I Anybody here read Winfree "Geometry of Biological Time"?

[Jimster41]

I'm not sure I'm following the whole intro chapter 1C "Phase Singularities of Maps". Is he saying that the existence of an oscillator somehow requires a singularity?

 

[The Bill]

He's talking about something much more specific than "the existence of an oscillator." He's talking about the properties of a continuous function from two input oscillators to a single output oscillator. He gives a topological argument about point(s) in phase space where the winding number of the output oscillator jumps discontinuously with small changes of the input phases.

 

[Jimster41]

Thanks, @The Bill. Sorry my question was so vague. I was in a fugue of confusion/frustration.

So the paragraph right after he say's "there's a contradition coming..." is the one I've had to read 10 times - I'm struggling to grasp what the incompatibility of those three axioms (the first one modified to remove "almost", as he says) means.

1. a small change in phi A or phi B results in a small change in phi prime
2. phi A and phi B are interchangeable
3. when phi A or phi B are 0, phi prime is zero.

Firstly I don't quite get why the example/thought experiment needs two parents (phi A and phi B). Doesn't it work the same way if you have just phi A?
Secondly, this relates to difficulty I have had understanding how there can be a topology that cycles but is not contained in higher dimensional space in which the curvature that supports cycling/recursion is defined - in other words a strictly 2D+1 space (not a 2D+1 manifold in a 3D+1 space). In such a 2D+1 manifold an ant walking in a straight line returns to her point of origin because the 3rd dimension supports the curvature needed (the ant can experience no curvature - just the eventual return to her origin as she walks in a line). When you shrink that third dimension to nothing in an effort to do away with it I get to a conclusion that feels like what Winfree is talking about. When the size of that dimension goes to some shrinking limit the curvature has to be like what - infinite, undefined, paradoxical? The ant is standing there staring at her own butt? There is some intrinsic uncertainty? Yet if that dimension were zero - there could be no curvature and no cycle.

 

[The Bill]

A change in winding number isn't a small change. Far from singularities, small changes in the input phases doesn't change the winding number of the output. When the phase space trajectories are deformed in a way that crosses the singularity, the winding number can change.

Winding numbers for trajectories are just like a loop of string wrapped several times around an infinite solid rod, or a steel ring, or such. The winding number around the rod can't change without cutting and reattaching the string, or teleporting a hank of string discontinuously through the rod, etc.

If you haven't read it, Strogatz's "Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering" is an excellent broad based introduction to this topic, and has a very good discussion of winding numbers.

 

[Jimster41]

I read Strogatz's "Sync" which is a lightweight popularization but fascinating all the same. Reason I got Winfree's book.

A change in winding number isn't a small change. Far from singularities, small changes in the input phases doesn't change the winding number of the output. When the phase space trajectories are deformed in a way that crosses the singularity, the winding number can change.

Winding numbers for trajectories are just like a loop of string wrapped several times around an infinite solid rod, or a steel ring, or such. The winding number around the rod can't change without cutting and reattaching the string, or teleporting a hank of string discontinuously through the rod, etc.

If you haven't read it, Strogatz's "Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering" is an excellent broad based introduction to this topic, and has a very good discussion of winding numbers.

I'm a bit confused by your reference to change in winding number. The point, I thought, is that the winding number is an integer. It's minimum value per change in the input phase must be 1 cycle and that really this thought puzzle is about the difficulty of having a map from a real to an integer while still assuming infinitesimal continuity. There is always a smaller move you can make with the input - which still must result in a minimum of 1 integer winding cycle. Or there is no non-zero change in the input that does not cause a minimum of one cycle.

Anyway thanks for the lead on that book. I will look for it.

 

[Jimster41]

Actually I realize now there is a strong statement (Theorem) on pg 27

"The only continuous map from the disk (e.g. The area inside triangle αβγα) to the circle (the possible values of φ') have winding number 0 around the border of the disk. If the winding number cannot be zero (e.g., because of axiom 3), then the map cannot be continuous no matter what the underlying physical or biochemical mechanism might be. In fact no manifold can be retracted to its whole boundary while leaving that boundary point-wise fixed without a discontinuity appearing somewhere"

Wish I had a good grasp of what exactly "manifold retraction" means...

If I understand this correctly. My question then is what does the ubiquty of periodicity in nature suggest about its (nature's) fundamental support? Not that the converse of this theorem is given but the cause of periodicity at step 1 (aka quantum mechanics) must be given some mechanism. Winfree hints at the connection on pg 30 but leaves it there. From here I guess he's going to just go crazy on all the wonderful semi-empirical puzzles (because they are biological-observations as well as mathematics) that exist. But what an awesome start. Is he still around?

 

[The Bill]

Wish I had a good grasp of what exactly "manifold retraction" means...

The first two pages of Hatcher's "Algebraic Topology" has a pretty good explanation of retractions. Plus, the book is available free as a PDF from the author's site: https://www.math.cornell.edu/~hatcher/AT/ATpage.html