Posting has been light as I have been gobsmacked by something I discovered in a book that I've had for almost twenty years. I've always had trouble understanding it. I'm a geologist, and find this sort of thing (pdf) challenging.
It has to do with these probability density plots I've been making in phase space. I developed the idea intuitively, but the publishing has always been a slog because I had difficulty presenting a theoretical justification of my approach. I had made a leap of faith that each area of high probability density in phase space was centred about an attractor of indeterminate type.
It was a bit of a fluke getting the paper published in Paleoceanography--the reviewers weren't sure they agreed with it but were willing to give it a go. In terms of number of citations it ranks among the least influential publications in the journal's history.
The discovery was an interpretation of Zeeman's classification problem. His idea was that given a system which is described by a vector field on a manifold (say, a 2-d plane, which is what I have been using, but any surface is possible) so that the trajectory of the time-evolution of the system indicates a flow along the vectors; and given the system is somewhat noisy, so that there is a small random component to evolution along the trajectory; then the end-state of probability densities from all possible initial states on the manifold will be an invariant property of the vector field. What you will end up with will be diffuse balls of higher probability around each of the attractors on the manifold in phase space.
I read this as a justification of my intuitive approach, and he's a real mathematician.
Edit: Reference
Zeeman, E. C., 1988. Stability of dynamical systems. Nonlinearity, 1: 115. doi.org/10.1088/0951-7715/1/1/005
It has to do with these probability density plots I've been making in phase space. I developed the idea intuitively, but the publishing has always been a slog because I had difficulty presenting a theoretical justification of my approach. I had made a leap of faith that each area of high probability density in phase space was centred about an attractor of indeterminate type.
It was a bit of a fluke getting the paper published in Paleoceanography--the reviewers weren't sure they agreed with it but were willing to give it a go. In terms of number of citations it ranks among the least influential publications in the journal's history.
The discovery was an interpretation of Zeeman's classification problem. His idea was that given a system which is described by a vector field on a manifold (say, a 2-d plane, which is what I have been using, but any surface is possible) so that the trajectory of the time-evolution of the system indicates a flow along the vectors; and given the system is somewhat noisy, so that there is a small random component to evolution along the trajectory; then the end-state of probability densities from all possible initial states on the manifold will be an invariant property of the vector field. What you will end up with will be diffuse balls of higher probability around each of the attractors on the manifold in phase space.
I read this as a justification of my intuitive approach, and he's a real mathematician.
Edit: Reference
Zeeman, E. C., 1988. Stability of dynamical systems. Nonlinearity, 1: 115. doi.org/10.1088/0951-7715/1/1/005
No comments:
Post a Comment