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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93 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
27 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
91 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: $$\dot{x}=DX(...
2
votes
1answer
137 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} \right) \...
1
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0answers
57 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 ...
0
votes
1answer
46 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
votes
1answer
31 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 $$\tilde{f}(w)=e^{i\alpha}w+w^2\...
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0answers
60 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 through ...
3
votes
0answers
197 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 u_i(k)$...
40
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5answers
728 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 $f(x)$...
6
votes
4answers
396 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 question/...
3
votes
1answer
170 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 ...
8
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2answers
208 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
votes
0answers
40 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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1answer
70 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 ...
3
votes
1answer
192 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
65 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 ...
2
votes
1answer
113 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 ...
0
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1answer
37 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} 0&1\\-1&...
2
votes
1answer
46 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
votes
0answers
40 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 \pi^{n}(\...
0
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1answer
52 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 chaotic....
0
votes
1answer
59 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
43 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 $\...
1
vote
1answer
29 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 $(...
1
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0answers
56 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 ...
0
votes
1answer
40 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
votes
0answers
39 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. ...
2
votes
1answer
51 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
vote
1answer
85 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
40 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= \...
-1
votes
1answer
26 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 {y}=-x+x^...
2
votes
2answers
155 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
votes
1answer
104 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 ...
7
votes
0answers
171 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 ...
1
vote
0answers
77 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 center). ...
0
votes
1answer
77 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 $k<2b$ ...
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vote
0answers
51 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}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
1
vote
1answer
46 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 $$K=\bigcup_{i=1}^...
0
votes
1answer
46 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 $a$, $\...
1
vote
1answer
100 views

Closed orbits of dynamical systems

Consider the system $$\dot{x}=x-rx-ry+xy, \qquad \dot{y}=y-ry+rx-x^2,\qquad r=\sqrt{x^2+y^2},$$ which can be written in polar coordinates $(r,\theta)$ as $$\dot{r}=r(1-r), \qquad \dot{\theta}=r(1-\cos ...
4
votes
1answer
137 views

Chaos in Newtons Method

Im trying to prove that Newtons method applied to ${\rm f}\left(\, x\,\right) =x^{2} + c$, is chaotic for $c > 0$. I know I need to prove: (a) The periodic points of ${\rm f}$ are dense in $X$, ...
1
vote
0answers
61 views

Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\...
2
votes
1answer
85 views

Particle under central force (Dynamics)

So I have this question: There is a particle moving in response to a central force per unit mass of $$F(r) = {\alpha\over r^2} + {\beta\over r^3}$$ where $\alpha$ and $\beta$ are constants. Initially ...
0
votes
1answer
63 views

Hysteresis Model with two real parameters

I would like to ask the following: I am trying to make a throughout analysis of a Hysteresis model in one dimension, with two real parameters: $\frac{dx}{dt}=f(x,ν,μ)=νx-x^3+μ$, where $ν,μ\in{\...
3
votes
0answers
137 views

Find all sinks/sources/saddles for a certain diffeomorphism

I'm trying to do the following exercise from Devaney's Introduction to Chaotic Dynamical Systems, exercise 2.6.1. The problem is this: Consider the diffeomorphism $Q_\lambda$ of the plane given by ...
2
votes
1answer
137 views

Bifurcation Example Using Newton's Method

I am studying dynamical systems as part of a research project. I have been using Newton's Method and studying the dynamic properties. Does anyone know where I could find a relatively simple example ...
1
vote
1answer
43 views

Show there is only one trajectory passing through each point

I have to show the following: Let $\varphi$ be a flow on the manifold M and suppose that that the orbits {$\varphi_t (x_0)$} & {$\varphi_t (x_1)$} intersect. Prove that the orbits coincide.
0
votes
1answer
230 views

What is the difference between disturbance and noise for dynamic systems

In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) \...
0
votes
1answer
302 views

Resolvent matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...