Attractors macroscopic

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

Fixed points are not the only possible states of dissipative, far-from-equilibrium systems more complex macroscopic attractors, like limit cycles or even strange attractors, are commonly observed [19]. Bistabilities between the different attractor types may occur and give rise to interesting transition rate processes when these systems are subjected to external noise. We examine some of the new features that enter the calculation of the transition rate by examining some specific examples of systems displaying bistability between a fixed point and a limit cycle, but the discussion can be generalized to other situations. 

On the micro-level of the brain, there are massively many-body-problems which need a reduction strategy to handle with the complexity. In the case of EEG-pictures, a complex system of electrodes measures local states (electric potentials) of the brain. The whole state of a patient s brain on the micro-level is represented by local time series. In the case of, e.g., petit mal epilepsy, they are characterized by typical cyclic peaks. The microscopic states determine macroscopic electric field patterns during a cyclic period. Mathematically, the macroscopic patterns can be determined by spatial modes and order parameters, i.e., the amplitude of the field waves. In the corresponding phase space, they determine a chaotic attractor characterizing petit mal epilepsy. 

Our position toward this problem is that as far as the validity of the macroscopic description is concerned, the important point is not Poo.n being broad or not, but rather Poo.a being close to poo or not. In other words, even if individual fluctuations reach anomalous values as a result of sensitivity to initial conditions the macroscopic description will remain valid, provided that in the overwhelming majority of cases this process is taking place on the system s (deterministic) attractor. Some rigorous partial results supporting... 

Oscillations and chaos are observed frequently in chemical systems. Most of the experimental investigations of these phenomena have been carried out for well-stirred systems where spatial degrees of freedom are assumed to play no role. If this is the case the system may be described in terms of chemical rate equations for a small number of macroscopic chemical concentrations. The periodic or chaotic attractors typically have low dimensions and can be characterized using the tools of dynamical systems theory [20]. Chemical systems may also display spatiotemporal oscillations and chaos. If spatial degrees of freedom are important the appropriate macroscopic model is the reaction-diffusion equation. The attractors may have high dimension and the theoretical description of such spatiotemporal states is less well developed. 

All of these phenomena arise out of the random reactive and elastic collision events in the system and a fundamental understanding of how macroscopic, self-organized chemical structures appear must be based on descriptions that go beyond the macroscopic, mean-field rate laws or reaction-diffusion equations. In this section we use the reactive lattice-gas method to examine how molecular fluctuations influence oscillatory and chaotic dynamics. In particular we shall show how system size, diffusion, reactions and fluctuations determine the structure of the noisy periodic or chaotic attractors. 

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