21

Economics offers many examples of simple models for various systems not based on physical principles. One nice example is Hotelling's line model, which gives rise to Hotelling's "law". It considers two firms deciding where to locate their business to capture the most customers along a stretch of road. Considering such models over a period of sequential ...


18

Theorem 1.1 in Richard Swartz's paper Obtuse Triangular Billiards I: Near the (2,3,6) triangle rules out easy proofs: He shows that, for any $\epsilon>0$ and any $N>0$, there is a triangle whose angles are within $\epsilon$ of $(\pi/2, \pi/3, \pi/6)$ and for which any closed path involves more than $N$ bounces. So we can't write down some finite list ...


17

The book Enns, Richard H., It’s a nonlinear world., Springer Undergraduate Texts in Mathematics and Technology. New York, NY: Springer (ISBN 978-0-387-75338-6/hbk; 978-0-387-75340-9/ebook). xii, 383 p. (2011). ZBL1214.00007. contains many nice examples, also for models of sports (Ch. 6) and war (Ch. 11).


15

Start from the simplest path, a triangle with angles $\alpha, \beta, \gamma$, and build the unique triangle for which this path is a billiard path. It's easy to see that the latter triangle has angles $\frac{\alpha+\beta}{2}, \frac{\gamma+\beta}{2}, \frac{\alpha+\gamma}{2}$ and is therefore acute. Any acute triangle can be obtained in such a way. The next ...


15

I asked Rich Schwartz, who is one of the experts in this area (as noted by the OP). Here, with Rich's permission, is his response: I am not sure why it is so hard. All I can really say is that, after a lot of experimentation, I can't really see any pattern to it. It might be hard in the same way that building the fountain of youth is hard: nobody ...


12

Main areas are dynamics of automorphisms (for example, Henon maps), dynamics of endomorphisms, dynamics of foliations, and local dynamics. Eric Bedford, Tien-Cuong Dinh, John Fornaess, Misha Lyubich, Nessim Sibony, John Smiley, and their students and collaborators. One-dimensional holomorphic dynamics, functions of several complex variables (especially ...


11

W. Klingenberg in [Riemannian Manifolds With Geodesic Flow of Anosov Type, Annals of Mathematics, Vol. 99, No. 1, 1974, pp. 1-13] proved among other things that a closed Riemannian manifold with Anosov geodesic flow does not have conjugate points, and hence the exponential map at any point is a covering map. In particular, the manifold is aspherical.


10

Given an entire function $f\colon\mathbb{C}\to\mathbb{C}$, the escaping set, $I(f)$, is the set of $z\in\mathbb{C}$ such that $f^n(z)\to\infty$. Per the Wikipedia article, the escaping set of a non-linear entire function is nonempty. The reference for this is On the iteration of entire functions by Eremenko.


10

$\newcommand\al{\alpha}$Let us drop the factor $1-\al$, by considering $$Y:=X_\infty/(1-\al)=\sum_{k=0}^\infty\al^k X_k.$$ By Kolmogorov's three-series theorem, this series will converge almost surely (a.s.) unless at least one of the tails of the distribution of $X_0$ is too heavy. Assume that the series indeed converges a.s. Then, obviously, $$Y\...


9

Here is a sketch of the "cheapest" way I know how to prove something like that. Filling in the details may still be a bit lengthy but should be essentially routine. To set things up, let's write $X$ for the space of oriented non-degenerate area $1$ triangles, so $\tau \in X$ is a pair $(v_1,v_2)$ with $v_i \in \mathbf{R}^2$ and $|v_1 \wedge v_2| = 2$. I then ...


9

It appears that there are applications of dynamical systems to music. See, e.g., Rick Bidlack, Music from chaos: nonlinear dynamical systems as generators of musical materials or Boon and Decroly, Dynamical systems theory for music dynamics or David Burrows, A Dynamical Systems Perspective on Music or just type dynamical systems in music into Google, as ...


9

My favorite example is the use of hidden Markov models to analyze defensive strategies in basketball, as explained in Characterizing the spatial structure of defensive skill in professional basketball, by Alexander Franks, Andrew Miller, Luke Bornn, and Kirk Goldsberry, Ann. Appl. Stat. Volume 9, Number 1 (2015), 94–121. Making various modeling ...


9

The solution $Z(t)$ of your differential equation with $Z(0) = Z_0$ satisfies $$ Z(t) (e^t + (1-e^t) Z_0) = Z_0 $$ In order for this to be periodic with period $p$, you'd need $(1-e^p) Z_0 (1-Z_0) = 0 $. $1-e^p = 0$ (for real $p$) only if $p=0$, while if $Z_0 (1-Z_0) = 0$ we have a fixed point.


8

The question is really about the iteration behaviour of the maps $(x,y) \mapsto (y, y^3-x)$ with various starting points. We have a fixed point $(0,0)$ and a $6$-cycle $$(1,0),(0, -1), (-1, -1), (-1, 0), (0, 1), (1, 1)$$ It appears that for something like $-0.797 < x < 0.797$, $(0,x)$ is on an invariant curve. For example, here are $10000$ iterates ...


7

The answer to a) is yes, and this was proved by Fatou in 1919. Sur les équations fonctionnelles Bulletin de la S. M. F., tome 48 (1920), p. 208-314. There are many generalizations of this fact. For one generalization, and further references you may look to Meromorphic functions with linearly distributed values and Julia sets of rational functions, Proc. AMS....


7

If a 1-dimensional foliation of a 3-manifold contains a Reeb component, then it is not geodesible. This follows from a theorem of Sullivan, see Section 2.5, Obstruction 1. A Reeb component is a saturated annulus embedded in the manifold with induced foliation looking like: One can find a 1-dimensional foliation of any 3-manifold containing such a Reeb ...


7

There are a lot of applications of ideas in dynamical systems to social media for things like event detection and forecasting. Some of the literature has to be taken with a grain of salt because of the difficulties in gathering data, but this paper looks nice: https://arxiv.org/pdf/1603.00074.pdf


7

Although it's presented as a game, the fox-and-duck problem may be cast as a control problem from the perspective of the fox. The aim of the fox is to catch the duck, while the duck conversely tries to escape the fox. The duck can swim in any direction on a circular pond with speed $v$. The fox, who cannot swim, can run around the pond with speed $\lambda v$ ...


6

This is an expansion on Igor's comment and fixes my previous mistake (see also Willie's answer). A direct computation gives $$ D(f) = 2f^{ab}\nabla_{(a}X_{b)} + X_a(\nabla^b\nabla_b\nabla^a-\nabla^a\nabla_b\nabla^b)f + f^a\nabla^b\nabla_bX_a . $$ The middle summand is the same as $R_{ab}f^a$, while the formula $\nabla^b\nabla_aX_b=\nabla_a\nabla^bX_b+R_a{}^...


6

I think the question is perhaps confusing, since the term Hopf vector field is usually reserved for your $X_i$ vector field (which is tangent to the fibers of the standard Hopf fibration). As I understand, you are instead referring to the vector field which, at a point $p$ with $P(p) = (a,b,c) \in S^2 \subset \mathbb{R}^3$, is given by $Y := aX_i + bX_j + ...


6

The most intuitive explanation I know is the following: suppose that you have a certain amount of mass (I usually picture a pile of sand) that is distributed over $\Sigma_A^+$ according to the density $g\,d\nu$, where $g$ is an arbitrary continuous function and $\nu$ is the eigenmeasure for the transfer operator $\mathcal{L}$. Then you move this mass around ...


6

edit I realized there is a simpler construction that achieves the same (examples of top. mixing zero-entropy homeos on $S^3$): Instead of the bi-directional flow through cuboids, have a flow from bottom to top on the cylinder $D^2 \times [0,1]$, and have it slow down to rate $0$ on the boundary. Also, instead of all the $C_i$ playing the same role, have a ...


6

Consider the shift space $X \subset \{0,1\}^{\mathbb{Z}}$ obtained by forbidding the words $1 0^m 1^n 0$ for all $m, n \geq 1$, and denote the shift map on $X$ by $\sigma$. Since a point of $X$ can contain at most three transitions from $0$ to $1$ or back, the only $\omega$-limit points of $X$ are the two uniform points (all-$0$ and all-$1$). However, $x = \...


6

It seems to me that the structure of ${\rm Aut}^{m}(C_{n})$ also depends heavily on the prime factorization of $n$, and I don't really see any reason to expect the answer to Q1 to be any more tractable than determining the structure of ${\rm Aut}(C_{n})$. For example (just to illustrate) , if we choose a prime $p$, greater than $3$, and then we take a pair ...


6

A list of predecessors as mentioned in my comment. I document pairs of $(m,n)$ for consecutive $m$ and their 1-step predecessors $n$ such that $f(n)=m$. The value $n=0$ indicates, that $m$ has no predecessor. I didn't reflect, that one $m$ can have two predecessors, but if $n/2$ is odd, then $n/2$ is a second predecessor.(This makes the table more ...


6

If all the particles remained in a bounded domain, the virial theorem would apply. In the case of a radial inverse square power law, it states that twice the asymptotic time average of kinetic energy of the system equals minus the asymptotic time average of its potential energy. However, while the kinetic energy is always nonnegative, the potential energy ...


5

The simplest and earliest example I know regarding the renormalization group idea is the following. Suppose we want to study some feature $\mathcal{Z}(\vec{V})$ of some object $\vec{V}$ which is in a set $\mathcal{E}$ of similar objects. Suppose that unfortunately this question is too hard. What can one do? The renormalization group philosophy is try to ...


5

This isn't in any way a complete answer but I do find it illuminating to see this behaviour in a numerical simulation. In particular we see the periodic flips that seem to appear out of nowhere. Note how angular momentum is temporarily stored in the $\omega_1$ and $\omega_3$ components of the angular velocity during the flips.


5

• Q1 (the first yellow boxed question in the OP): The current status of the "persistence problem" has been discussed by Ilyashenko in Complex length and persistence of limit cycles in a neighborhood of a hyperbolic polycycle (2014), see also Persistence Theorems and Simultaneous Uniformization (2006). As discussed on page 285 of the 2014 paper, the ...


5

There is also the classic Palis, de Melo: Geometric Theory of Dynamical Systems.


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