0
votes
2answers
67 views

Sketching phase portraits [on hold]

I am trying to answer this question: I would like to know how I go about drawing a phase portrait. All of the examples in my notes are simply the solution with no explanation, and this method of ...
0
votes
0answers
74 views

Is there a general formula for the angles of reflection in a rectangular billiards table?

While writing a program, I encountered a problem in which I needed to calculate the angles of reflection of in a rectangular billiards table. Let's say that we look at the table from above, and that ...
0
votes
0answers
18 views

transversal homoclinic points on a higher-dimensional torus

In many sources (for example introduction to chaotic dynamical systems by Devaney) one can find a proof of the fact that the transversal homoclinic points (points which lay on both stable and unstable ...
4
votes
1answer
78 views

The Starry Rebound

An (infinitely small) ball starting out in the middle of a 5 pointed star table (outer 5 points 10m radius, inner 5 points 5m radius) has a starting angle of a random value from 0 to 360 degrees. The ...
5
votes
0answers
328 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
4
votes
1answer
145 views

small circle inside embedding of complete graph in the plane

On the web, I found this beautiful drawing of the complete graph on 13 vertices: It is on the Geometry Daily tumblr page. A computer scientist drew a more interactive version up to about 40 ...
0
votes
0answers
67 views

basin of attraction(?) 3d, radius and depth

I'm not so sure I'm using the right terms in this question, but I will try to explain: Say I have a 3d surface, x axis and y are positives, z goes from 0-1. That surface has many "basins"(??), i.e. ...
8
votes
2answers
155 views

Is it possible to capture a light ray in a solar panel?

A while ago I was wondering how we could use mathematics to increase the efficiency of solar panels. The kind of mathematics I was thinking about in particular was Dynamical Billiards. Though I think ...
6
votes
1answer
131 views

Trajectories on the $k$-dimensional torus

Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$. My intuition clearly tells me that the set $$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$ is dense in $\mathbb ...
1
vote
2answers
97 views

Ring-shaped mirror chaos points

I came across this problem in the context of spectroscopy today. Because of it's simplicity, I'm sure it's a question that's been posed by some mathematician ages ago, I just can't figure out how to ...
2
votes
2answers
240 views

Broken glass geometry

If topology is called rubber-sheet geometry, would it be accurate to describe the "cut and shuffle" topic of "piecewise isometries" as broken glass geometry ? Isometry sounds more geometrical than ...
4
votes
2answers
949 views

Create a trapping region for Lorenz Attractor

I would like to show that the quantity: $-2\sigma\left(rx^{2}+y^{2}+b\left(z-r\right)^{2}-br^{2}\right)$ is negative on the surface: $rx^{2}+\sigma y^{2}+\sigma\left(z-2r\right)^{2}=C$ for some ...