CHAOS - Chaotic Oscillatory Behaviour

Variations in the values of the parameters will produce attractive oscillations. The equations are 

The variables are X and its first and second derivatives. No specific nomenclature is necessary. 

Vary A, Al, B, and B1 individually to see the influence of each. Plot the variables versus time and as phase planes. 

Experiment with time variations of the parameters, for example, with sine wave variations for B1. 

Karba and Zupancic (1992) also quote the following chaotic system equations 

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The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

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. 

Chaotic features are quite common in human physiology. Interval between heartbeats varies chaotically. Complex oscillatory features can be easily observed in ECG and EEG. It has been pointed out that chaos in body signals health and periodic behaviour can foreshadow disease [41]. 

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