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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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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0answers
17 views

Clarifying understanding of Hamilton-Jacobi Theory in Hamiltonian Dyanmics

I am just trying to clarify my understanding of a few things regarding Hamilton-Jacobi Theory: My understanding is that in using the Hamilton-Jacobi Equation, we want to try and find a generating ...
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11 views

Help in finding the functional form of the probability density function

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
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18 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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39 views

what's the recent trend in the field of dynamical system? [on hold]

I'm studying dynamical system,and I want to know some research topics in the field of dynamical system,such as bifurcations and chaos theory.Could you recommend me some materials or references?Thanks ...
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35 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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13 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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41 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
31 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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32 views

Fix points and stability [closed]

Find fixed (equilibrium) points of the map $f(x) = x^3 − \dfrac{x}{9}$ and classify their stability: asymptotically stable (attracting), unstable (repelling), neither. Illustrate your answer by the ...
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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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10 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 ...
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20 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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21 views

Topological conjugacy in Hénon map

$\textbf{Definition:}$ $\textit{(Topologically conjugate)}$ Let $f:A\rightarrow A$ and $g:B\rightarrow B$ be two maps. $f$ and %g% are said to be topologically conjugate if there exists a ...
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12 views

Simultaneously solving trigonometric equations

Let $N\in\mathbb N$. Given $\theta_1,\ldots, \theta_N\in [0,2\pi)$ I would like to prove that there exist $\rho\in\mathbb R_+$ and $\varphi\in[0,2\pi)$ such that $$ f_\ell(\rho,\varphi):=\theta_\ell ...
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43 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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23 views

Discretization of a continuous time-invariant linear system

I have the following autonomous system $$\dot{x}(t) = Ax(t)$$ where $x \in \mathbb{R}^2$ and $A$ is a constant matrix with suitable dimensions. When I discretize this system under a sampling time of ...
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50 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
46 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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33 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
32 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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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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30 views

How do I verify the stability of this fixed point?

The difference equation I am given is: $ x_{n+1} = rx_n(1-x_n)$ or the map $f(x) = rx(1-x)$ I am asked to verify that "the nontrivial 2-cycle is stable for $3 < r < 1 + \sqrt(6)$ and unstable ...
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1answer
35 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
33 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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38 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
32 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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30 views

Why can't the general solution of separable first order ODE cross the stationary solution?

For example, if we have the following Cauchy problem: $y'=y^2-4, y(0)=0$ In class, our professor told us that $y=-2,2$ are the two stationary solutions, but how could it be, since our initial point ...
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1answer
39 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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21 views

KSE and Shannon entropy

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

Morphisms of $\mathsf{Meas}$ and Dynamical Systems

The morphisms of a category $\mathsf{Meas}$ whose objects are measure spaces are defined to be equivalence classes of a.e-equal measurable maps that pull back null sets to null sets. Why is pulling ...
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14 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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83 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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1answer
21 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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25 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 ...
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27 views

What is an open property?

From an academic paper, "the existence of elliptic or hyperbolic 2-periodic orbits is an open property". I have never seen the term "open property" used before, moreover the paper gives no ...
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Definiton of invariant curve

What is the definition(s) of an invariant curve? What book should i read to get a better idea of their use in dynamical systems. Are there any defining features i should be aware of especially with ...
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34 views

Limit of a Discrete Dynamical System, Part 2

In my previous post (i.e., Limit of a Discrete Dynamical System) the following system was considered: ...
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38 views

Lagrangian Dynamics Practice Question

working on some exam practice questions, and just needed a bit of help to check my answer to the last part of this question: A particle of unit mass is projected around the inner surface of an ...
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1answer
27 views

Find the expression for the path length on the surface of the cylinder with constant

Hi just working on some past exam questions for an upcoming quiz for a class in Lagrangian dynamics, and I am a bit stuck on this question (which seems quite different from what we have done in ...
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30 views

Using the Lyapunov-Perron method to find the local stable/unstable manifolds

Hello Stack Exchange community. I am currently having an issue finding the local stable/unstable manifolds of this system. After going at it for a few hours I believe the person who wrote this ...
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31 views

Homogeneous function in Complex plane and its Periodicities

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number. It is easy to ...
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How to find an orbit in implicit form for a first order non-linear system of differential equations?

How to find an orbit in implicit form for a first order non-linear system of differential equations? Say $x'= x - xy$, $y'= y - 2xy$ is our system. How do we find an orbit of it in an implicit form?
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100 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, ...
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18 views

Find approximate solutions of the dynamics of the pendulum by expanding the Lagrangian up to quadratic order near equilibrium pt

If I have the following Lagrangian: $$L = \frac{1}{2}m(l-a\theta)^2\dot{\theta}^2-mga\cos\theta+ mg(l-a\theta)\sin\theta$$ and I know the equilibrium is at $\theta = \frac{\pi}{2}$, how do I "expand ...
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28 views

Solve these linear Lagrange Equations

I am working on some practice questions and just need a bit of help with understanding the last part of the this question and the solutions. My question is really about the last part, but here is the ...
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20 views

example of time invariant system and connection to memoryless

textbooks give abstract examples of time invariant and non-time invariant (time sensitive) systems. can you please give an intuitive example of a time invariant system and one which is not? obviously ...
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1answer
19 views

Frequency response of unit impulse function

Could someone throw some light on how to get the frequency response of unit impulse function. I am not from EE, but I need it for my wavelet study.
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1answer
44 views

Question about Pollicott-Yuri's proof of Rudolph theorem

On the book Dynamical System and Ergodic theory by Pollicott and Yuri there is a proof of the Rudolph x2 x3 theorem (page 153). It looks very clean in comparison with the original proof but I didn't ...