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Deterministic and Random Evolution (Mathematics Research Developments)

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The first notes for this text were written during the summers of 2008–2010 when I taught a short course on mathematical modeling at the University of New Mexico. The audience consisted mostly of undergraduate mathematics students, and an aim of the course was to interest them in math at the graduate level.

The students had some basic knowledge of ordinary differential equations and numerics. I tried to build on this foundation, but instead of increasing the technical skills of the studen I tried to lead them to more fundamental questions. What can one model with differential equations? What is determinism? If the universe evolves deterministically, what about free will and responsibility? Does it help if there are elements of randomness in the laws of evolution?

Of course, these are deep questions, and in this text we can only scratch the surface in our discussion. Nevertheless, mathematics — even at a rather elementary level — may help to clarify what is at stake. Throughout, I try to put the discussion into historical context. For example, the text contains a rather detailed description of the derivation of Kepler’s laws of planetary motion using ordinary differential equations. After all, Newton’s great success in deriving these laws were an important starting point of the scientific revolution and a deterministic world view. It made classical mechanics a model for all sciences.

Even if a deterministic description of an evolution is possible, there are often practical limitations of predictability because of the exponential growth in time of any uncertainty in the initial condition. Iteration with the logistic map gives an example. However, even if the accurate determination of future states is impractical, the average behavior of a system may still be very robustly determined. The logistic map again serves as an example. Do we have a similar situation for weather and climate? We cannot predict the weather two weeks in advance, but it may still be possible to determine the average weather 30 years from now.
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