatures (see (Casati et al, 2005) for details)and then integrating the equations of motion. Numerical results (Casati et al, 2005) demonstrated that, in the small uj regime, the heat conductivity is system size dependent, while at large uj, when the system becomes almost fully chaotic, the heat conductivity becomes independent of the system size (if the size is large enough). This means that Fourier law is obeyed in the chaotic regime. 

The title chaos is an unfortunate misnomer. As shown above, there is considerable structure displayed in the onset and interruption of chaotic behaviour. Even within the chaotic regime the value of x evolves according to completely defined rules—one value explicitly determines the next, with 100 per cent certainty. There is no randomness, no element of chance uncertainty or irregularity. If we know the rules and can measure a given starting condition exactly, even a chaotic pattern can be predicted exactly. 

Irregular behaviour of concentrations and the correlation functions observed in the chaotic regime differ greatly from those predicted by law of mass action (Section 2.1.1). Following Nicolis and Prigogine [2], the stochastic Lotka-Volterra model discussed in this Section, could be considered as an example of generalized turbulence. 

Taylor, H.S. and Zakrzewski, J. (1988). Dynamic interpretation of atomic and molecular spectra in the chaotic regime, Phys. Rev. A 38, 3732-3748. 

Figure 11.17 The four snapshots show the evolution and breakup of a spiral wave pattern in 2-dimensional simulated cardiac tissue (300 x 300 cells). The chaotic regime shown in the final snapshot corresponds to fibrillation. Reprinted from [587] with permission from Lippincott, Williams and Wilkins.

Undoubtedly, the most promising modehng of the cardiac dynamics is associated with the study of the spatial evolution of the cardiac electrical activity. The cardiac tissue is considered to be an excitable medium whose the electrical activity is described both in time and space by reaction-diffusion partial differential equations [519]. This kind of system is able to produce spiral waves, which are the precursors of chaotic behavior. This consideration explains the transition from normal heart rate to tachycardia, which corresponds to the appearance of spiral waves, and the fohowing transition to fibrillation, which corresponds to the chaotic regime after the breaking up of the spiral waves, Figure 11.17. The transition from the spiral waves to chaos is often characterized as electrical turbulence due to its resemblance to the equivalent hydrodynamic phenomenon. 

A note of caution, however, is in order now. While it is indeed true that for a generic r in the chaotic regime the length of an analytical formula for the prediction of Xn from xq grows exponentially in n, exceptions do occur at special values of r. For r = 4, e.g., a clever argument by Ulam and von Neumann (1947) shows that the iterates of (1.2.1) can be written explicitly as... 

Because of the apparent chaos in Fig. 6.5, simple analytical solutions of the driven SSE system probably do not exist, neither for the classical nor for the quantum mechanical problem. Therefore, if we want to investigate the quantum dynamics of the SSE system, powerful numerical schemes have to be devised to solve the time dependent Schrddinger equation of the microwave-driven SSE system. While the integration of classical trajectories is nearly trivial (a simple fourth order Runge-Kutta scheme, e.g., is sufficient), the quantum mechanical treatment of microwave-driven surface state electrons is far from trivial. In the chaotic regime many SSE bound states are strongly coupled, and the existence of the continuum and associated ionization channels poses additional problems. Numerical and approximate analytical solutions of the quantum SSE problem are proposed in the following section. 

Fig. 6.5. Poincare section of a microwave-driven surface state electron in the chaotic regime.

In conclusion we hope that, despite the experimental diflBculties discussed in the previous sections, SSE experiments will eventually be performed in the chaotic regime, experiments that put the very existence of critical fields, as well as their numerical values estimated in (6.4.10), to an experimental test. 

Fig. 9.10. Nearest neighbour distribution of the eigenangles of the S matrix in the chaotic regime (a) and the regular regime (b). (Adapted from Bliimel and Smilansky (1988).)...

These equations are similar to the Lorenz equations and can exhibit chaotic behavior (Haken 1983, Weiss and Vilaseca 1991). However, many practical lasers do not operate in the chaotic regime. In the simplest case /j, y, K then P and D relax rapidly to steady values, and hence may be adiabatically eliminated, as follows. 

T. Pierre, G. Bonhomme, and A. Atipo Controlling the chaotic regime of nonlinear ionization waves using time-delay autsynchronisation method, Phys. Rev. Lett. 76, 2290 (1996). 

In the phase space, the trajectory followed by the system never passes again through the same point, but remains confined to a finite portion of this space (fig. 4.10) the system evolves towards a strange attractor (Ruelle, 1989). The unpredictability of the time evolution in the chaotic regime is associated with the sensitivity to initial conditions two points, initially close to each other on the strange attractor, will diverge exponentially in the course of time. 

H. S. Taylor and J. Zakrzewski, Phys. Rev., A38, 3732 (1988). Dynamic Interpretation of Atomic and Molecular Spectra in the Chaotic Regime. See also references therein. 

K. Sohlberg and R. B. Shirts, /. Chem. Phys., 101,7763 (1994). Semidassical C andzation of a Nonintegrable System Pushing the Fourier Method into the Chaotic Regime. 

It should be noted that chaos control can only be obtained if deterministic chaos is involved. In case of (i) chaotic laser (ii) diode (iii) hydrodynamic and magneto-elastic systems and (iv) more recently myocardial tissue, feedback algorithm has been successfully applied to stabilize periodic oscillations. Quite recently, in order to stabilize periodic behaviour in the chaotic regime of oscillatory B-Z reaction, Showalter [14] and co-workers (1998) applied proportional feedback mechanism. Feedback was applied to the system by perturbing the flow rate of cesium-bromate solutions in the reactor keeping the flow rate of malonic acid fixed in these experiments. This experimental arrangement helped the stabilization of periodic behaviour within the chaotic regime. 

Computational studies have indicated that chaotic behavior is expected in classical mechanical descriptions of the motion of highly excited molecules. As a consequence, intramolecular dynamics relates directly to the fundamental issues of quantum vs classical chaos and semiclassical quantization. Practical implications are also clear if classical mechanics is a useful description of intramolecular dynamics, it suggests that isolated-molecule dynamics is sufficiently complex to allow a statistical-type description in the chaotic regime, with associated relaxation to equilibrium, and a concomitant loss of controlled reaction selectivity. 

It is a promising fact that the routes to chaos are qualitatively universal there exist sequences of events that occur when a parameter of the right-hand side is changed until the chaotic regime is reached. 

Unstable cycle is the result of a strange (chaotic) attractor. It represents a set of cycles of complicated geometry, which attract the nearby passing trajectories. To predict the behavior of trajectories for chaotic systems over a long period is impossible, because of the very high sensitivity to the initial conditions of a reaction. In experiments the initial conditions are usually given with limited accmacy. It is also very difficult to reproduce mnnerically the trajectory for the chaotic regime of a reaction due to the approximate nature of mnnerical methods. 

From a practical point of view one wishes to be able to control frictional forces so that the overall friction is reduced or enhanced, the chaotic regime is eliminated, and instead, smooth sliding is achieved. Such control can be of high technological importance for micromechanical devices, for instance in computer disk drives, where the early stages of motion and the stopping processes, which exhibit chaotic stick-slip, pose a real problem [56]. 

The periodic windows in the U-sequence exist in the "chaotic regime (e.g., the regime of Section 3.2) that follows the accumulation point of the period doubling sequence [50]. There are no chaotic intervals, yet the set of bifurcation parameter values for which the behavior is chaotic has positive measure. [For a typical map probably about 85% of the measure (of the bifurcation parameter range in which the U-sequence occurs) corresponds to chaotic states, and 15% to periodic states.]... 

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