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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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} ...
2
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3answers
431 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 ...
0
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1answer
58 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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1answer
265 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 ...
4
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1answer
81 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
93 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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1answer
68 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
169 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 ...
5
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2answers
121 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 ...
3
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3answers
223 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
55 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 ...
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1answer
61 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
votes
1answer
458 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
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1answer
39 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
32 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
97 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
31 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 ...
2
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1answer
163 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 ...
7
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1answer
367 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
51 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 $$ ...
1
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2answers
134 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
149 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
35 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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2answers
256 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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0answers
55 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
55 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
114 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 ...
2
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0answers
58 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
69 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 ...
1
vote
1answer
58 views

A question on Bernoulli measures and mixing

this is a question on ergodic theory. Suppose I have an integer $N \geq2$ and a probability space $(\sum^{+} , B, \mu_{p})$, where $\mu_{p}$ is the Bernouilli measure with respect to probability ...
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1answer
79 views

Hopf bifurcation phase portrait orientation

Say for a system $$\dot{x}=y$$$$\dot{y}=-x+\mu y -y^3$$ I have confirmed a hopf bifurcation occurs at the origin and that the branches are stable i.e., a stable limit cycle and the origin being stable ...
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1answer
150 views

Nonlinear first order system of ODEs

While solving some physical problem, I have obtained the following system of differential equations with boundary conditions: $$\left\{\begin{matrix} \frac{d\phi_1}{dz}=\frac{m^2}{\lambda}- ...
0
votes
1answer
72 views

Finding discrete maps with prescribed cycle-structure (functional digraph-structure)

I apologize in advance for the naive nature of the following questions. I am also thankful to suggestions for improving the direction of the questions instead of direct answers. Let $f: \mathbb N \to ...
0
votes
1answer
78 views

Show a continuous bijection cannot have periodic points of prime period greater than 2

Suppose f:R↦R is a continuous bijection. Show that the system x_n+1=f(xn) cannot have periodic points of prime period greater than 2. Hint: Use Sharkovskii's Theorem to reduce the problem to the case ...
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3answers
303 views

Showing that the function $C(b)$ is a compact set for $|b| > 1$

I am reading "An Invitation to Dynamical Systems", and one of the challenge problems is to prove that $C(b)$ is a compact set where $C(b)$ is defined as the set of all numbers that can be expressed in ...
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0answers
47 views

$\cos(2\arccos(\frac{a}{a+1})x$

I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using ...
4
votes
0answers
176 views

Holonomic constraints and degrees of freedom

Back in my undergrad I learned that in a dynamical system, if I add a holonomic constraint, I subtract one degree of freedom from the space of configurations. But one can think of situations in which ...
5
votes
1answer
123 views

A question on ergodic theory: topological mixing and invariant measures

This is a question on dynamical systems. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. ...
2
votes
1answer
94 views

Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...
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0answers
72 views

Understanding a proof concerning flows on tpological spaces

Let $M$ be a topological space and to each point $x\in M$ let there be an open interval $I(x)=(I_-(x),I_+(x))\subset\mathbb{R}$. To make it shorter, we set $E:=\bigcup_{x\in ...
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1answer
38 views

Function with many variables, diffrentiating. Hamiltonian and Langragian functions

If I have: $$H(q,p,t)=\dot{q}_ip_i-L(q,\dot{q},t)$$ How can I obtain the following relations: $$\mathrm{d}H=\dot{q}_i\mathrm{d}p_i-\dot{p}_i\mathrm{d}q_i-\frac{\partial L}{\partial t}\mathrm{d}t$$ ...
0
votes
1answer
58 views

FermiPasta-Ulam problem

Consider $H(q,p) = \frac{1}{2} \sum\limits_{j=1}^{n+1} {(p_j^2 + (q_{j}-q_{j-1})^2)}$ $H(q,p) $ is the Hamiltonian considered in the FermiPasta-Ulam problem. Consider canonical transformation $Q = ...
7
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1answer
225 views

Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
14
votes
1answer
576 views

How to make a smart guess for this ODE

I am dealing with a strange problem currently, we have a differential equation $$y(x)^2 = \pm \sqrt{-A \cos(x) - B \cos^2(x)+y'(x)-C},$$ where $C, A$ and $B $ are parameters. (The case that either ...
0
votes
1answer
60 views

Level sets of a conserved quantity are trajectories of differential equation

If we have a differential equation $\mathbf{\dot{x}}=\mathbf{F}(\mathbf{x})$ and we have conserved quantity $E(\mathbf{x})$, which means $\dot{E}=0$, then I don't understand why level sets of $E$ are ...
15
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0answers
315 views

Notions of stability for differential equations

Consider a system of differential equations $$\dot{x} = f(x,u)$$ $$y = h(x,u)$$ where $x(t), u(t)$ are vectors in some $\mathbb{R}^n$. We define the infinity norm of a function in more-or-less in the ...
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0answers
109 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
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0answers
35 views

Making foliations C^1 smooth

I am not an expert in foliations, but I'm interested in the following conjecture of Cantwell and Conlon (from "An interesting class of $C^1$ foliations"): "If $F$ is a transversely orientable ...
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1answer
85 views

Free Vibrations - Simple Harmonic Motion

As an engineering enthusiast I have been practicing with numerous model assignments to see how well I could deal with dynamics if I were in an educational environment. Most problems seem simple to ...
3
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1answer
52 views

Commuting functions on the closed interval have the same value somewhere

We are given two commuting continuous functions $f,g:[0,1]\to[0,1]$. How can we prove that $f(x)=g(x)$ for some $x\in[0,1]$? A trivial observation is that if one of the two functions is a ...