3
votes
0answers
48 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
1
vote
2answers
39 views

Regarding iterated maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$, and strictly decreasing norm

Consider a map $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, and the iteration $(x,y) \mapsto g(x,y)$. Say that the origin is an asymptotically stable fixed point and there is a region $A$ in the first ...
3
votes
1answer
42 views

Can we construct a Koch curve with similarity dimension $s\in[1,2]$?

We can make a Koch curve $K$ with similarity dimension $s\in \mathbb Q \cap [1,2]$ by writing $s=\frac{p}{q}$, and constructing such a generator that by scaling with the factor of $2^q$, we'd find ...
2
votes
3answers
135 views

What is the topological dimension of the Peano curve?

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it's on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the ...
1
vote
0answers
43 views

Continuation fixed points of parameter dependent Newton

Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in ...
2
votes
1answer
47 views

Why do we require a finite number of subsets for self-similarity?

Here is how my text defines self-similarity: We call $M \subset \mathbb R^d$ self-similar if there are $T_1, \ldots, T_m \subsetneqq M$ and similarity maps $\alpha_1, \ldots, \alpha_m$ such that ...
6
votes
1answer
131 views

Confluent Heun equation. Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
15
votes
1answer
423 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2-y'(x)+C = -A \cos(x) - B \cos^2(x),$$ where $C, A$ and $B $ are parameters. (The case that either $A$ or $B$ ...
2
votes
1answer
102 views

Properties of this Schrödinger equation / Sturm-Liouville problem.

Given the ODE $$\Psi''(\theta) + \eta \cos(\theta) \Psi(\theta) + \xi \cos^2(\theta) \Psi(\theta)= \lambda \Psi(\theta),$$ where $\theta \in [-\pi,\pi]$, $\Psi(-\pi)= \Psi(\pi)$ and $||\Psi ||_{L^2} = ...
0
votes
0answers
23 views

measurability restriction operator

Let $M\subset \mathbb{R}^k$ compact. For every $x\in M$ we define $L(x): \mathbb{R}^m \rightarrow \mathbb{R}^m $ a linear isomorphism Let $G_n (\mathbb{R}^m)=\{ W: W\ \mbox{is subspace of} \ ...
0
votes
1answer
26 views

Order in taking limits?

So I am reading this physics paper that they define Kolmogorov entropy for dynamical systems as follows: $$K=\lim\limits_{\epsilon\to 0}\lim\limits_{T\to \infty}\frac{I(\epsilon,T)}{T}$$ They ...
4
votes
2answers
231 views

Proving Nonhomogeneous ODE is Bounded

I am trying to prove the following: Given a solution $x(t)$ of the IVP $\dot x=A(t)x+h(t)$, where $A(t), h(t)$ are continuous on $0<t<\infty$, prove that x(t) is bounded for $t\ge1$ if both ...
0
votes
0answers
16 views

Looking for methods/results for explicitly bounding iterations of rational functions

This is a cross-post of http://mathoverflow.net/questions/155775/looking-for-methods-results-for-explicitly-bounding-iterations-of-rational-funct But I received no answer there to the actual ...
3
votes
0answers
18 views

inequality involving lifts of a positive oriented homeomorphism of the circle

Let $\pi: \mathbb R \to S^1$ be the natural projection and let $f:S^1 \to S^1$ be a positive oriented homeomorphism. We say that $F: \mathbb R \to \mathbb R$ is a lift of $f$ is $\pi \circ F = f \circ ...
1
vote
0answers
24 views

Stability and Asymptotic Stability of Rational Matrix Solutions

If $X(t)$ is a fundamental matrix solution of $\dot{x}=A(t)x$ on $a<t<\infty$ and suppose the entries of $X(t)$ are rational functions of the variable t in the form $x_{ij}=p_{ij}(t)/q_{ij}(t)$. ...
1
vote
1answer
33 views

About dependence on initial conditions

I need some help with this question relating periodic points and dependence on initial conditions: $\bullet\;$Let I be a real interval, and $g:I\to I$ a derivable function. If $x_0$ is an ...
5
votes
2answers
296 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...
0
votes
0answers
55 views

Bounding Solutions of System of Inhomogeneous Equations using Gronwall Inequality

I am trying to use Gronwall's Inequality to show that for the system $x'=A(t)x+g(t)$, $x(1)=\left( \begin{array}{c} 1\\ 1\\ \end{array} \right)$, where $A(t)=\pmatrix{ \frac{1}{2t^4} & ...
0
votes
1answer
124 views

About the tent map and eventually fix points

I need some help with this question: Given the tent map, $T:[0,1]\to[0,1]$ defined as $T(x)=2x$ if $0\leq x\leq\frac{1}{2}$ and $T(x)=2-2x$ if $\frac{1}{2}\leq x\leq1$, I'm trying to check that the ...
0
votes
1answer
55 views

Proving a flow of a dynamical system is complete

I am trying to prove that the flow (defined as a mapping from R^n to R^n) of a dynamical system (a R^n system with first order derivatives) is complete. However I am stuck after I have proven that a ...
1
vote
1answer
107 views

Local stability + global attractivity = global asymptotic stability?

I was wondering how could I prove such a property stated in [Angeli, 2004]. For instance, consider the system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is Lipschitz continuous. Claim. ...
1
vote
1answer
36 views

Mapping one integral curve onto another

Let $v$ be the vector field $\sum_{i=1}^n x_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be the projection map $(x_1,\ldots,x_n)\rightarrow x_1$. Show that $v$ ...
0
votes
0answers
23 views

Construction of time flow with three fixed points

I'm trying to construct a time flow $f:\mathbb{R}\times [-1,1]\to [-1,1]$ (ideally smooth) such that $-1,0,1$ are fixed points (that is $f(t,x)=x$ for all $t\in\mathbb{R}$ and $x=-1,0,1$) of which ...
4
votes
0answers
36 views

One-parameter group for spheres

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
4
votes
1answer
302 views

One-parameter group of diffeomorphisms generated by vector field

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
2
votes
0answers
65 views

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...
0
votes
1answer
48 views

How to prove that a given set is strongly invariant?

Consider a dynamical system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is of class $\mathcal{C}^1$. For any time $t$, denote a solution issuing from $x$ by $X(t,x)$. Consider the proper ...
0
votes
1answer
63 views

Flow, dynamical systems

We have $f:\mathbb{R}^N \to \mathbb{R}^N$ lipschitz and $x(\cdot ;x_0 )$ is the unique solution of $$x'(t)=f(x(t))$$ $$x(0)=x_0$$ Every solution of this ODE is global. Show, that $\Phi_t ...
1
vote
1answer
42 views

A set of functions which is open in the space $C^1[0,1]$

Let $f:[0,1]\to [0,1]$ be a $C^1$ and increasing function such that $i)$ If $f(p)=p$ then $|f'(p)|\ne 1$ I want to prove that there exist an $\varepsilon>0$ such that if $g\in C^1$ and ...
0
votes
1answer
65 views

If $F:\mathbb{R}\to\mathbb{R}$ is a lift of the circle homeomorphism $f$, show that $F^n$ is a lift of $f^n$ for every $n\in\mathbb{Z}$.

By a lift of a circle homeomorphism $f$, I mean that $F:\mathbb{R}\to\mathbb{R}$ satisfies $\pi\circ F = f\circ \pi $ where $\pi$ is the natural projection $\pi:\mathbb{R}\to\mathbb{R}/\mathbb{Z}$. ...
0
votes
1answer
70 views

Showing the limit of a flow must be an equilibrium point under certain restrictions.

I'm stumped on how to approach this one: Consider the autonomous ODE $\dot{x} = f(x)$, $x \in \mathbb{R}^n$ with initial condition $x(0) = x_0$ and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ (at ...
5
votes
2answers
87 views

A continuous function that when iterated, becomes eventually constant

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and let $c$ be a number. Suppose that for all $x \in \mathbb{R}$, there exists $N_x > 0$ such that $f^n(x) = c$ for all $n \geq ...
4
votes
0answers
66 views

A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor ...
0
votes
0answers
72 views

functions taking in 2x, taking limit of some f(x) as 2x --> 0

Suppose $(X,d)$ is a compact metric space, with $f: X \rightarrow X$ continuous. For each $n \in \mathbb{N}$, the metric $d_{n}(x,y) = \max_{0 \leq k \leq n-1}d(f^{k}(x),f^{k}(y))$ measures the ...
5
votes
2answers
110 views

“Constrained” numerical solutions of ODEs with conservation laws?

Hi know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
3
votes
1answer
228 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
2
votes
2answers
200 views

Milk and Coffee will they ever finish

Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coffee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass ...
3
votes
2answers
91 views

Show the map $f(x)=\frac12 (x+1/x)$ has an attractive fixed point in $(0,\infty)$

From numerical test, I know $x=1$ is an attractive fixed point of the function $$ f(x)=\frac12 \left(x+\frac{1}{x}\right), $$ on $(0,\infty)$. Is there a way to prove it? Since $$ ...
4
votes
3answers
100 views

Repeated nested roots

Quite some years ago, I remember being asked the following question: Suppose $\alpha = \sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}$, what is $\alpha$. The solution was given by squaring $\alpha$ and solving ...
2
votes
0answers
128 views

Bernoulli shift on $S^\mathbb{Z}$

Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
1
vote
0answers
47 views

Monotonicity of Poincaré's Map

thanks for reading. Consider a one-dimensional dynamical system $\dot{x} = f(t,x)$. Let's call $\phi(t,t_0,x_0)$ the solution passing through $x_0$ at time $t_0$ (where $t$ is the time argument of ...
2
votes
1answer
232 views

Tent map: show that $x$ is a periodic point IFF it is a rational number of the form $\frac {m}{p}$ where $m$ is even and $p$ is odd

Let us consider the tent map: $f [0,1] \rightarrow [0,1]$ where $f(x) = 2x$ if $0\leq x \leq \frac{1}{2}$ and $f(x) = 2(1-x)$ if $\frac{1}{2}\leq x \leq 1$. Show that $x$ is a periodic point IFF it ...
2
votes
3answers
143 views

Dynamical Systems problem — a function of ONLY period 3

I am trying to construct an example of a function $f: [0,1] \rightarrow [0,1]$ such that it has a periodic point of period 3 and NO other periodic points. Any ideas? how can I even start envisioning ...
2
votes
0answers
65 views

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$?

Is the half iterate of $2\sinh(x)$ - expanded from fixed point $0$ - analytic near the real line $\mathbb{R}$? I know this function has 2 other fixed points apart from $0$, so I'm not sure. Also ...
4
votes
1answer
94 views

Cylinders and dyadic Intervals

Let be $\Sigma_{+}=\{0,1\}^{\mathbb{N}}$ the space of sequences of $0^{'s}$ and $1^{'s}$. Consider the following surjective application $\phi:\Sigma_{+}\to [0,1]$ given by $$ ...
3
votes
1answer
291 views

Prove that the orbit of an iterated rotation of 0 (by (A)(Pi), A irrational) around a circle centered at the origin is dense in the circle.

I think the title of the question says it all. I unfortunately did not seem to conclude anything. Some ideas I had: It is easy to show that (given $T$ is the rotation) $\{T^n(\theta)\}$ is a set of ...
4
votes
1answer
275 views

Method of isoclines

I have this exercise and I do not know how to solve it. By using the method of isoclines represent the integrals of equation corbes nonautonomous $x'=x^2-t$. There are some indications: Let $P = ...
3
votes
1answer
310 views

Convergence of fixed-point iteration for convex function

Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying $$ f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty. $$ I am confident to be able to argue that ...
1
vote
1answer
174 views

Implicit Function theorem and Bifurcation points

So let us say we have a function $\dot{x} = f(x,r)$ that has some critical point at $(x_0,r_0)$ such that $f(x_0,r_0)=0$. The question now is: when is this a bifurcation point? I understand that ...
3
votes
1answer
122 views

Solution space to a functional equation

This question comes from my attempts at understanding an example presented by Bill Gasarch on his blog. The example is of a continuous strictly increasing function whose derivative is zero almost ...