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Strange Attractors

Conclusion

Despite the preeminence of simple, Euclidean shapes as the theoretical basis for much of our geometrical understanding it appears that the intricate and complex fractal is the norm in nature. If we think about it, there are uncountably many factors that can affect a natural system. Even the example of the pendulum is an approximation; real pendulums don't have perfectly rigid supports and their damping parameters are time and position variant, unlike our model. Even the fractal fern leaf only gives an impression of the real object. The leaves of real ferns have a different structure to the leaflets. Neither do they look the same at all magnifications. There are physical limitations which must be applied to the mathematics behind these systems.

Chaos and fractal theory do give us the tools to approximate real systems by mathematics and to tell us how good those approximations are. For instance, by running weather models on a supercomputer using slightly different initial conditions forecasters can tell how predictable the weather will be (Gleick, 1987). If the results diverge widely between the two sets of data, then the weather is in an unpredictable mode, otherwise they know their forecasts will be pretty good.

The are some forecasts that the weathermen can make with great certainty. We all know that, baring a cosmic disaster, in ten years time the air temperature will be less that one hundred degrees Celsius and above the freezing point of helium. We also know that if we a swinging our non-linear pendulum in Melbourne then we don't expect it to suddenly appear in Sydney. We can say this because we know that these systems will be confined to their attractors. What we don't know is exactly where on the strange attractor the system is at any given time and hence where it will exactly be in the future.

 

Links

Introduction - The Lorentz Butterfly

The Driven Non-linear Pendulum

IFS Fractals

Conclusion

References

Conmputer Programs

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