In dynamical systems, the motion of a particle in some geometric space, governed by some time dependent rules, is studied. The process can be discrete (where the particle jumps from point to point) or continuous (where the particle follows a trajectory). Dynamical systems is used in mathematical ...

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16 views

A M-system is a E-system?

(X,T)is a transitive system, if the minimal point is dense in X,then we call (X,T) is a M-system. if there exist a full measure m(i.e. supp(m)=x )and m is a T-invariant measure,then we call (X,T) ...
2
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37 views

suggestion for lyapunov function

Consider differential equation \begin{align}x'&=-t(x+y)\\ y'&=-y+x-y(y^2-6).\end{align} Can some one suggest a lyapunov function for it. I have examined $V(x,y)=x^2+y^2$ , ...
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15 views

How to find the value of a parameter such that the map has a period-doubling bifurcation?

For example: $f(x)=x_{n+1}=\mu+x_n^2$. Is it when $|f'(x^*)|=1$, where $x^*$ is a fixed point of the system? In this case, $\mu=1/4$? Also how to determine whether it is supercritical or ...
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1answer
42 views

To show solutions of a linear system lie on parabolas in phase space.

Given a linear system $\dot{x}=x$ $\dot{y}=2y$ To show solutions of a linear system lie on parabolas in phase space. Which solutions (if any) do not lie on parabolas? It is the second question ...
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35 views

Prove that if $\lambda_j$ are the eigen values of $Df(\bar x)$, and if $\lambda_j<1$, then $\bar x$ is assymptotically stable.

We study the discrete dynamical system in $\mathbb{R^n}$ with differentiable function $f(x)$: $$x_{n+1}=f(x_n)$$ $1.$ Assume that $\bar x$ is a fixed point and consider small perturbations around ...
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1answer
56 views

if $f$ is weakly mixing then $f^n$ is ergodic?

if $f$ is weakly mixing then $f^n$ is ergodic?I think this is false but I cant find a counter example because I dont know transformations weakly mixing but not mixing.can you prove or give a ...
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12 views

Expressing function as superposition of suitable functions

I came across the following paragragh in the paper entitled "simulation of Power-Law Relaxations by Analog circuits: Fractal Distribution of Relaxation Times and Non-integer Exponents" In linear ...
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1answer
49 views

Bi-infinite sequences of $0$'s and $1$'s

Consider the set $\Sigma=\{0,1\}^\mathbb{Z}$, i.e. the space of bi-infinite sequences of 0's and 1's, and the left-shift $\sigma:\Sigma\to\Sigma$. Define a distance in $\Sigma$ as follows: ...
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1answer
33 views

Differential system in 4 nonlinear equations

I have some problems in solving the following differential system. To simplify notation, I write $x$ for $x(t)$ and $x'$ for $\frac{\text{d}}{\text{d} t} x(t)$ $$ \begin{cases} w'=w^{a+1} & \\ ...
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14 views

Estimates for iterations of unimodal maps

I've been trying to read Jakobson chapter on Ergodic Theory of One-Dimensional mappings (Encyclopaedia of Mathematical Sciences Volume 2, 1989, pp 179-199) and a I have a small question about one of ...
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2answers
102 views

Proving a function is chaotic on an interval

I'm self studying from the book 'First course in chaotic dynamical systems' and am having a hard time grasping how to prove that a function is chaotic. For the function we have $T(x) = 2x$ for $x ...
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1answer
19 views

Is $T:(x,y)\mapsto(x+\alpha, y+x)$ mod $1$, expansive on $\mathbb{R}^2 / \mathbb{Z}^2$?

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha, x+y\right) \mod 1 $$ One ...
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1answer
47 views

Show that $S= \{ \left(\frac{i}{k},\frac{j}{nk} \right) : 0 \leq i < k, 0 \leq j < nk \} $ is an $(n,\epsilon)$-spanning set

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) ...
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1answer
15 views

Show that $T^n(x,y)=\left(x+n\alpha \mod 1, y+nx+\frac{n(n-1)}{2}\alpha \mod 1 \right)$

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$. $$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) ...
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1answer
48 views

Show that $\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $ using Birkhoff Ergodic Theorem

Show that for Lebesgue-almost every $x \in [0,1)$, the geometric mean $$\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $$ exists and has common value. What is this? (no ...
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1answer
46 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
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26 views

Conditions to guarantee unique limits of trajectories.

For a real function $f$ on $\mathbb R^n$, such that no trajectories of the gradient escape to infinity, what are necessary and/or sufficient conditions so that each trajectory limits to a unique point ...
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2answers
36 views

Show that $ \lim_{n \rightarrow \infty} \frac{|f(T^n(x))|}{n}=0 $

Let $T: (X, \mathcal{A},\mu) \rightarrow (X, \mathcal{A},\mu)$ be ergodic wrt a measure $\mu$ on $(X,\mathcal{A})$. Show that for any $f \in L^1(X,\mathcal{A})$ and $\mu$-almost every $x \in X$ we ...
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1answer
59 views

Show that $Per_n(f)$ of periodic points of period $n$ is finite

Prove that if $f: X \rightarrow X$ is an expansive topological dynamical system of a compact dynamical system $X$, then the set $Per_n(f)$ of periodic points of period $n$ is finite. Any ideas of how ...
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1answer
34 views

A polynomial equality for the square of a self-adjoint positive contraction in $L^2$ — from Krengel's book Ergodic theorems

Another mystery from Ulrich Krengel's textbook - Ergodic Theorems (first mystery). This time it's from page 190, in the proof of theorem 2.7. He takes $P=T^2$, where $T$ is a self-adjoint positive ...
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23 views

Normal forms of vector fields

Consider the $m$-parameter family defined by \begin{align} \left(\begin{array}{c}\dot{x}_1\\\dot{x}_2\end{array}\right)&=J\textbf{x}+ \left(\begin{array}{c} ...
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2answers
73 views

Actual Classification re Nielsen-Thurston Theorem (how to)?

according to Nielsen -Thurston Classification: http://en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston_classification If $S$ is compact and orientable surface, then any homeomorphism is isotopic to ...
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0answers
21 views

Question about Chebyshev Polynomials in Beardon

I happen to be reading through Beardon's book, Iteration of Rational Functions, and I have come across a statement I don't quite believe. He uses it a little later on, so I'm concerned with clearing ...
2
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1answer
65 views

Prove there is no an Analytic Centre Manifold

I must prove that the differential equation below does not have an analytic centre manifold: $$ \dot{x}=x^3, \dot{y}=2y-2x^2 $$ I try: The linearisation of the system at the origin is: ...
2
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1answer
121 views

Solutions of a periodic non-autonomous system

I must find solutions for the system $$ \left( \begin{matrix} \dot{x_1}\\ \dot{x_2} \end{matrix} \right) = \left( \begin{matrix} \cos(t) & -\sin(t)\\ \sin(t) & \cos(t) \end{matrix} ...
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41 views

Wronskian and roots

I am given an ODE $$-y''(x) + q(x) y(x) = \lambda y(x),$$ and let $y_1,y_2$ be two solutions to this ODE on $[a,b]$ to two different values $\lambda_1 \neq \lambda_2$(on the right side of this ...
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1answer
41 views

Linear dynamical systems

Show that, if the real system $$\dot{x}=\left(\begin{array}{cc}\alpha&-\beta\\\beta&\alpha\end{array}\right)x$$ is diagonalised over the complex numbers $\mathbb{C}$, such that the ...
2
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1answer
28 views

Transformations of diffeomorphism $f(z)=e^{i\alpha}z+\overline{z}^3+z^2\overline{z}$ that eliminates $\bar z^3$

Find a transformation of the form $z=w+a\overline{w}^3$ such that $$f(z)=e^{i\alpha}z+\overline{z}^3+z^2\overline{z}$$ where $\alpha\neq2\pi p/q,\ q=1,2,3,4,$ becomes ...
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26 views

Change of Variables_Dynamical Systems

I have to show that the vector field $$X(x)= \left( \begin{matrix} x_2+ax_1^2+bx_1x_2+cx_2^2 \\ dx_1^2+ex_1x_2+fx_2^2 \end{matrix} \right) $$ is transformed to $$\tilde{X}(y)= \left( \begin{matrix} ...
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0answers
9 views

unboundedly finite-to-one factor map between transitive topological dynamical systems

$\pi: X \to Y$ is called a factor map from a topological dynamical system $(X,T)$ (a compact metric space together with a self homeomorphism $T$) to $(Y,S)$ if it is continuous, onto, and commutes ...
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38 views

Finding the index of a linear vector field at the origin

For the linear vector field $f(x,y) = (f_1, f_2) = (ax+by,cx+dy)$, show that the index with respect to the origin is $\pm 1$ depending on whether $ad-bc > 0$ or $ad-bc < 0$. I've gone ...
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0answers
116 views

Dynamic programming problem for discrete linear time varying system

I'm working on a linear time varying discrete(LTV) multi input multi output(mimo) system. I formulate the problem description in the following way $$x_i(k+1) = x_i(k)\cdot A_i(k) + B_i(k)\cdot ...
37
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5answers
505 views

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of ...
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4answers
360 views

prove conjecture; the limit of iterating is $\sqrt{z^2 - 2}$

$$\lim_{n \to \infty} f_n(x)=x-\frac{1}{nx}\;\;\; g(x)=f_n^{on}(x)$$ The conjecture is for values of $|x|>\sqrt{2}$: $g(x) = \sqrt{z^2 - 2}$ This question comes from another matstack ...
3
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1answer
61 views

Is this a correct application of Poincaré-Bendixson?

Consider a non-vanishing $C^1$ vector field $f$ on a neighbourhood of the annulus with radii $1$ and $2$ in $\mathbb{R}^2$. The vector field is transversal to the boundary of the annulus and points ...
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2answers
183 views

eventually constant maps

Let $f:[0,1]\to [0,1]$ be a continuous function with a unique fixed point $x_{0}$ Assume that $\forall x\in [0,1], \exists n\in \mathbb{N}$ such that $f^{n}(x)=x_{0}$. Does this implies ...
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0answers
34 views

Equality of measure sets of dynamical system

This is a homework question I have been crunching my brains on for a lot of time, but unfortunately I'm stuck. I would greatly appreciate any help! The problem is as follows: We have some continuous ...
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40 views

Stadium billiard reflection angles

Given a boundary and a massless particle with constant velocity with a certain direction, a billiard consists of an experiment where the particle collides with the walls conserving its velocity ...
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1answer
44 views

singularity and degeneracy of an ODE

I have trouble distinguishing the difference of singularity and degeneracy in the context of ODE theory. Could anyone give me a couple of examples in illustrating the difference of singular point and ...
2
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1answer
67 views

Definition of Markov partition?

My teacher handed out an excerpt from a book by Robinson on chaotic dynamical systems. The excerpt is from a chapter on Markov partitions, and the following part has me confused: Let $$f(x)= ...
4
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1answer
59 views

periodic solution of $x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$

Assume differential equation $$x''-\ (1-\ x^2-\ (x')^2)\ x'+x=0$$ I want to discusse about non-constant periodic solution of it. Can someone give a hint that how to start to think. And does it have ...
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0answers
13 views

Locally Linear Systems-repeated $\lambda$

For a locally system whose corresponding linear system has repeated eigenvalues, the type of equilibrium point cannot be determined. I know that the locally Linear system equilibrium can possibly be a ...
2
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1answer
77 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
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1answer
33 views

Proving F is an Integral of the Linear Map L

In the question, I'm asked to show that \begin{align*} F\begin{pmatrix}x\\y\end{pmatrix}=x^2+y^2 \end{align*} is an integral for the linear map \begin{align*} L(\text{x})= \begin{pmatrix} ...
2
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1answer
36 views

“Evenly” dense orbit?

I want to prove the following: let $a$ be an irrational constant and $m$ an integer. Then $$\lim_{n \to\infty} \frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi m i (x+ka)} = \begin{cases} 0, & m\not=0 \\ 1, ...
3
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0answers
33 views

Poincare map trouble

Consider $ X' = F(X)$, $F \in C^1(\mathbb{R}^2)$. Suppose that the system has an orbit $\mathcal{O}_p$ and $\Sigma$ an transversal section in $P$. Show that if $$\pi^{n+1}(\Sigma) \subset ...
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1answer
32 views

About the Sharkovsky Forcing Theorem

(Sharkovsky Forcing Theorem ). If $m$ is a period for $f$ and $m⊲ l$ , then $l$ is also a period for $f$. I have the following question: Let $f$ be a such map having a period three, So $f$ is ...
0
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0answers
33 views

Orbit closure is uncountable, unless there is a periodic element.

Let $a = (a_i)_{i \in \mathbb N}$ be a sequnece over some finite alphabet $\Sigma$. We may define on the space $X = \Sigma^{\mathbb N}$ a shift operation by $(Sx)_i = x_{i+1}$. Let $A$ be the orbit ...
0
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1answer
48 views

Lemma 1.7 in chapter 5 in Ergodic Theorems by Ulrich Krengel.

The lemma is on page 181. Here's the lemma and its proof: I don't understand how did they get: $$\lim | S^{n_k+1}f(\omega)|\leq \gamma (\alpha - \epsilon)+(1-\gamma)\alpha$$ The term that is ...
1
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0answers
38 views

Reducing a system of differential equations

Let $\mathbf F$ be a system of 1st order differential equations in $n>3$ variables $$\mathbf{F} : \mathbb{R}^n \to \mathbb{R}^n$$ $$\frac{d\mathbf{u}}{dt} = \mathbf{F}(\mathbf{u})$$ such that ...