28
votes
3D Billiards problem inside a torus
These so-called "whispering gallery modes" are familiar from studies of microcavity lasers; they can trap the light indefinitely, only limited by diffraction; this web site by Jens Nöckel nicely ...
- 188k
16
votes
Accepted
3D Billiards problem inside a torus
As to the fact that this improbable scenario arose from chaos, it seems that numerical error really was at fault (my bisection code was faulty).
This behavior is more to be expected.
EDIT: According ...
- 589
10
votes
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
Here are a few simple remarks and one warning that are too long for a comment box. I surmise that you know most of them yourself but I'll just make them in case some reader finds any of them "non-...
- 61.9k
8
votes
Proven chaos in logistic maps
For all values of $r > r_0 \simeq 3.5699...$, the topological entropy of the logistic map is strictly positive [1]. That means that there is an uncountable set of points whose orbit accumulates on ...
- 18.7k
8
votes
Proven chaos in logistic maps
By choosing $r=4$, the logistic map is topologically semi-conjugate to the doubling map on $\mathbb{R} / \mathbb{Z}$: the solution takes the form $ x_{n}=\sin ^{2}(2^{n}\theta \pi)$. Thus any ...
- 9,501
8
votes
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
This is not a solution, but some additional numerical evidence for the non-convergent behaviour of the sequence $r_n$.
The original recurrence implies for the two new variables
$$
b_n ~=~ 1- r_n^2,~~...
- 1,676
7
votes
Proven chaos in logistic maps
There is a nice article by Mikhail Lyubich in the October 2000 edition of Notices of the AMS, "The Quadratic Family as a Qualitatively Solvable Model of Chaos" that provides a nice summary ...
- 8,903
7
votes
On Mathematical Foundations of Football
There is a great article by Connes (translated in Symmetries,
Eur. Math. Soc. Newsl. 54, 11-18 (2004).
ZBL1176.00001. - link to free pdf, p. 11-18) motivated by fair pairings in soccer competitions, ...
- 1,134
6
votes
Accepted
On Mathematical Foundations of Football
Typing "mathematics of soccer" into the internet led to J.A. Tenreiro Machado, António M.Lopes, On the mathematical modeling of soccer dynamics, Communications in Nonlinear Science and Numerical ...
- 39.9k
6
votes
Accepted
Are all Torus Links in fact Lorenz links or not?
The point is that the two papers use slightly different definitions of "Lorenz Links".
The newer paper defines Lorenz links as links on the Lorenz template. With this definition all torus links are ...
- 10.4k
6
votes
Accepted
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
OK, finally something about the measure. I'll put it into a separate answer again for readability. If the moderators frown upon such practice, they are welcome to merge in any way they find ...
- 61.9k
6
votes
Bounding proportion of phase space which is chaotic
You will want to study the Poincaré map, by numerically integrating the equations of motion on a grid of starting positions. The regular regions show up in the Poincaré section as regions of zero ...
- 188k
5
votes
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
Just another illustration (cw) - color-coded reciprocals of distances from the origin after 14 iterations:
I believe this shows that there is no easy answer...
5
votes
Accepted
Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?
Explicit solutions for arbitrary $r$ exist in various forms:
Logistic map: an
analytical solution (1995) represents the solution as a power of
a transfer matrix.
An explicit solution
for the ...
- 188k
5
votes
Is this a new strange attractor?
I just figured I added some Mathematica code and a picture for the attractor in the question.
...
- 15.8k
5
votes
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
I still cannot figure out what is going on in the measure-theoretic sense, but the topological dynamics has become a bit clearer. Formally we will show that the set of points that stay sufficiently ...
- 61.9k
4
votes
Questions about a return map
We have $f(u) < u$ for $0 < u < 1$ and $f(u) > u$ for $u > 1$, so the fixed point $1$ is unstable. Similarly $f(u) < u$ for $u < -1$ and $f(u) > u$ for $-1 < u < 0$ ...
- 54.2k
4
votes
Accepted
State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"
This question was answered recently by Geschke, Grebík, and Miller:
S. Geschke, J. Grebík, and B. D. Miller, "Scrambled Cantor sets," Proceedings of the AMS 149 (link to the arxiv version)
...
- 18.5k
3
votes
Chaotic complex dynamics and Newton's method
"Repeatedly iterating $g^{-1}$" does not really make sense, since your map is not injective.
What is usually done when one talks about "backwards iteration" in this context is that ...
- 6,548
3
votes
Accepted
Is this a new strange attractor?
In another forum a user drew my attention to the publication of
J. C. Sprott, Some simple chaotic flows, Phys. Rev. E 50, R647-R650 (1994), doi:10.1103/PhysRevE.50.R647, author pdf.
This shows that ...
- 101
3
votes
Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)
I'm not sure if these examples are generalizable for your purposes (I do symbolic dynamics, and the examples I like the most probably have nothing to do with quantum mechanics...), but:
Every ...
- 2,621
3
votes
Accepted
Metric entropy and topological entropy
It's true for a very simple reason: the entropy of a dynamical system with respect to a (not necessarily ergodic) invariant measure is the average of the entropies of its ergodic components.
- 17k
3
votes
Example of a Chaotic discrete dynamical system in dimension 2
Kiki, regarding question 1), any Anosov diffeomorphism on $\mathbb{T}^2$ is Devaney chaotic.
Regarding question 2), the are minimal (i.e. every orbit is dense) Li-Yorke chaotic diffeomorphisms on $\...
- 1,060
2
votes
Gutzwiller trace formula
First of all, if you are just looking for intuition, you can easily find more accessible accounts than this paper (such as the book by Gutzwiller, or even Wikipedia).
Also, you can look up the '...
- 2,552
2
votes
Example of a Chaotic discrete dynamical system in dimension 2
For another answer to Question 1, consider Julia sets of holomorphic functions of one complex variable (e.g., polynomials or rational maps of degree $\geq 2$, or transcendental entire functions). Any ...
- 6,548
2
votes
How to analytically prove chaos
For specific systems at specific parameters, it can often be infeasible to give a pen-and-paper proof of chaos. However there is a rich literature of using computer-assisted-proofs based on interval ...
- 177
2
votes
Oscillator with discrete number of amplitudes?
I'm not sure about what your requirements are. In particular, I didn't get whether you need that the phenomenon arises from a chaotic oscillator, or it was just your try to search among those.
However,...
- 4,459
2
votes
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
This is not an answer, but I think it provides nice insights.
First reformulation.
Starting from @Karl Fabian reformulation
$$ b_{n+1} = 4g_n b_n^2, \ \ g_{n+1} = \frac{1}{b_n} -1 - g_n $$
$$ g_0 = \...
- 2,152
2
votes
Accepted
Solution of an ODE upon singular perturbation
Let me change the notations to fit the mathematical literature. I will denote by $x(t) \in \mathbb{R}^{3N}$ the positions of the particles, by $y(t) \in \mathbb{R}^{3N}$ their velocities and by $0 <...
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2
votes
Stable periodic orbits for three equal masses
The proposer gives as their definition of stability the standard notion
of Lyapunov stability. Unfortunately, there are no known solutions for the planar or spatial three-body problem which are ...
- 8,336
Only top scored, non community-wiki answers of a minimum length are eligible