Summary

If a dynamical system becomes chaotic at some parameters of physical interest then that is usually an unwelcome occurrence. Firstly, it means that the system in the chaotic state is extremely sensitive to initial conditions, and hence the errors inherent in real-life measurements of finite precision. The extreme sensitivity implies a lack of practical predictability of the long-term behaviour of the system even though the equations of motion are completely deterministic ! Nevertheless long-term unpredictability due to chaos does not preclude short-term predictability and usefulness. Furthermore, one might be able to exploit the sensitive dependence of chaotic systems to initial conditions by controlling those systems: very small perturbations to the system can be used to completely change its long-term behaviour. Some attempts in this direction are currently underway, for example, for control of cardiac chaos (See Ref.[10]). However chaos might be useful in some cases: It has been reported that epileptic seizures are the result of the electrical brain waves syncronising, while their natural state is "chaotic"! It is very important to note that chaos arising in deterministic systems is quite different from noise in random (stochastic) systems. This is because one still sees order and patterns in the phase space plots for chaotic systems which is absent for random systems. In fact one can say that chaos (in deterministic systems) is order camouflaged as disorder ! An important technique in distinguishing whether a given time-series has a deterministic underlying basis is through the method of attractor reconstruction. For low-dimenisonal systems that are at least approximately equivalent to a unimodal one-dimensional map, the approach to chaos is through period-doubling bifurcations. The Feigenbaum constant summarizes the universality of this class of systems. Chaos in higher dimensional systems with many degress of freedom can occur in different ways from that in low-dimensional systems. In the previous chapter we saw how the geometric complexity of nature could be described by simple algorithms and models that generate fractal structures. That is, complex patterns and natural order can arise from a simple underlying basis. In this chapter we see the flip side : very simple deterministic dynamical models can lead to behaviour that looks random. However the apparent disorder is actually chaotic rather than random as the order is hidden now in the phase-space of the system. What this suggests is that some of the apparent irregularity observed in nature or man-made systems might be ammeanable to deterministic modelling instead of appealing to random processes.