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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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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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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58 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
39 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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44 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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35 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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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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34 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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36 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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110 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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KSE and Shannon entropy

Is there a theoretical connection between Kolmogorov-Sinai and Shannon entropies? What is it?
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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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34 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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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
23 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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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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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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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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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
57 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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51 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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38 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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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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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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48 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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1answer
91 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
28 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
62 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 ...
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Reference for Algebraic Groups in Ergodic Theory

It seems that the theory of algebraic groups is used in ergodic theory. I was hoping someone could recommend an introduction to algebraic groups that assumes a knowledge of commutative algebra ...
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1answer
27 views

Dynamical Systems Question on Definition

I'm working through some old notes on Dynamical systems, and I see a definition that I'm not familiar with. I'll call it property B, because I'm not sure what else to call it. In the notes, we assume ...
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references about homoclinic or heteroclinic orbit

Could you recommend me some references about the connecting orbit manifolds of differential equation? And what about the methods of studying the existence of heteroclinic or homoclinic orbit in ...
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60 views

Stability of critical points on dynamical system

Consider a generic dynamical system \begin{cases}\dot{x} &= f(x,y)\\ \dot{y} &=g(x,y) \end{cases} with a critical point in $P=(x_0,y_0)$ (there may be other critical points somewhere else). ...
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1answer
57 views

stability of linearly perturbed linear nonautonomous system

I have a linear time-varying linearly perturbed ODE of the form: $$ \dot{x} = [A(t)+B(t)]x $$ where $A(t)$ is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. ...
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Invariant set under the flow defined by Schroedinger equations

I have to show that the set of functions of the form $$\psi(x,t)=c(t)^{-1}e^{\frac{-(x-q(t))^2}{2c(t)^2}}e^{ip(t)x}\hspace{1cm}c(t),p(t),q(t)\in\mathbb{R}$$ is invariant (as set) under the flow ...
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Is an Invariant set Connected?

Let an autonomous dynamical system is characterized by the state equation $$ \dot x(t) = f(x(t)),\quad x(0)=x_0 $$ with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, ...
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1answer
73 views

Linearising a dynamical system at an arbitrary point (affine dynamical systems)

Usually, one linearises a dynamical system around a fixed point, in order to determine whether the fixed point is stable. That is (assuming the fixed point it at the origin), one approximates $$ ...
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1answer
41 views

About a Property of maximal solutions of separable ODE's $y'=g(x)h(y)$ for locally Lipschitz $h : U\to\mathbb R$, $U$ open

Theorem: Let $\varphi : (a,b) \to \mathbb R$ be a maximal solution of the IVP $$ y'(x) = g(x) \cdot h(y(x)), \quad y(x_0) = y_0 \quad (1) $$ with continuous functions $g : I \to \mathbb R$ and $h : U ...
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1answer
38 views

Find out Fixed Points

Consider a set $M$ of all possible square matrices of dimension $k$ over a finite field $F_p$. Consider a map $f$ defined on $M$ as $f(X)=X^2+C$ where $X \in M$ and C is an arbitrary fixed matrix from ...
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2answers
81 views

A matrix $G$ with all eigenvalues with nonzero real part. Then $t\mapsto |\exp(tG)x |$ is unbounded

I am trying to see why this is true. A book I am reading has this claim without any verification and I'm trying to see why it is true. Let $G$ be an $n\times n$ matrix all of whose eigenvalues have ...
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260 views

How did Ulam and Neumann find this solution?

In the book "Chaos, Fractals and Noise - Stochastic Aspects of Dynamics" from Lasota and Mackey the operator $P: L^1[0,1] \to L^1[0,1]$ $$ (Pf)(x) = \frac{1}{4\sqrt{1-x}} \left[ ...
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Lyapunov Exponent for Logistic map $x_{n+1}=bx_n(1-x_n)$

The Lyapunov exponent for logistic map $x_{n+1}=f(x_n) $ where f is given by $f(x)=bx(1-x)$ is given by: $$\lambda = \frac{1}{n}\sum^n_i log(b(1-2x_i))$$ where the index $i$ refers to $i$th iterate ...
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Diffeomorphism and hyperbolic points

Suppose $f$ is a diffeomorphism.Prove that all hyperbolic periodic points are isolated. I tried using the mean value theorem using two diferent periodic points (assuming the periodic points arent ...
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21 views

$\Phi(\cdot ,x):I_x \rightarrow M$ injective?

I don't understand we the map $\Phi(\cdot ,x):I_x \rightarrow M$ from the excerpt from below of the lecture notes of my professor has to be injectiv. (Here $M$ denotes the domain of the function on ...
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1answer
27 views

A question on periodic points and recurrent points

I have been reading the book "dynamical systems and semisimple groups an introduction". In this book, a point of a topological $G$-space $X$ is a periodic point if $G/G_x$ is compact, where $G$ is a ...
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Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
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1answer
40 views

Global existence of a dynamical problem.

Prove that all the solutions to the system $$ \begin{cases} \dot x= e^{-y^2}\sin(x^n+y^n), \\ \dot y= x^n\sin(x^n+y^n), \end{cases} $$ where $n$ is a fixed natural number, are defined on ...
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Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function