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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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 ...
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Showing forward invariance for an ODE

Say I have an ODE defined by $$\dot{x} = f(x)$$ I want to show that the set $$G = \{ x | g(x) \geq 0\}$$ (where $g$ is differentiable) is forward invariant, i.e. $x(0) \in G \rightarrow x(t) \in G$ ...
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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 $$ ...
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1answer
19 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 ...
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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 ...
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27 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 ...
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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 ...
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1answer
40 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 ...
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39 views

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 ...
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2answers
64 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 ...
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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 ...
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18 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. ...
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29 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 ...
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1answer
34 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 ...
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26 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 ...
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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 ...
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1answer
16 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 ...
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30 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 ...
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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 ...
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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 ...
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30 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 ...
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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 ...
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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 ...
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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&...&\\ ...
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1answer
50 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 ...
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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, ...
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4answers
96 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, ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
35 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 ...
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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
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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 ...
4
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1answer
110 views

Is chaos theory really a theory? Why not just call it non-linear dynamics?

This may just be semantics, but it's always confused me. What is the thesis of Chaos Theory? I have read an entire book about it, and as far as I can tell, its just a bunch of analytical techniques, ...
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18 views

Solutions of unstable switched systems

Consider $n$ affine systems, $\dot{x} = A_i x + a_i$, where $\sigma(A_i)\subset \mathbb{C}^{+}$ for each $i\in \{1,\dots,n\}$. Here $A\in\mathbb{R}^{k \times k}$, $a\in\mathbb{R}^k$, ...
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1answer
23 views

Difference between slow and inertial manifolds?

Could anyone provide a clear definition of the basic difference between two of the invariant manifold types - the inertial and slow manifolds?
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2answers
64 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 ...
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Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ is a vector of 1s ...
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23 views

Estimating the distance to the Julia set of a rational map

Suppose that $f \colon \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ is a rational map of degree $d \ge 2$. Let $z_0$ be a point in the Fatou set $F(f)$. I'm interested in finding an estimate for the ...
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2answers
44 views

Techniques for finding period points

Consider the tent function $f_2$ given by: $$f_2 = \begin{cases} 2x, & 0\leq x\leq \frac{1}{2} \\ 2-2x, & \frac{1}{2} < x \leq 1 \end{cases}$$ How do I find the periodic points of this ...
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Equicontinuous and distal factors

Let $(X, \{T_g\}_{g \in G}), (Y, \{S_g\}_{g \in G})$ be topological dynamical systems, with $G$ a group, $(X, d_X), (Y, d_Y)$ compact metric spaces and $\{T_g\}_{g \in G}, \{S_g\}_{g \in G}$ groups of ...
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48 views

An Introduction to Dynamical Systems: Continuous and Discrete

Does anyone know where I can find the manual for the book in the title by R. Clark Robinson? It is a very difficult book to follow along with and my professor doesn't speak a lick of English. I'm ...
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2answers
33 views

If $f(x) \cdot x < 0$ for all $x \in \partial B_R(0)$, then the IVP $x' = f(x)$, $x(0) = x_0$ has a global solution.

I have a homework problem that asks If $f : \mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and satisfies $$ f(x) \cdot x < 0 \quad \quad \text{for all } x \in \partial B_R(0) ...
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3answers
30 views

Simple derivate question

In a paper I am reading about dynamics systems, they set the following variables: $a(\theta) = \ddot{\theta}$, $b(\theta) = \dot{\theta}^2$ Where $\dot{\theta}$ and $\ddot{\theta}$ are the first ...
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1answer
29 views

Non-uniform contraction

Let $(X,d)$ be a metric space. A map $f: X \longrightarrow X$ is called contracting if there exists a $\lambda < 1$ such that for any $x, y \in X$ $$d(f(x),f(y)) \leq \lambda d(x,y)$$ It is well ...
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12 views

Integrability and decoupling of variables

Can the following set be integrated in closed form? If not, can they be expressed as two separate second order coupled ODEs of x and y after eliminating z? I find it difficult to do when b is ...