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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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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141 views

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

I am analyzing the iterated sequence of the function $\lambda \sin( \pi x)$ for $x, \lambda \in [0,1]$, where $x_n=f(x_{n-1})$ for a paper I am writing. I know that all fixed points of this function ...
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1answer
120 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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222 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
58 views

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

If $f(x)=2x \mod1$ 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 a ...
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64 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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102 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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63 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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79 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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30 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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36 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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56 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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48 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
75 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
83 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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6k 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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55 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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70 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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29 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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68 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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45 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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92 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: ...
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67 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 ...
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239 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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49 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
158 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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1answer
38 views

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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47 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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79 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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40 views

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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2answers
47 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
35 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
51 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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28 views

How do I make this statement about trajectory curve mathematically precise?

Suppose that $p(t) = (x(t),y(t),z(t))$ is a continuous trajectroy in $R^3$ defined on $t \in [0,t_1]$ Assume that $p(0) = (x_o,y_0,z_0)$, where $(x_o,y_0,z_0)$ is a point in $R^3$. Let $B$ be a ...
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1answer
53 views

Assertions about measures with computers

Let's consider the Lebesgue measure ($\mu$) over the closed interval $[0,1]$. As you know, $\mu(\mathbb{Q} \cap [0,1]) = 0$. In other way, as far as I know the computer just can represent accurately ...
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52 views

Looking for advice with the following integral

I have the following integral to evaluate: $$\frac{1}{f(t)} \int_0^t s^m f(s) \sin(ps) \mathrm{d}s \quad m,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. Even using ...
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2answers
73 views

Periodic Points and One Dimensional Maps Homework Help

Let f be the tripling map $f(x)=3x\mod(1)$. I need to make a table that includes the following for $n\le6$: number of points in Fix($f^n$), number of points in Fix($f^n$) of lower period, number of ...
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144 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 ...
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1answer
84 views

Fixed Points and Graphical Analysis

For the following functions (i) find the fixed points and (ii) use the graphical analysis to determine the dynamics for all points on R: $f(x)=2sin(x)$. I have no problem finding the fixed points, ...
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123 views

How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or ...
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82 views

Fixed points and periodic orbits of $F(x)=x^2-1.1$

A question asked me to find the fixed points of $F(x)=x^2-1.1$, then use the fact that these points were also solutions of $F^2(x)=x$ to find the cycle of the prime period 2 for F. How do I go about ...
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1answer
40 views

What to do when regular approaches fail on linear, non-homogeneous ODES.

In my research problem, I have come across the following form of a time varying, non-homogeneous ordinary differential equation. $$\dot x + \frac{k_1}{t} x = k_2t^{3n}\sin(bt) + k_3 t^n \sin(bt) - ...
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69 views

Bride Groom Problem

Let's consider a system of $n$ men and women. Each woman is paired with one man (there are only pairings between a woman and a man in this system). There are $n!$ possible distinct pairings. I refer ...
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1answer
49 views

Operator induced by continuous function and measures

If $X$ is a compact metric space, and $T:X \rightarrow X$ is continuous map, what would be meant by $T_\ast$ is the operator on measures induced by $T$? Allow $\mu$ to be some Borel regular normed ...
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1answer
237 views

Not uniquely ergodic transformation

Could you teach me an example of NOT uniquely ergodic but ergodic transformation? And when any continuous, measurable, and ergodic transformation on a topological space X is uniquely ergodic, how ...
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32 views

KSE and Shannon entropy

Is there a theoretical connection between Kolmogorov-Sinai and Shannon entropies? What is it?
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223 views

Circle rotation number invariant under topological semi-conjugacy.

For a circle homeomorphism $f: S^1 \rightarrow S^1$ we can define the the rotation number $$ \rho(f) = \lim_{n \rightarrow \infty} \frac{1}{n}(F^n(x) - x) \mod 1, $$ for a lift $F:\mathbb{R} ...
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3answers
109 views

Is there a closed form solution for this differential equation?

I was trying to solve the following ODE, but I cannot find an easy way anywhere. I also tried using Mathematica, but it also does not provide me with a solution. $\frac{dx}{dt}=-k_1 x+(1-x)k_2 ...
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2answers
36 views

Geometric translation of a theorem about stability of equilibrium point

In the book Nonlinear Systems by Hassan Khalil, there is a theorem about the stability of equilibrium point ‎ which asserts that : Theorem :‎ Let‎ ‎$X=0$ ‎be an equilibrium point for‎ ‎$‎\dot x = ...
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45 views

Examples of dynamical systems over various spaces

Let's define a dynamical system as follow : ‎ A dynamical system is a triple‎ ‎$(T, X, ‎\varphi‎) $‎‎ where T is a time set, X is a state space, and‎ ‎$‎\varphi : T ‎\times X ‎‎\rightarrow X ...