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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343 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 ...
10
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250 views

Intuition for the Yang-Baxter Equation (was: Giving relations via formal power series)

I'm reading a book (Yangians and Classical Lie Algebras by Molev) which regularly uses (what appear to me to be) clever tricks with formal power series to encapsulate lots of relations. For instance, ...
9
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60 views

Differential equations with dense solutions

Consider the differential equation $P(y',y'',y''',y'''')=0$ on $\mathbb R$, where $P(x,y,z,w)$ is the homogeneous polynomial of degree $7$ given by $$ ...
9
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179 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert ...
9
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280 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 ...
7
votes
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159 views

Eigenvalues of a quasi-circulant matrix

The following matrix cropped up in a model I am building of a dynamical system: $$A= \begin{bmatrix} 1 - \alpha & \alpha/2 & 0 & 0 &\cdots & 0 & 0 & \alpha/2\\ \alpha/2 ...
6
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146 views

Transforming the solutions of $\dot x = f(\mathbf{x}, t)$ and $\dot x = B(\mathbf{x}, t)f(\mathbf{x}, t)$ into each other

How can I prove the following theorem? If the function $B(\mathbf{x}, t)$ is strictly positive, then the solutions of the two differential equations $\dot x = f(\mathbf{x}, t)$ and $\dot x = ...
6
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245 views

Stability for Nonlinear System

I am trying to assess the (Liapunov) stability of the equilibrium at $(0,0)$ of the system \begin{align*} x_1' &= -4x_2 + x_1^2 \\ x_2' &= 4x_1 + x_2^2. \end{align*} I plotted the phase ...
6
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75 views

Separable non-linear ODE (with radicals)

I am trying to solve the equation $$ \frac{dy}{dt}=\sqrt{(\gamma-1+\frac{2\alpha\beta}{2\alpha-1})e^{-2\alpha y}-\frac{2\alpha\beta}{2\alpha-1}e^{-y}+1} $$ [1] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...
6
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419 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
5
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50 views

Nuclear operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
5
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101 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 ...
5
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0answers
110 views

Baker's map is ergodic

Define Baker's map by \begin{align} f(x,y) = \begin{cases} (2x,y/2) & \mbox{ if } (x,y)\in[0,1/2]\times[0,1] \\ (2x-1,y/2+1/2) & \mbox{ if } (x,y)\in[1/2,1]\times[0,1] \\ \end{cases} ...
5
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92 views

Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At ...
5
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148 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
5
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90 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: ...
5
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137 views

Earnshaw's theorem

Proposition Suppose $U\colon\Omega\to\mathbb R$ is a non-constant harmonic function, i.e. $U\in\mathcal C^\omega$, i.e. analytic, and $\Delta U=0$, where $\Omega\subseteq\mathbb R^n$ is a region. ...
5
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292 views

Difference between dynamic system and complex system

I know this might not be an easy question, I've already read the wikipedia page, and there is an interesting view: Therefore, the main difference between chaotic systems and complex systems is ...
5
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230 views

Proving that $f$ has points with prime period 3

Exercise: Let $f:\mathbb{R}\longrightarrow\mathbb{R} $ be continuous, $n>3$, and $x_1,x_2,\cdots,x_n$ be points such that $x_1<x_2<\cdots<x_n$. Show that if $f(x_i)=x_{i+1}$ for ...
5
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67 views

Definition and some elementary properties of the “vector turn map”

This is actually a follow up question. I have to apologise for the length of it. I didn't anticipate to be that long. I hope that it will be proved interesting nevertheless. I used the name "vector ...
4
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36 views

a dynamical systems view of the central limit theorem?

I have seen many heuristic discussions of the classical central limit theorem speak of the normal distribution (or any of the "stable distributions") as an "attractor" in the space of probability ...
4
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40 views

Is there a classical analog of Bloch's theorem?

In quantum mechanics, having a spatially periodic Hamiltonian imposes a lot of structure on solutions of Schrodinger's equation (e.g. band structure), primarily due to Bloch's theorem. In perfect ...
4
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37 views

a weak notion of flow in a metric space

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is ...
4
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68 views

Structure of first-coordinate-projection of set of solutions of “elliptic” diophantine equation $xy(6-(x+y))=6$

Say that a rational number $a$ is good iff there is a rational number $b$ with $ab(6-a-b)=6$, or equivalently iff $a^4 - 12a^3 + 36a^2 - 24a$ is the square of a rational number. Denote by $G$ the set ...
4
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53 views

Help on designing a dynamical system

I would like to build a four-dimensional dynamical system that has the following behavior: Here, $x_1, x_2, x_3$ and $x_4$ are the four dimensions, and each axis has a fixed point that should be a ...
4
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186 views

Has this difference equation :$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$ been studied before?

I posted this problem before in MO but only I would like to know if this difference equation :$$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are ...
4
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79 views

About the existence of a regular solution in a chaotic system

Show that for some choice of $x_1\in\mathbb{R}$, the sequence given by: $$ x_{n+1} = n-3^{x_n} $$ satisfies: $$ \lim_{n\to +\infty} x_n= +\infty. $$ I was able to prove the statement by showing ...
4
votes
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67 views

Bottlenecks in Dynamical Systems

Consider the equation $\dot{x} = r+x^2$. When $0 < r \ll 1$, this system experiences a bottleneck effect. Then the time $T$ spent in this bottleneck can be approximated by: $$T_{bn} = ...
4
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88 views

Connection between possibility of non-monotonic solutions to first-order delay differential equations and 1-d discrete dynamical systems?

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the ...
4
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78 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} ...
4
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78 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
4
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0answers
31 views

properties of certain semigroup action on $\mathbb{Z}/p\mathbb{Z}$

Suppose we have a polynomial $f \in \mathbb{Z}/p\mathbb{Z}[x]$, $f(x) = x^2 - x$. We are interested in elements $n \in \mathbb{Z}/p\mathbb{Z}$ such that after repeated application of f they ...
4
votes
0answers
107 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} ...
4
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191 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 ...
4
votes
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92 views

Determining whether the origin is an attracting fixed point for a scalar system

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE \begin{equation} \frac{dx}{dt} =-x \end{equation} which gives the system ...
4
votes
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277 views

Cross Product - Moments :: Dynamics

Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the ...
4
votes
0answers
101 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
4
votes
0answers
95 views

An element of $\ell^2$ wanted

I am looking for an element $x=(x_0,x_1,x_2,\cdots)$ in $\ell^2$ such that the sequence $z_n, n=0,1,2,\cdots$ defined by $$z_n=2^n(x_n, x_{n+1},\cdots)$$ is dense in $\ell^2$. It seems that this is ...
4
votes
0answers
116 views

What is this bifurcation?

I have a discrete dynamical System $x_{n+1}=f(x_{n},x_{n-1},x_{n-2},x_{n-3},x_{n-4},\lambda)$ with a paramteter $\lambda>0$, and where all $x_{n}$s are in [0,1]. $f$ is actually a larger ...
4
votes
0answers
230 views

Rewriting the advection-diffusion equation

This is mostly a reference request question, although I certainly appreciate any insights and/or comments. Let us assume $p:R^n×(0,∞)\to \mathbb R$ is a scalar concentration, $u\in R^n$ is the ...
4
votes
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148 views

Bifurcation in 3 dimensions (simple)

I am Doing a project i have a toy system that describes a bifurcation in 3 dimensions i am posting this in part because i can no longer understand what i have written down ( its been awhile) i have ...
4
votes
0answers
421 views

How to prove stability of this dynamic system?

I'm trying to prove stability of the following dynamic system but I think my Mathematics knowledge is not deep enough. My dynamic system consists of a state vector $x \in \mathbb{R}^n$. The system ...
4
votes
0answers
133 views

The Hausdorff dimension of the set of solutions of a system of coupled differential equations

I am interested in the relationship between non-linear differential equations and the Hausdorff, or fractal, dimension of the set of solutions. For example, the Lorenz Attractor, which is a standard ...
3
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0answers
21 views

How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter ...
3
votes
0answers
67 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
3
votes
0answers
107 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
3
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31 views

Omega-limit set consists of one point. Does this mean the orbit tends to this point as $t$ grows?

The $\omega$-limit set is defined in this wikipedia article. My question is: If we have an orbit $x_t$ and the $\omega$-limit set of this orbit contains one element $w^*$, does this imply ...
3
votes
0answers
55 views

Asymptotic analysis of non-linear ODE

I'm looking for references on the topic of asymptotic analysis of non linear ODE's of the sort $$ x'' + x'x = 0 $$ This specific case has an analytic solution (with some $\tanh(\cdots)$ involved) and ...
3
votes
0answers
47 views

Understanding Proof of Poincare Recurrence Theorem

I'm trying to follow a proof in my book of the Poincare Recurrence Theorem, but I have three questions about this proof: Theorem Let $(X,\Sigma,\mu$ be a finite measure space, $f:X\to X$ be a ...
3
votes
0answers
78 views

Two Matlab ODE solvers, two different results

I am solving a system of ODEs using Matlab. One particular set of parameters caused the solver to fail, so I worked my way through the different solvers Matlab provides. I was surprised to find that ...