New answers tagged dynamical-systems
1
vote
Accepted
How can we know the $f$-invariant measures?
You can show easily (say by induction) that
$$
f^n(v)=\frac{A^nv}{|A^nv|}
$$
for each $n\in\mathbb N$. Writing $v=(x,y)$ we have
$$
A^nv=\begin{pmatrix}
a & 0 \\
0 & 1/a
\end{pmatrix}^nv=(a^nx,...
- 17.4k
0
votes
References on Kolmogorov-Sinai (measure-theoretic) entropy and convergence of measures
In his paper "INVERSE LIMITS, ENTROPY AND WEAK ISOMORPHISM
FOR DISCRETE DYNAMICAL SYSTEM" James R. Brown shows that entropy respects inverse limit of measure-preserving systems.
- 484
3
votes
Accepted
Other invariant measures than Lebesgue measure?
Hint: If $f:X\to X$ is a function and $x\in X$ is such that $f^p(x)=x$ for some $p\in\mathbb{Z}_{\geq1}$, then the average of Dirac measures
$$
\dfrac{\delta_x+\delta_{f(x)}+\cdots+\delta_{f^{p-1}(x)}}...
- 11.1k
1
vote
Accepted
Is there a source out there which works out a Hartman-Grobman-type theorem on a manifold?
But, of course, it is a purely local result. For manifolds, see for instance Ch. 2, Theorem 4.1 in
Palis, Jacob jun.; de Melo, Welington, Geometric theory of dynamical systems. An introduction. Transl....
- 103k
1
vote
Equation for the Logarithmic Spiral from a vertex to the Brocard Point
Fig. 1 : With the triangle taken as second example in the answer given by @disgraced. I have an error in my code because there should be an agreement on the curves. I will attempt to spot it and ...
- 84.5k
1
vote
References on Kolmogorov-Sinai (measure-theoretic) entropy and convergence of measures
This is nowhere close to a full answer, but a place you can read about this is in Peter Walters book, "An introduction to ergodic theory". Specifically, he discusses in chapter 8.1 ...
- 7,930
4
votes
Accepted
Equation for the Logarithmic Spiral from a vertex to the Brocard Point
Given $\triangle ABC$, you first compute the Brocard angle $\omega$ from the equation:
$ \cot \omega = \cot A + \cot B + \cot C $
Next, if you scale $\triangle ABC$ by a scale factor $\alpha \lt 1$ ...
- 25k
0
votes
Compute the number of ways the frog can move from A to B.
Alternative approach, which is based on partitioning an integer.
To partition $~5~$ into terms all less than 3:
P5a: 2 - 2 - 1
P5b: 2 - 1 - 1 - 1
P5c: 1 - 1 - 1 - 1 - 1
To partition $~6~$ into ...
- 38k
0
votes
Compute the number of ways the frog can move from A to B.
This is not a 3D DP problem (in an algorithm sense) given the constraint that the frog can't move $k$ steps in the same direction consecutively ($k=3$ in this case), because the state of the frog is ...
- 36
2
votes
Accepted
What are the $f$- invariant measures?
Not a complete answer. Just calculating the fixed points for $f$ in case $A=\left( \begin{smallmatrix} a&0\\ 0&1/a \end{smallmatrix} \right)$.
When
$f\left(\begin{smallmatrix} x\\ y \end{...
- 1,128
2
votes
Accepted
Find all $\tilde{f}$-invariant measures on $S^1.$
Before starting, let me point out that it may be an impediment to insist on identifying the one point compactification of $\mathbb R$ with $S^1$. Instead, keep in mind that they are homeomorphic; ...
- 126k
4
votes
The origin of the name homological equation
Almost ten years after asking this question, I ended up stumbling upon an attempted answer sort of by chance. It comes from the book Normal Forms and Unfoldings for Local Dynamical Systems, by James ...
- 486
2
votes
Accepted
Show that the only $f$- invariant probability measure is the delta measure at $0.$
Let $\mu$ be a Borel probability measure on $X$ and suppose that $\mu$ is $f$-invariant.
By Poincare recurrence, for any Borel set $E\subset X$, we have that
$$\mu\bigg(\bigcup_{N=1}^\infty\bigcap_{n\...
- 15.8k
1
vote
Accepted
If $\frac{d}{dt}f(x(t))=-|\nabla f(x(t))|^2$,$x(t)$ is bounded,standard arguments give that $\nabla f(\xi)=0$ at every accumulation $\xi$ of $x(t)$
The dynamical system that is being considered here is typically called a gradient system. You can see the reason for this name by applying the chain rule on the time derivative of $f(x(t))$ (here and ...
4
votes
Compute the number of ways the frog can move from A to B.
An elementary (if tedious and error prone) method.
Note: I am including this at the OP's request. In practice, I would automate the search...the pencil and paper method is a bit too error prone to be ...
- 72.5k
6
votes
Compute the number of ways the frog can move from A to B.
The bivariate generating function for the words on the alphabet $\{ L, U \}$ not containing the subwords $LLL$, $UUU$ is
$$
g(x,y) = \frac{1}{1-x-y + \frac{x^3}{1+x+x^2} + \frac{y^3}{1+y+y^2} }
$$
...
- 10.2k
0
votes
If $\frac{d}{dt}f(x(t))=-|\nabla f(x(t))|^2$,$x(t)$ is bounded,standard arguments give that $\nabla f(\xi)=0$ at every accumulation $\xi$ of $x(t)$
The following is a counterexample.
Let $f(x) = |x|^2/2$, $x(t)=e^{-t}x_0$, where $x_0$ is a non-zero element of $\mathbb{R^n}$. Then,
$x(t)$ is defined on $[0,\infty)$ and is bounded. The closure of ...
- 320
2
votes
Accepted
Unique ergodicity
Consider the discrete space $\{1,2,3,4\}$ and let $T$ be the cyclic permutation $(1,2,3,4)$, i.e. $T(1) = 2$, $T(2) = 3$, $T(3) = 4$, $T(4) = 1$. It is easy to see that this is uniquely ergodic for ...
- 458k
3
votes
Accepted
Convergence of a differential equation as $t \to \infty$.
This solution does not exist for $x = \frac{1}{2}\ln 3$, so analyzing the limit as $x\to\infty$ does not tell you anything meaningful about the solution behavior when considering the IVP starting at $...
- 7,092
1
vote
Topological Conjugacy of Arnold's Cat Map
When stuck, an important method in dynamics is to try to prove something stronger. In this case,
$$\Phi(x,y) = (x-b,y+b-a) \mod 1$$
is an affine automorphism of the torus such that $\Phi\circ g = f\...
- 11.1k
0
votes
Why does the output of a stable LTI system approach the input mathematically speaking?
Think of of the Laplace Transform applied to the ode with null initial conditions and $n \ge m$.
$$
G_a(s)\hat y = G_b(s)\hat x\Rightarrow \hat y = \frac{G_b(s)}{G_a(s)}\hat x
$$
Now if the zeros of $...
- 34.7k
0
votes
Why does the output of a stable LTI system approach the input mathematically speaking?
Assuming that the input $x$ is very slowly moving relative to the real parts of the eigenvalues (and their implied decay rates or half-life), then the excitation of the left side from time step to ...
- 129k
1
vote
Accepted
Proof of Lemma 4.1.3 in 'Translation Surfaces' by Masur and Athreya
You seem to be right; simply merging the last two sentences and editing as follows seems sufficient: "... or $q'$ does, in which case $p$ returns to $\beta$ at time $t_1-t_0$."
For reference ...
- 11.1k
1
vote
Accepted
Incorrect use of Bendixson-Dulac theorem?
Your proof looks correct to me. The system does not have a limit cycle as the trajectories are not bounded.
I looked over the paper to see if I could find the error.
The paper's authors argue as ...
- 907
Top 50 recent answers are included