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Oliver Sheridan-Methven Chaos Supervisor - DL Pageites Word Count - 6460 X 唴 _ F(X 1 ) = λ(-X 0 2 + X 0 ) Figure2 λ What the logistics map symbolises for chaotic dynamics is the fact that in some of even the most simple systems; which can have just two or three factors effecting it, where one of these factors is adjustable, then it can be adjusted to produce any of three system states. First of these states is the state where nothing is happening and any initial state will soon settle into a fixed state. The second is where the system will fall into an active steady state where it will either be doing a fixed motion or will be cycling through states regularly (demonstrated in a later experiment). The final implication of the logistics map is that if the correct parameter (λ) is chosen, then the system will be unpredictable and will fall into a system of chaos that will appear to have no pattern and therefore looks random [2,7]. When a system appears random the challenge is to then spot some sort of underlying pattern or trend; and to manipulate the data until such a pattern (and thus predictability) arises. Whilst many chaotic systems may not have an underlying pattern, a significant amount does. The task is to first find any strange attractors; which are points or trajectories through phase space. Phase space is just a graph of how the system is behaving as time goes by (Figure3) and where a certain point represents a characteristic, such as velocity or angle or potential energy etc. Thus one point could represent a system having a lot of kinetic energy when it is at such an angle whilst another point signifies another state the system can be in such as low kinetic energy but high temperature [1,2,3]. Thus if all paths through phase space always end up reaching a certain point then that point is a strange attractor. Similarly if a path continues orbiting a point in a circle for example, then the centre is a strange attractor. Finally, if all paths converge eventually into a certain path/trajectory (be it a straight line or a bizarrely shaped curve as in Figure3), even if initially off that path, then that trajectory is known as a strange attractor. Page 4 of 18
Oliver Sheridan-Methven Chaos Supervisor - DL Pageites Word Count - 6460 Figure3 Strange Attractors Figure3 shows how the measurements of a system and how it is behaving over time (top row); e.g. velocity with time, translate into phase space diagrams (bottom row) of height with gradient. Here it is evident that where the top row may be unclear in displaying how the system is behaving and what it will do with time (most prominent in the top right plot), when transformed into a phase space diagram it can become apparent what the system is doing. Thus apparent randomness, in what appears to be a chaotic system, can be translated into predictability in phase space, allowing for a chaotic system to be made predictable [1,3]. The idea of there being underlying patterns in phase space which produce predictability, seems counterintuitive for chaotic systems. The fact that "randomness" can have patterns seems slightly illogical (and to an extent impossible) and thus an experiment was conducted to try to demonstrate this concept with real data collected from the lab, and not produced from theory. A very basic system encountered which was unexpectedly chaotic was that of the pendulum. For an equivalent James Gleick used the image of a child on a swing, which will be extended upon in the explanation of the experiment. If a child is on a swing and the swing is not moving then as time goes by the child will stay on the swing in the same position, which is obvious. If a child is then pushed once in the swing then they will oscillate to and from the same height, just at different angles on the swing (demonstrating the conservation of energy). Finally if friction is acknowledged then eventually the energy will dissipate from the system and the child will eventually come to a halt in the bottom of the swing position again. When these three processes are plotted in phase space (Figure4) it is much clearer to see the behaviour of the system and to spot the strange attractors in the system. NB even though the system is not yet chaotic, it has strange attractors in phase space [2]. In the frictionless system the swing will carry on swinging and the phase space will form a loop (Fiugre4) which Strange Attractors Figure4 Page 5 of 18
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