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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51 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
33 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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0answers
21 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} ...
2
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2answers
69 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
19 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
56 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
103 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} ...
2
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0answers
38 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
39 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
27 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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0answers
25 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
7 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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0answers
28 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
99 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 ...
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5answers
481 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
343 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
43 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 ...
7
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2answers
164 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 ...
3
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0answers
33 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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0answers
34 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
34 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
41 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)= ...
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1answer
57 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
11 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
68 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
27 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
votes
1answer
30 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
32 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 ...
0
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1answer
26 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
27 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
40 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 ...
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0answers
37 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 ...
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1answer
25 views

A sufficient condition that domain of solution of differential equation became $\mathbb R$

If $ f:\mathbb R^n\to \mathbb R $ be bounded and continous then differential equation $$x'=f(x)$$ has a solution with domain $\mathbb R$. outlook of proof : if the maximal domain of solution is ...
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0answers
42 views

No periodic solution using Bendixson's criterion and Global analysis.

Theorem: Let $Z:U\subset \mathbb{R}^2\rightarrow \mathbb{R}^2$ a $\mathcal{C}^1$ field defined in a simply connected set $U$. If $\mathrm{div} Z(x)\neq0$ for all $x\in U$, then $Z$ does not have any ...
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1answer
38 views

Dynamical System , Series : can't find the general terms

I have a dynamical system defined as follow : $$V_{n+3} - 6V_{n+2} +12V_{n+1} - 8V_n = 8, ~ \mbox{with}~ V_0=V_1=V_2=1$$ I have to find $V_n$ = ? So I began by solving this equation : $$x^3 -6x^2 ...
0
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0answers
38 views

Homeomorphism between the group of $S(O)_{2}$ and the $S_1$.

During an exam I had to prove the following: "Let there be a dynamical system of $n=2$ dimensions and let the eigenvalues that correspond to it, to be imaginary with their real part equal to zero. ...
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0answers
19 views

Coupled Discrete Dynamical Systems in Complex plane.

Consider two dynamical systems $$Z_{n+1}=f(W_n, Z_{n-1})$$ and $$W_{n+1}=f(Z_n, W_{n-1})$$ where $z_0, w_0,z_{-1}, w_{-1}$ are given. The function $f, g$ are defined from $\mathbf{C}^2$ to ...
0
votes
1answer
27 views

Behavior of Non-Hyperbolic Equilibria?

So I'm working on a differential equation problem concerning epidemics - we're using the Kermack-McKendrick model. I've reached a point where I need to sketch phase portraits near my equilibria, ...
1
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1answer
45 views

Nature of Equilibrium Points

I would like to prove the following: "The nature of the equilibrium points (i.e. stability/instability) of a one-dimensional differential equation remains invariant under the effect of the ...
1
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0answers
29 views

Cesaro bounded.

The exercise is from Ulrich Krengel's book, Ergodic Theorems, on pages 173-174. First preliminary notions: a function $h$ with $T^*h=h$ is called harmonic, where $T$ is a contraction in $L_1$. $Y= ...
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1answer
19 views

Undamped Forces

I want to make sure I am doing this problem correctly, especially when it comes to drawing the potential function V(x). Consider the system of differential equations: $$\dot {x}=y$$ $$\dot ...
2
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2answers
95 views

Closed orbits of vector fields under perturbation

Consider a vector field $V$ on an annulus $U$, say. Also, assume that the vector field $V$ has a closed orbit. I am looking for a reference that gives stability results of the following type: If the ...
0
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1answer
36 views

Stable and unstable manifolds of fixed points

I want to make sure I understand the definition of these terms. If someone could correct me or let me know if I am right I would appreciate it. The stable manifold of a fixed point is the set of ...
0
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0answers
26 views

Gronwall's inequality for LTI scalar systems with an input

Suppose we have a dynamical system such that $$\dot{x}(t)\le-\alpha x(t) + u(t),$$ with $\alpha>0$ for all $t\ge0$. Can we say that $$x(t) \le e^{-\alpha(t-t_0)}x(t_0) + \int_{t_0}^t ...
5
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0answers
58 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
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0answers
35 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same ...
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1answer
63 views

Question about having periodic solution.

Assume $a>0$ and $b>0$ and $g(x)=0$ when $|x|>1$ , $g(x)=k$ when $|x|\le1$ . Now show that in system of differential equation $$x'=y $$ $$y'=-[2b-g(x)]ay-ay^2$$ if ...
1
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0answers
47 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
1
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1answer
34 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means ...
0
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1answer
25 views

Omega limit set of omega limit set $\omega(\omega(a))$

Consider a dynamical system with a flow $\phi(t;a)$, and let $A\subset \mathbb{R}^n$. The omega limit set of $A$ is defined as the union of all $\omega(a)$ over all $a\in A$. Since for a given $a$, ...