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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Complex Analysis Dense Set Problem

The Problem: Suppose $f(z) = e^{i\theta}z$. Show that if $\theta$ is not a rational multiple of $\pi$, then the orbit of $ z \in \mathbb{C}$ is dense in the circle with radius $|z|$ and at the center ...
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37 views

Example of polynomial in dynamics

I am looking for an example of a post-critically finite polynomial $P$ (i.e. all critical points have finite orbit), which has both the following: a critical point on its Julia set (such as the ...
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1answer
37 views

Notation question in linear estimators (Kalman filter)

I'm just learning about Kalman filters, and I'm trying to understand some notation. The book that I am reading through sets up a system with the state-space realization: $$\dot{x}(t) = A(t)x(t) + ...
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74 views

Numerically solving a system of partial integro-differential equations in Matlab [closed]

Given the following system of partial integro-differential equations - $\frac{dS(t)}{dt}=\Lambda-\mu S(t)-\beta S(t)F(t),\\ \frac{\partial I(t,\omega)}{\partial t}+\frac{\partial ...
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1answer
67 views

Construction of Rauzy Fractals with substitutions without a fixed point

The formal definition of a Rauzy fractal can be found at the beginning of this paper Using Sage-math-cloud, I can generate Rauzy fractals of substitutions that I choose. Should I choose the ...
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71 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
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31 views

What are the general steps to turn a PDE into a dynamical system $\dot x(t)= Ax(t) + Bu(t)$

It is said that every boundary value PDE such as the heat equation can be turned into dynamic system of the type $\dot x(t)= Ax(t) + Bu(t)$ with appropriate I.C. Can someone elaborate as to how to ...
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7 views

Determine points with bounded orbits of discrete dynamical system defined by quadratic polynomial and chaos

Consider the discrete dynamical system definded by the function $f(x) = ax^2+bx+c$ for real parameters $a,b,c$ with $a \neq 0$, $(b-1)^2 \geq 4ac$. How does the set $\Lambda$ of all $x \in \mathbb R$ ...
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59 views

Averaging for nonlinear systems

I am trying to figure out how the following result has been obtained. Consider a function $J:\mathbb{R} \longrightarrow \mathbb{R}$ and a dynamical system: $$ \dot{ \hat{x} }(t) = k a \sin ( \omega ...
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59 views

A Hamiltonian vector field on $\mathbb{R}^{4}$ which has closed orbit but does not have critical point

Is there a polynomial function $H:\mathbb{R}^{4} \to \mathbb{R}$ without critical points but the corresponding hamiltonian vector field possess at least one closed orbit?
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42 views

An equivalent definition of the rotation number of a circle homeomorphism

Let $f : \mathbb S^1 \to \mathbb S^1$ be an orientation-preserving homeomorphism. The classical definition of the rotation number is the following: we lift $f$ to a homeomorphism $F : \mathbb R \to ...
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20 views

Finding a Ljapunov function for discrete dynamical system with 3 variables.

Consider the discrete dynamical system given by $(x_{k+1},y_{k+1},z_{k+1}) = f(x_k,y_k,z_k)$, where $f(x,y,z) = (x(1-ay),y(1-b+ax),z+by)$ with $a,b \in (0,1)$ are parameter and we are only interested ...
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1answer
32 views

Nonconstant solutions of discrete predator and prey model and Perron-Frobenius

Consider the discrete dynamical system given by $x_{n+1} = A x_n$, where $A = \begin {pmatrix} a & -b\\c &d\end {pmatrix}$ and $x_n = \begin {pmatrix} u_n\\v_n\end {pmatrix}$. Are there ...
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15 views

A 2-variable stochastic difference equation exhibiting 2 stable orbits with switching?

I have some social science data to which I would like to fit a stochastic difference equation in two variables. I will describe the dynamics of the system that I have observed. I am hoping someone ...
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19 views

Inverse evolution of a dynamical system

Background Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = ...
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1answer
38 views

A simple question on forward and backward invariant sets

A subset $A\subset X$ is forward invariant if $f^{t}(A)\subset A$ for all $t\ge 0$ and backward invariant if $f^{-t}(A)\subset A$ for all $t\ge 0$. I want to show that the complement of a forward ...
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21 views

Topological Semi conjugacy between Henon map and Logistic Map

I am currently teaching myself dynamical systems and have come across a problem I am not quite able to figure out. More specifically, I am unable to find a conjugator function to establish a semi ...
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28 views

Definition of Hamiltonian system through integral invariant

I've read that Poincare's integral invariance can be used as a definition of a Hamiltonian system. That is to say, if $g^t$ is a phase flow satisfying $$\oint_{\gamma} \omega = \oint_{g^t \gamma} ...
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16 views

Approximating non-continuous dynamic system by smooth functions

I have a dynamic system $\dot{y}=f(y)$, with $y\in \mathbb{R}^4$ and $f=(f_1(y), f_2(y), f_3(y), f_4(y)$. Here, $$ f_1(y)=\left\{\begin{array}{cc} -1 & \text{ if } y_2>0 \\ [-1,1]& ...
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40 views

Krylov-Bogoliubov theorem in discrete and continuous time

Consider the following two similar frameworks (the first one in discrete time, the second one in continuous time) in which I am trying to apply the Krylov-Bogoliubov theorem. 1) (Discrete time.) ...
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1answer
23 views

Convergence of a sequence of subspaces

Let $E_n\subset \mathbb R^n$ be a sequence of subspaces. What does it mean $E_n$ convergence to a subspace $E\subset \mathbb R^n$? I saw this when reading about hyperbolic sets. Where can I read ...
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36 views

Periodic phase curves

I'm currently reading Arnolds "Mathematical Methods of Classical Mechanics" and I'm having a hard time solving some of the problems in Chapter 2. I think that the following problem is fairly simple ...
2
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2answers
53 views

stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples

I'm currently self-studying dynamical systems. I'm trying to summarize what can be said about the stability of equilibrium points for an $n$-dimensional non-linear system of differential equations: ...
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1answer
19 views

System of separable diff. eqns, explicit solution and curves, Lotka-Volterra model

In the book on p.68 is a system of differential equations for a Predator-Prey model (Lotka-Volterra) given as: $$ \dot x=x(\alpha-c\gamma) \\ \dot y=y(\gamma x -\delta) $$ On the next page, it is ...
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1answer
31 views

Infinite closed shift-invariant set of binary sequences with zero entropy?

Does there exist an infinite closed shift-invariant $X \subset \{0,1\}^\Bbb Z$ with zero topological entropy? How to think of an example? Will periodic points of shift$|_X$ have zero entropy?
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66 views

How do i find formula for the recurrence relation :$x_{n+1}= x^2_{n}-x^2_{n-1}$ with :$x_{-1}=0,x_{0}=\frac{3}{4}$? [closed]

How do i find formula for the recurrence relation according to the below initial conditions: $x_{n+1}= x^2_{n}-x^2_{n-1}$ with :=$x_{-1}=0,x_{0}=\frac{3}{4}$ ? Note:$n=0,1,2\cdots$ Thank you for ...
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36 views

A continuous function with a dense set of periodic points but without sensitive dependence on initial conditions

The Question: Give an example of a continuous function, $f$, on the interval, $I$, such that the set of periodic points of $f$ is dense in $I$, but f does not have sensitive dependence on initial ...
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1answer
25 views

Difficulty understanding the concept of writing an L-System?

I've recently tried my hand at L-Systems, but I'm having some difficulty wrapping my head around it. I watched this video on the subject which is pretty good, but I had a question around the 1:43 ...
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22 views

Does an exponential bound on a Lyapunov candidate imply asymptotic stability?

If I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
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1answer
17 views

Local center manifold theorem.

Local center manifold theorem, under certain assumptions, state that for the \begin{cases} \dot x = Cx+F(x,y) \\ \dot y = Py+G(x,y)\\ \end{cases} there exist a function $h(x)$ such that ...
2
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45 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
1
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1answer
33 views

About expansive homeomorphim

We say $(X,f)$ is expansive if there is $c(f)>0$ such that if $d(f^{n}(x), f^{n}(y))< c(f)$ for every $n\in Z$ then $y=x$. Let $(X,f)$ is expansive with constant $c(f)$ and for infinite set ...
2
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28 views

Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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0answers
37 views

Trajectories of predator prey equation

I am studying the predator prey equation recently, and here is an example: Let $x'=x(1-0.5y)$ and $y'=y(-0.75+0.25x)$. This is a predator prey equations. Then ...
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8 views

Transform stiff systems to non-stiff

Many dynamic systems tend to be stiff, so an explicit integrator is unstable. The solution is to use an implicit integration scheme. I am curious if there is some way to change the dynamics of the ...
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26 views

Linear Operator on real (positive definite) symmetric matrix; Generalization of Lyapunov theorem

I am wondering if there is any results on a somewhat "generalization of Lyapunov theorem". By which I mean, as we know from Lyapunov theorem, for a Lyapunov operator on real symmetric matrix, $L_A: ...
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18 views

Incidence matrix of invertible substitution

A statement I am reading says: An invertible substitution $\sigma$ over $\{1,2\}$ is non-primitive iff $M_{\sigma}$ (it's incidence matrix) has one of the following forms:$$\left( \begin{array}{cc} 1 ...
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0answers
32 views

Books on Catastrophe Theory

I'm looking for a technical introduction to catastrophe theory, preferably something short. I have a good background so graduate level texts are welcome. Thanks in advance.
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45 views

Convergence of a particular fixed point iteration scheme

Setup I have the following non-linear system of equations: $$ \mathbf{x} P(\mathbf{x}) = 0 $$ where $\mathbf{x} \in \mathbb{R}_{>0}^n$ is a probability distribution, i.e., $\sum_i x_i = 1$, and ...
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0answers
13 views

Two vector fields are cojugate but not take orbits

Let X and Y be C1 vector feilds on R^m. Suppose that 0 is an attracting hyperbolic singularity for X and Y. Show that there exists a homemorphism h of a neighborhood of origin which conjugate the ...
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2answers
90 views

Fundamental Matrix (Floquet theory)

Let $\begin{pmatrix} \dot{x}_1 \\\dot{x}_2\end{pmatrix}=A(t)\begin{pmatrix}x_1\\x_2 \end{pmatrix}$ where $$A(t)=\begin{pmatrix}\alpha(t)+\cos(t)&\sin(t)\\ -\sin(t)& ...
1
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1answer
21 views

Show that if f in Diff^r(M), r >=1, is structurally stable then all the fixed points off are hyperbolic.

i think since f is structurally stable so there exists an open nbd u containig of g then f and f are topoligy equivalent.i think since hyperbolic fixed pints dence and open there exists neighberhood v ...
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20 views

Question about periodic points in shift spaces

Let $A$ be a finite set endowed with the discrete topology. Then, the pair $(A^{\mathbb{Z}}, \sigma)$ is said to be the full shift over the alphabet $A$ where $A^{\mathbb{Z}}$ is endowed with the ...
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2answers
233 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
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1answer
69 views

Trapping region for $\ddot x + (x^2-2) \dot x + x + \sin(x) =0$

I need to show that the system $$\ddot x + (x^2-2) \dot x + x + \sin(x) =0 $$ Have a periodic orbit. I always use polar coordinates to find a trapping regio, but with the sine term, I am kinda lost. ...
3
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1answer
54 views

How to solve $x^2+x+a=0$ with fixed point iteration?

So when the constant is negative, iteration of $f=\sqrt{-a-x}$ converges quite easily. Also the derivative is less than 1 as long as $-2 \lt a \lt {1 \over 4}$, I don't think that's relevant as the ...
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1answer
50 views

Analyzing the singularity of ODE system

It is asked to analyze the singularities of the system $$\dot{x} = y e^y$$ $$\dot{y} = 1-x^2$$ I've found that the singularities are (1,0) and (-1,0) The linearization of the sysyem give the matrix ...
5
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95 views

Properties of join of open covers

I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological ...
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1answer
49 views

From multivariable system transfer function matrix to state space representation

I have the transfer function matrix $H(s) = \begin{bmatrix} {1\over s+1} & {2\over s+2} \\ {-2\over s^2+3s+2} & {2s\over s+1} \\ \end{bmatrix}$ And I want to ...
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0answers
39 views

Time period of oscillations of a point about the function's minimum value?

How am I to go about the following problem? Please do not explicitly solve it. Let $E_0$ be the value of the potential function at the minimum point $\xi$. Find the time period $T_0=\lim_{E\to ...