By Symposium in Pure Mthmts, Thomas J. Jech (ed.)
Publication by way of SYMPOSIUM IN natural MTHMTS
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Extra info for Axiomatic Set Theory, Part 1
Yes, even more easily than in ordinary differential equations. Also in linear partial differential equations you can have chaos. Quantum mechanics does formally involve linear partial differential equations, so from this point of view, you would expect chaos. But then we have quantization which limits the complexity. My question here is, in the Copenhagen case, we do not have simultaneous information about a particle’s position and momentum. So if you want to model three electrons which are not free but trapped in a potential trough, you could expect chaos, classical Hamiltonian chaos, with momentum plotted versus position, for example.
After the 1970s and 1980s, and still around the year 2000, we find a lot of papers on chaotic behavior which taken together give us a rather precise mathematical definition of what is going on and how to find new examples. This led to very nice achievements in controlling chaos and also in anti-controlling chaos—since both methods, suppression and facilitation, are important for applications. My question now in 2013 is, when we look back some years and decades in the wake of many papers published, do you think that the field of chaos has been ‘‘saturated’’ to date because we do not see very big achievements anymore?
The ‘‘fractals’’ of Benoit Mandelbrot turn out to be implications, not of chaos but of hyperchaos. So if we have one more dimension than we need for chaos—four interacting variables rather than three—, we can produce fractals—beautiful never-stopping self-similar structures. In other words, the explanation of why nature is fractal is because chaos theory gives birth to fractals when you go from chaos to hyperchaos, which is deterministic chaos acting simultaneously and independently in two mutually slanted directions.