[math-fun] Topology of the Lorenz System
A question was asked whether or not the "template" of the Lorenz strange attractor could have the same topology of the YGB surface, and I said, for typical parameters, the answer is No. See Figure 3 of: https://arxiv.org/pdf/1610.07079.pdf Cut along the midline and glue together the two loose ends, then you have a mobius strip. Repeat: When self-intersecting a mobius strip on an edge, there is a choice between two alternatives. The Lorenz template features an intersection with a point-like join between two cycles. The YGB surface involves a line-like join between two cycles (Try drawing graphs with two T-nodes and two cycles). Is there a chaotic system of ODE's whose attractor template is topologically equivalent to the YGB surface? The other issue is almost-periodic analysis of the Lorenz system, when considered as two joined oscillation disks. Has anything been written on Period Functions? Does this question sound interesting to anyone else? --Brad
I don't know either --- but the paper looks interesting. WFL On 1/22/21, Dan Asimov <dasimov@earthlink.net> wrote:
What is the YGB surface?
—Dan
On Thursday/21January/2021, at 7:27 AM, Brad Klee <bradklee@gmail.com> wrote:
A question was asked whether or not the "template" of the Lorenz strange attractor could have the same topology of the YGB surface
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YGB Surface is a two-dimensional, non-orientable, genus-2 manifold embedded in R^3. It can be obtained by twisting a mobius strip into a figure eight and allowing one edge to cut across the surface in such a way that one cycle of length L=2 becomes two mirror-image cycles of length L=3/2. The two cycles must have a line-like adjacency, allowing a third symmetric cycle of length L=1. In a previous email I called these cycles the "three shortcuts". For an illustrations see: Surface: https://0x0.st/-iRj.png Cycle Graph: https://0x0.st/-iR9.png (Be nice! Only my second day at watercolor, first attempt at YGB) The following page has lots of nice computer-generated pictures: https://chaoticatmospheres.com/mathrules-strange-attractors Easy to see non-trivial topology due to chaotic oscillation disks getting squashed together. Lots of analysis could be done. So far as I know, probably no one has even started how to characterise period functions on a particular disk. This topic could be a nice follow up to my dissertation. Some people complained it dealt only with integrable systems, and that is another "obvious reason why No". --Brad On Fri, Jan 22, 2021 at 8:41 AM Fred Lunnon <fred.lunnon@gmail.com> wrote:
I don't know either --- but the paper looks interesting. WFL
On 1/22/21, Dan Asimov <dasimov@earthlink.net> wrote:
What is the YGB surface?
—Dan
participants (3)
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Brad Klee -
Dan Asimov -
Fred Lunnon