Spatio-temporal chaos

We shall describe some of tire common types of chemical patterns observed in such experiments and comment on tire mechanisms for tlieir appearance. In keeping witli tire tlieme of tliis chapter we focus on states of spatio-temporal chaos or on regular chemical patterns tliat lead to such turbulent states. We shall touch only upon tire main aspects of tliis topic since tliere is a large variety of chemical patterns and many mechanisms for tlieir onset [2,3, 5,31]. 

The cores of the spiral waves need not be stationary and can move in periodic, quasi-periodic or even chaotic flower trajectories [42, 43]. In addition, spatio-temporal chaos can arise if such spiral waves break up and the spiral wave fragments spawn pairs of new spirals [42, 44]. 

Figure C3.6.14 Space-time (y,t) plot of the minima (black) in the cubic autocatalysis front ( )(y,t) in equation C3.6.16 showing the nature of the spatio-temporal chaos.

Hence, the system s dynamics is fully relaxational, i.e. the evolution of the system is characterized by a monotonic decrease of the Lyapunov functional and the approach to a final stationary state, which excludes the possibility of oscillatory instabilities, spatio-temporal chaos etc. 

Equation (158) is a paradigmatic model for studying spatio-temporal chaos... 

Figure 20. Numerical solutions of eq.(158) spatio-temporal diagram showing spatio-temporal chaos (upper figure) and a snapshot of the solution at a particular moment of time.

For more senior readers, the dynamic model should be used not only to simulate the dynamic behavior of this heterogeneous distributed system but also to investigate the possible spatio-temporal chaos associated with it. Important note As for the previous exercise, it is stated everywhere that all biokinetics and permeation parameters as well as other parameters are to be obtained from the literature. However, whenever the reader has the experimental facilities to determine any of these parameters, he/she is encouraged to obtain these values. This will be useful and add to the educational benefits of this mathematical modeling and computer simulation exercise. 

Coming back to limit cycle oscillations shown by systems of ordinary differential equations, this simple mode of motion still seems to deserve some more attention, especially in relation to its role as a basic functional unit from which various dynamical complexities arise. This seems to occur in at least two ways. As mentioned above, one may start with a simple oscillator, increase [x, and obtain complicated behaviors this forms, in fact, a modern topic. However, another implication of this dynamical unit should not be left unnoticed. We should know that a limit cycle oscillator is also an important component system in various self-organization phenomena and also in other forms of spatio-temporal complexity such as turbulence. In this book, particular emphasis will be placed on this second aspect of oscillator systems. This naturally leads to the notion of the many-body theory of limit cycle oscillators we let many oscillators contact each other to form a field , and ask what modes of self-organiza-tion are possible or under what conditions spatio-temporal chaos arises, etc. A representative class of such many-oscillator systems in theory and practical application is that of the fields of diffusion-coupled oscillators (possibly with suitable modifications), so that this type of system will primarily be considered in this book. 

Reaction-diffusion systems are expected to show spatio-temporal chaos in various circumstances. A few specific cases will be discussed. They include the turbuhzation of uniform oscillations, of propagating wave fronts and of rotating spiral waves. 

People often speak of chemical turbulence whereby either of two distinct chaotic phenomena may be meant. One is the spatially uniform but temporally chaotic dynamics exhibited by the concentrations of chemical species, while the other involves spatial chaos too. For chemical turbulence in the latter sense, our attention is usually focused upon systems in which the local dynamics itself is non-chaotic, while such non-chaotic elements are coupled through diffussion to produce spatio-temporal chaos. In fact, if the local elements were already chaotic, the fields composed of them would trivially exhibit spatio-temporal chaos. Hence non-trivial chemical turbulence involving spatio-temporal chaos may be called diffusion-induced chemical turbulence. 

A direct simulation of the coupled system of the partial differential equations, Eq. 29, with appropriate boundary conditions (v = 0, n prescribed at the confining plates, etc.) is at the limits of the supercomputers of today. It will turn out that in the liquid crystal systems a rich scenario of patterns, including spatio temporal chaos, develops already near threshold so that perturbational calculations are useful. 

Fig. 24. Turing-Hopf spatio-temporal chaos . Snapshots (a) and (c) are separated by 15 seconds corresponding to a half-period for the oscillation in the Hopf-hole . (b) Time-averaged image over one period of oscillation. The small scale mosaic corresponds to the Turing mode while the larger uniformly gray patches correspond to Hopf-holes . View size 4.2 mm x 4.2 mm. Experimental conditions as in Figure 23 but with [CH2(C00H)2] = 8.5 X 10 M.

We shall now analyze the effect of fluctuations in the regime of diffusion-induced spatio-temporal chaos. The simplest model exhibiting this behavior is the Brusselator introduced already in Section 3, Equations (16), in the region of parameter values for which inequality (20) is violated. One can readily verify that this is indeed the case if >2 = 0 and a = 2. In the sequel... 

Depending on the pressure and temperature, periodic patterns develop on the surface, such as the target-like patterns or characteristic spiral waves exemplified in Fig. 5.26 these propagate at front speeds of a few pm/s.The core of a spiral often arises at a surface region with enhanced defect density. Ertl and coworkers were also able to observe a break-up of the spirals and the onset of spiral turbulences and of spatio-temporal chaos. 

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