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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3
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+100

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...
0
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0answers
9 views

Is the conjugacy map between two distinct circle homeomorphisms unique?

Suppose $f,g,h_1,h_2$ are circle homeomorphisms with $f≠g$ and $fh_i = h_ig$ for $i=1,2$. Does it follow that $h_1 = h_2$? I restrict $f≠ g$ because I noticed that if $$f(x) := g(x) := R_\alpha(x) ...
0
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0answers
48 views
+50

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
7
votes
2answers
338 views

A proof of a theorem of Liouville

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

Convergence to a fixed point

When the following system is given: $x(k+1)=r-rx(k)$ where $r>=0 $ is a parameter Can someone explain why the fixed points are given by: $x(k+1)=x(k)=x^*$, so $x^*=\frac{r}{1+r}$? and how to ...
1
vote
2answers
26 views

$\lim\limits_{t\to\infty}t-x(t)=0\ ?$

Let $\displaystyle\cases{ x'=\frac{t-x}{1+t^2+x^2} & \cr x(1)=1 }$ be the Initial value problem, prove or disprove $\lim\limits_{t\to\infty}t-x(t)=0$ We've already proved that: for $t>1, ...
0
votes
0answers
31 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
4
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0answers
21 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
4
votes
2answers
66 views

Fun with Newton's Method - Infinitely many cycles

I'd like to preface this problem by saying that I have absolutely no clue if it is solvable or not. This is just the result of some musings, and I'm looking for either some guidance, or to be pointed ...
0
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0answers
8 views

Question on stable manifolds

If $x\in M$ is a hyperbolic fixed point of a diffeomorphism $\phi:M\to M$, then the stable manifold $$ W^s=\{y\mid \lim_{n\to\infty}\phi^n(y)=x\} $$ is the image of an injective immersion $$ ...
1
vote
1answer
44 views

All fixed points of a function are globally stable or unstable.

I am analyzing the function $\lambda \sin( \pi x)$ for $x \in [0,1]$ for a paper I am writing. I know that all fixed points of this function are either globally stable or unstable but I am not sure of ...
2
votes
1answer
23 views

Repairing solutions in ODE

Recently I encounter something interesting that I hope to hear from your opinions: Suppose we are given a ODE $\frac{dy}{dx}=y$, with no initial condition. Naively, we divide both sides by $y$ and ...
0
votes
1answer
32 views

About quadratic map

let us consider the following quadratic map: $$s_{n}=s_{n-1}²+c$$ $$(*)$$ There is several papers disscuting the dynamics of (*). I want to know the behavior of this map for $c=-2$ and I am asking ...
0
votes
1answer
28 views

How I can find a similar expression for $x₀>1/2$

For the logistic map http://mathworld.wolfram.com/LogisticMapR=2.html the formula (4) in the link is valid only for $x₀<1/2$. How I can find a similar expression for $x₀>1/2$. The same question ...
6
votes
2answers
115 views

Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as: $$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, ...
2
votes
2answers
71 views

What kind of bifurcation occurs for $\mu=-1$ for $f_\mu(x)=\mu+x^2$?

Let $f_\mu(x)=\mu+x^2$. What bifurcation occurs for $\mu=-1$? Pretty straight forward, but I'm having a hard time with this entire section in my book. It's not making any sort of sense and the ...
0
votes
1answer
38 views

Uniform Wiener-Wintner Theorem - proof

I am looking for proof of uniform version of Wiener-Wintner theorem: Let $(X, \mathcal{A}, \mu, T)$ be an ergodic measure preserving system. For $f \in L^1(\mu)$ which is orthogonal to the ...
1
vote
1answer
35 views

Problem with itinerary of a coding problem with infinite 1's

If $f(x)=2x \ mod \ 1$ on $[0,1)$. Then if we code $x \in [0,1)$ with its itinerary w.r.t. the partition $P_0=[0,1/2)$ and $P_1=[1/2,1)$. Can you show that there is no point $x$ whose itinerary has ...
1
vote
1answer
305 views

stable, center and unstable manifolds

‎let ‎‎$‎f:M‎\rightarrow ‎M‎$ ‎be a‎ ‎‎‎‎diffeomrphism and‎‎‎ ‎$ ‎\Lambda‎ $ a‎‎ ‎hyperbolic ‎set. ‎We ‎can ‎give a‎ ‎characterization ‎of ‎(local) ‎stable ‎and ‎unstable ‎manifolds ‎by‎: ‎‎for ...
0
votes
0answers
20 views

Conjugacy of linear systems with one zero eigenvalue

I have a question from Hirsch, Smale, and Devaney's "Differential Equations, Dynamical Systems and an Introduction to Chaos." Consider all linear systems with exactly one eigenvalue equal to 0. ...
3
votes
0answers
30 views

Sharkovskii's theorem in two dimensions?

Question A weak form of Sharkovskii's theorem in $1$D dynamical systems states that, if a continuous function $f:I\to I$ does not include a periodic point of least period $2$ on $I$, then ...
2
votes
1answer
35 views

1 dimensional flows and phase portraits

I have a flow defined by $\dot{x} = x-x^4+1 :=f(x)$. I need to sketch its phase portrait. Firstly, I have to find its fixed points, these occur at $f(x)=x$. So, $x^4=1 \Rightarrow x= \pm1$. Next, I ...
1
vote
1answer
28 views

Non-integrable systems

If a Hamiltonian system in $\mathbb{R}^{2n}$ has $n$ suitable first integrals, then it is called an integrable system, and the Arnold-Liouville theorem tells us all sorts of nice things about the ...
1
vote
0answers
27 views

Are stable manifold for gradient flows embedded submanifold?

Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image ...
0
votes
2answers
37 views

Derivative of a projective transformation

Assume $A$ is a matrix from $R^{n\times n}$, $A:R^n\rightarrow R^n$. Then $A$ induces a projective transformation $f:RP^{n-1}\rightarrow RP^{n-1}$. For example, $\\$ $$\begin{pmatrix} 4 & 0 ...
0
votes
0answers
31 views

Dynamical System Problem

If (X, f) and (g, Y ) are dynamical systems (with semigroup |N_0 lets say) and π : Y → X is a semiconjugacy, then periodic points for g are periodic for f. Give an example that the opposite is not ...
3
votes
1answer
44 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
0
votes
1answer
20 views

Find for which r this system converges to a fixed point

Given the following (discrete time) system $x(k+1)=r-rx(k)$ where $ r>=0 $ is a parameter Find for which $r>=0$ all solutions of this system converge to a fixed point Verify if there exist ...
0
votes
0answers
33 views

A question of integral from Krengel's book in Ergodic Theorems.

As the picture depicts, I don't understand how did he get the RHS of: $$\int_0^{2X(\omega)} t^{-1} \psi(dt) \leq m(\log^{+} 2X(\omega))^{m-1} \int_{0}^{ 2X(\omega)} t^{-1} dt$$ Presumably it ...
0
votes
0answers
8 views

Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
0
votes
2answers
316 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
0
votes
0answers
12 views

Basin of attraction

Let $$g(x)=\frac{2}{5}x^3-\frac{7}{5}x$$. The fixed points are 0 and $$\sqrt6$$. There is a period-2 orbit of 1 and -1. The critical points are $$\sqrt\frac{7}{6}$$ a. calculate the Schwarzian ...
0
votes
0answers
23 views

Stability of a fixed point

Determine the stability of all the fixed points of the following functions: So basically I understand how to find fixed points and whether they are attracting/repelling...but I am confused on how to ...
1
vote
1answer
23 views

Clarifying understanding of Poisson Brackets in Hamiltonian Dynamics

I'm just reading through my textbook and would like to clarify my understanding of 'Canonically related variables'. In my textbook, it says that if $Q_i$, $P_i$ are related to $q_i$, $p_i$ by a ...
1
vote
1answer
21 views

Show that the system is controllable (i.e. prove P has full rank)

Given the matrix: $$A = \begin{pmatrix}m&1&0&0&0\\ 0&m&1&0&0&...\\ 0&0&m&1&0&...\\ 0&0&0&m&1&...&\\ ...
1
vote
1answer
51 views

Mathematica Question regarding NSolve

I'm trying to solve for 2-cycles for the equation $$x=xe^{r[1-x]}$$ using Mathematica. I've tried using NSolve, FindRoot, and Solve. When I use NSolve I input it as $$\textrm{NSolve}[xe^{r[1 - xe^{r[1 ...
0
votes
4answers
97 views

On finding the equilibrium solutions to a system of differential equations

I am asked to find all equilibrium solutions to this system of differential equations: $$\begin{cases} x ' = x^2 + y^2 - 1 \\ y'= x^2 - y^2 \end{cases} $$ and to determine if they are stable, ...
0
votes
0answers
19 views

Why this operator is not fredholm?

Define $f:S^{2}\to \mathbb{R}$ by $f(x,y,z)=z$. Let $D:=D_{\nabla f}$. As I learned from the following post this operator is not counted as a fredholm operator.( I did not underestand, ...
1
vote
1answer
40 views

Newton Map in Dynamical Systems

Let $$p(x)=(x^2-1)(x^2-4)=x^4-5x^2+4$$. Let $$N(x)=Np(x)$$. Notice that N(x) goes to +-infinity at $$a1=-\sqrt{2.5}$$ $$a2=0$$ $$a3=\sqrt{2.5}$$ a. Sketch the graph of N(x). So the Newton map would ...
1
vote
1answer
36 views

Dynamical Systems: Transition Graphs

Let f be a continuous function defined on the interval [1,4]with f(1)=4, f(2)=3, f(3)=1, and f(4)=2. Assume that the function is linear between these integers. a. Sketch the graph of f b. Label ...
0
votes
0answers
105 views

Hopf bifurcation for a delayed DE

Consider the system of delay differential equations given by $$ \begin{array}{lcl} x^{\prime}(t) &=& (1+\alpha_1)y(t-\tau)-\alpha_2\alpha_3x(t)-\alpha_2(1-\alpha_3)z(t),\\ y^{\prime}(t) ...
0
votes
1answer
21 views

Ergodic behaviour for bounded random dynamical system

Considering an iterated system described by $$ X_n =\gamma_nX_{n-1} , $$ where $\gamma_i$ are non-negative i.i.d. variables. It is easy to show that the expectation will grow unbounded for $\mathbb ...
0
votes
0answers
52 views

Connection between the Cantor set and the Tent map $T_3$

I would like to prove part c) using the ternary expansion as shown in the second half of the image. I understand how to compute up to what is shown in the image. I am struggling to understand the ...
0
votes
0answers
17 views

Rotational invariant curve definition

I have a question about the definition of rotational invariant curve. The definition I have is the following "By an invariant curve for a twist map $F$ we mean a simple closed curve that is invariant ...
4
votes
1answer
271 views

Are continuous chaotic systems necessarily uncomputable?

I have seen the claim in a recent unpublished paper that chaotic dynamics are necessarily uncomputable. This follows, they argue, from the sensitivity to initial conditions shown in chaotic systems. ...
1
vote
0answers
26 views

How can I prove these two fields are locally topologically conjugated?

The problem is to prove that the fields $x'=x$ and $x'=x^3$ are locally topologically conjugated in the origin. I found that the corresponding flux for the first equation is $\phi(x_0,t)=x_0e^t$.The ...
3
votes
1answer
97 views

Definition of the higher dimensional mapping tori

This is proving harder to search for than I imagined. The usual definition of a mapping torus $\mathcal{M}_h$ associated to a homeomorphism $h\colon X\rightarrow X$ on a topological space $X$ is the ...
0
votes
0answers
58 views

Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: ...
4
votes
2answers
46 views

Iterating a periodic function

I'm curious about what happens if you iterate a function that is periodic. What happens to the period? For example, consider iterating a function like $\sin(x)$ or $\tan(x)$ several times. It should ...
-1
votes
1answer
37 views

How to find a transfer function the transfer function from angular velocity to the current used by the motor? [closed]

I need to find the transfer function of a control system with the input being an angular velocity (from a sensor measuring the rotation of a wheel of a vehicle) and the output being a current used to ...