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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70 views

The abelian group of homeomorphism

Let $G$ be a subgroup of the group of homeomorphisms on the circle, and we suppose $G$ is abelian, if every element of $G$ has a fixed point on the circle, does it imply that $G$ has a common fixed ...
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50 views

Is there a method to list all periodic points for a funcion?

I search for a method that finds all periodic points of a given function e.g. $f(x)=x-x^2$ on its domain. You may explain some methods for a part of functions e.g. polynomials or $\mathcal{C}^k$ or ...
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25 views

Hill's problem for moon trajectories.

When we work with the three-body problem, we have a parameter $\mu$ that shows the ratio of the two biggest bodies with $\mu\in(0,1)$. This let's us do practical applications easily. For example we ...
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1answer
63 views

A good function to fit this data

I'm computing the angle of intersection between to curves (the invariant manifolds of a dynamical system). I do this with a numerical algorithm, but I would like to fit a function with this data. ...
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33 views

Orbit , trajectory, dynamical system

The orbit of φ(t) through x(0) is the set O(x (0)) ≡ {φ t (x 0 ) : −∞ < t < ∞}. This is also called the trajectory through x(0) What's the different between Orbit and Trajectory ?? Please help ...
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17 views

Mathematical treatment of mass action kinetics with detailed balance (as opposed to complex balance)

I'm looking for a mathematical treatment of mass action kinetics under the assumption of detailed balance. In particular, I'm interested in the proof that the free energy is a global Lyapunov ...
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37 views

Chaotic dynamical systems with hard proposition

How I can prove this proposition ? proposition : Consider a network with a fixed connection topology, having in each node identical members of the family of m-modal functions f in the real interval. ...
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0answers
19 views

Linear time varying definition of lyapunov stability

I came across the following alternate definition for Lyapunov stability of continuous linear time varying (CLTV) systems in a textbook: A CLTV system is sait to be stable in the sense of Lyapunov ...
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1answer
42 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
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1answer
26 views

Is $ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $ dense in the rectangle $ [- A,A] \times [- B,B] $?

What conditions must $ a $ and $ b $ satisfy in order for the curve $$ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $$ to be dense in the rectangle $ [- A,A] ...
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2answers
40 views

Regarding iterated maps $\mathbb{R}^2 \rightarrow \mathbb{R}^2$, and strictly decreasing norm

Consider a map $g:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, and the iteration $(x,y) \mapsto g(x,y)$. Say that the origin is an asymptotically stable fixed point and there is a region $A$ in the first ...
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1answer
75 views

Important topics in Matrix Analysis

I'm doing a course in Matrix analysis, and I'm supposed to prepare a presentation about any topic in Matrix theory. We already covered the book "Matrix Analysis" by Horn, so preferably I need a topic ...
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22 views

Looking for discrete non linear dynamic system solution hints

I am studying a networking congestion control problem for which I would like to solve the following non linear discrete first order dynamic system (hope I got that correctly, I am no mathematicien ...
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0answers
19 views

Flow with divergence-free generating vectorfield: Conservating volume

A function $g\colon M\to M$ is called to conserve volume if for any Jordan-measurable subset $J\subset M$ it is $\text{vol}(g^{-1}(J))=\text{vol}(J)$, Now I would like to show that for a ...
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2answers
35 views

Finding a Lyapunov function for a given system of equations

I've got the following system of equations: $$ \begin{cases} x_1'=-8x_1^3-x_2 \\x_2'=-4x_2-4x_1^3 \end{cases} $$ I'm trying to check, if the equilibrium point in $(0,0)$ is stable or not. I am ...
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2answers
37 views

What is a Parabolic Fixed Point?

I know the definitions of hyperbolic and elliptic fixed point (or equilibrium). However, when I google I find references to 'parabolic elliptic points' but not a proper definition.
2
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1answer
29 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
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0answers
23 views

Topological entropy, spanning sets and expansiveness of simple maps on a torus

I am trying to solve the following problem. Take the torus $\mathbb{T}^{2}$ and define the map $T(x,y)=(x + \alpha$ mod 1, $x+y$ mod $1)$, where $(x,y) \in [0,1]^{2}$. By induction, we have ...
3
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1answer
54 views

Can we construct a Koch curve with similarity dimension $s\in[1,2]$?

We can make a Koch curve $K$ with similarity dimension $s\in \mathbb Q \cap [1,2]$ by writing $s=\frac{p}{q}$, and constructing such a generator that by scaling with the factor of $2^q$, we'd find ...
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224 views

Estimating a dynamical system's behavior without using Liapunov theorem

Assume that we have the following dynamical system $$x'=(\epsilon x+2y)(1+z)$$ $$y'=(-x+\epsilon y)(1+z)$$ $$z'=-z^3$$ Then how can I show that any solution that started from the region $z>-1$ ...
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66 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
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3answers
146 views

What is the topological dimension of the Peano curve?

The Hausdorff dimension of the Peano curve is know to be two. And I assume it to be a fractal since it's on the List of fractals by Hausdorff dimension. Moreover: According to Falconer, one of the ...
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1answer
45 views

A simple question about discrete dynamical system

Given a self-mapping $f:X\to X$, $X$ is a Hausdorff space, and $f$ is continuous and topologically transitive, if $X$ is infinite, then $X$ contains no isolated points. Why? I don't know why? Please ...
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177 views

The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
3
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1answer
27 views

Does Hyperbolic + Not Asymptotically Linearly Stable imply Not Asymptotically Stable?

Topic: Stability of Autonomous Non-linear ODEs I'm wondering whether having a hyperbolic critical point that's not asymptotically linearly stable (ALS) in the linearisation of a system implies that ...
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2answers
63 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
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0answers
20 views

Non-integrable systems

If a Hamiltonian system in $\mathbb{R}^{2n}$ has $n$ suitable first integrals, then it is called an integrable system, and the Arnold-Liouville theorem tells us all sorts of nice things about the ...
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3answers
125 views

How do we solve $c_1^d+\ldots+c_n^d=1$ for $d$?

The question is motivated by the definition of self-similarity dimension for self-similar sets: Let $M \subset \mathbb R^d$ be self-similar. That is, there are $T_1, \ldots, T_m \subsetneqq M$ and ...
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2answers
82 views

How did self-similarity come into mathematics?

As far as I understand the interest in self-similarity was born outside of mathematics. The textbooks I came across give a few objects as examples (tree, broccoli, river, etc) yet it's clear that the ...
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3answers
130 views

Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
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0answers
43 views

Continuation fixed points of parameter dependent Newton

Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in ...
2
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1answer
49 views

Why do we require a finite number of subsets for self-similarity?

Here is how my text defines self-similarity: We call $M \subset \mathbb R^d$ self-similar if there are $T_1, \ldots, T_m \subsetneqq M$ and similarity maps $\alpha_1, \ldots, \alpha_m$ such that ...
4
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1answer
417 views

Is the two-dimensional Koch curve space-filling?

Say, we'd like to make a Koch curve with self-similarity dimension of two. A Koch curve with the following generator seems to be two-dimensional, since if we double its size by scaling we'll find ...
0
votes
1answer
30 views

What determines the rotation direction in a $2$ D Dynamical system?

Say we have a non linear system $\dot{x}=f(x)$ and we linearise this around an equilibrium point $x_0$, to obtain the linear sytem: $\dot{x}=Df(x_0)\cdot x$. Where $Df(x_0)$ is the jacobian (and in ...
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0answers
26 views

Finding the homomorphism that links the linear part of a dynamical system to the nonlinear part.

here is a picture of my problem Basically what i have is that i was told i could find this homomorphism by doing the following ...
0
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1answer
33 views

Intuition or wisdom for stability and instability properties of locally linear system. Boyce, p513, Table 9.3.1

Our instructor requires us to memorize this table for our differential equations exam. So I wonder if anyone has some deeper intuition or observation to help with this? For example, I noticed ...
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0answers
21 views

what is the particle size effect in chaos?

I am studying dynamical system. As far as I know, chaotic behavior can be developed for one particle moving in a certain potential in 3D, and for this case, the position of the particle will be the ...
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1answer
38 views

What connections between machine learning and dynamical systems?

I have a background of ("pure") dynamical systems and ergodic theory, but I am switching to machine learning. Can some machine learning questions be treated from a dynamical systems/ergodic theory ...
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1answer
277 views

Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$ a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right). $$ The first ten terms are: $0.75$ ...
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0answers
50 views

iterates of generalized matrix system

This may be somewhat of an underspecified question but I'll nonetheless give it a try. In the context of applied work, I've recently come across systems of the form $$ ...
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2answers
44 views

Memoryless processes and independence

this is a mere question of definition, that one surely can figure out by conventional means, but maybe someone can just quickly give me the definition. What is a memoryless process? Following the ...
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0answers
27 views

Application of Poincare recurrence to Baker's map?

Please see figures at http://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe/wiki/projects/Recurrence.html. I heard that one of the applications of the Poincare recurrence theorem (which I do not ...
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1answer
18 views

Sensitivity Constants for Linear Expanding Maps

Let $E_m:S^1 \rightarrow S^1$ be the linear expanding map $E_m(x) = mx$ mod 1 (under the identification $[0,1] \sim S^1$). A sensitivity constant $\Delta$ is a positive real if for all $x, \in S^1$ ...
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76 views

What's the intuition behind definition of chaotic function?

I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos" ...
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41 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
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46 views

Set of diffeomorphisms on a manifold

It is well known that given a compact smooth boundaryless manifold $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r \geq 1$, is open in $C^{r}(M)$, the set of continuous functions (for ...
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1answer
22 views

Generlized Entropy compared to Generalized Dimension

I am currently reading the following paper by F.Takens: Multifractal analysis of dimensions and entropies. This paper discusses two different measures. One is generalized entropies and the other is ...
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1answer
26 views

Show that stable spriral has index of 1.

How do you go about generally showing that the index at the origin for different linear systems is equal to +1? For instance, the center of a stable spiral. For a specific system, I could show how ...
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47 views

About Network Dynamics

Consider the 6-node ring with only one bit active (000001). This is shown in the following figure as a hexagon with one circle filled. If the active bit is traveling in the counter clockwise ...
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1answer
58 views

locate any bifurcation in the $2D$ system?

bifurcation for the following $2D$ system: $$\left\{\begin{matrix} x′=ux-y+x^3\\ y'=bx-y \end{matrix}\right.$$ I have got $ux-y+x^3=0,\ y=bx$, then $x=0\ \ \text{and}\ x = \pm \sqrt{b-u}$. But I ...