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 ...

learn more… | top users | synonyms (1)

8
votes
0answers
426 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
8
votes
0answers
214 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, ...
7
votes
0answers
95 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 ...
6
votes
0answers
370 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
votes
0answers
168 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 ...
5
votes
0answers
64 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
votes
0answers
163 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
votes
0answers
151 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 ...
5
votes
0answers
55 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] $y(0) = 0$; $t_{0}=0$; $\alpha$, ...
4
votes
0answers
132 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
54 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
36 views

One-parameter group for spheres

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
4
votes
0answers
66 views

A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor ...
4
votes
0answers
86 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
125 views

two fixed points, same fractional iteration

Suppose a function f(z) has two fixed points, one repelling, and the other attracting. Call the repelling fixed point f(-1)=-1, and the attracting fixed point f(+1)=+1. I'm interested in functions ...
4
votes
0answers
45 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
votes
0answers
177 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
0answers
271 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 ...
3
votes
0answers
48 views

Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated! Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism ...
3
votes
0answers
171 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 $z>-1$ ...
3
votes
0answers
55 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} ...
3
votes
0answers
36 views

Ergodicity and Appropiate Partition of the Space

I'm trying to solve the following problem. Let $(X, \mathcal{B}, \mu)$ be a probability space and let $T \colon X \to X$ be a measure preserving function. Prove that if $T^n \colon X \to X$ is ...
3
votes
0answers
43 views

Showing iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
3
votes
0answers
18 views

inequality involving lifts of a positive oriented homeomorphism of the circle

Let $\pi: \mathbb R \to S^1$ be the natural projection and let $f:S^1 \to S^1$ be a positive oriented homeomorphism. We say that $F: \mathbb R \to \mathbb R$ is a lift of $f$ is $\pi \circ F = f \circ ...
3
votes
0answers
50 views

Dynamical Systems: Attractivity and Stability

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE $$dx/dt=-x$$ which gives the system $$S_tx=x_0e^{-t} $$ For the compact ...
3
votes
0answers
23 views

Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
3
votes
0answers
40 views

Henri Poincaré writings

I have heard that Poincaré writings were very intuitive in its approach and not very formal in the arguments. I'm searching for something like this to complement my study of dynamical systems. I ...
3
votes
0answers
49 views

I need urgent help with understanding my notes; period 2 solutions

Consider the following functions: $X_{n+1} = 2aX_na$ on $0 \leq X_n \leq \dfrac{1}{2}$ $X_{n+1} = -2aX_n + 2a$ on $\dfrac{1}{2} \leq X_n \leq$ We want to find if there are period 2 solutions for ...
3
votes
0answers
52 views

temporal operators: interpreting them topologically in a dynamic topological system

There's a paper that i've been reading recently called "Dynamic Topological Logic" which can be found at: http://individual.utoronto.ca/philipkremer/onlinepapers/DTL.pdf. I have a question about ...
3
votes
0answers
129 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 ...
3
votes
0answers
86 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 ...
3
votes
0answers
98 views

Differential equation $y'(t) = 1-y(t) e^{y(t)-1}$

I am interested in finding a clean explicit solution (if possible) to the differential equation $$ y'(t) = 1-y(t) e^{y(t)-1}, $$ where $0 \le t < 1$ and $0 \le y \le 1$. This can obviously be ...
3
votes
0answers
87 views

Are geodesic flows on surfaces with negative curvature Anosov?

I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
3
votes
0answers
98 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 ...
3
votes
0answers
88 views

Equivalence of two definitions of Lyapunov exponents

Yesterday I asked a question on eigenvalues of $A^{T}A$. Eigenvalues of $A^{T}A$ The reason why I asked this question was that I'd seen in articles two different definitions for Lyapunov exponents ...
3
votes
0answers
104 views

Hopf bifurcation

Consider the scalar linear retarded equation (with infinite delay) $$ x^{\prime}(t)=-ax(t)+b\int_{0}^{\infty}x(t-\tau)f(\tau)d\tau\;, $$ where $a,b\in\mathbb{R}$, $\tau\in\mathbb{R}^{+}$, and $f$ is ...
3
votes
0answers
408 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
3
votes
0answers
119 views

The measures in Furstenberg's correspondence

In the paper Inverting the Furstenberg correspondence (IFC), the author defines a function $D_{A}(\sigma)$ on the Basic clopens of Cantor space, $2^{\mathbb{N}}$, where $A$ is a finite binary string ...
3
votes
0answers
102 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 ...
2
votes
0answers
33 views

perturbation theory solution of forced Duffing's equation

Question: Find the leading order of the asymptotic expansion for large t: $\frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=Fcos(\frac{1}{3}\big(1+\varepsilon\omega)t\big)$ I have ...
2
votes
0answers
18 views

For all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for all $p \in \Delta_1$

Let $X_1$ and $X_2$ fields in $\Delta_1,\Delta_2$ subset open in $\mathbb{R}^n$. Then, for all topological conjugation $$h: \Delta_1 \rightarrow \Delta_2$$ we have to $h(\omega(p))=\omega(h(p))$, for ...
2
votes
0answers
18 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 ...
2
votes
0answers
34 views

Period 3 window in chaos

Li and Yorke established the fact that period 3 implies chaos, which implies that if we have a period 3 orbit in a system then we have a chaotic system. I have seen that in bifurcation diagrams there ...
2
votes
0answers
23 views

Identification of (Centers of) Cycles in a Discrete Time Dynamical System

I am studying dynamics on nonlinear Discrete Time Dynamical System of the form $$ \vec{X}_{t+1} = D(\vec{X}_t), $$ where D is some nonlinear function. I was looking for a (relatively) quick ...
2
votes
0answers
45 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 ...
2
votes
0answers
31 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 ...
2
votes
0answers
53 views

Physical interpretation of Ergodicity.

If $R_{\alpha}:[0,1] \to [0,1]$ is defined by $$R_{\alpha}(x)=x+\alpha $$ then $R_{\alpha} $is called a circle rotation, and it is known that $R_{\alpha}$ is ergodic iff $\alpha$ is irrational. I ...
2
votes
0answers
30 views

toplogical entropy of general tent map

Measure theoretic entropy of General Tent maps The linked question made me wonder how to calculate the topological entropy of a general tent map. Let $I=[0,1]$ and $\alpha \in ( 0,1)$. Define $T: I ...
2
votes
0answers
32 views

Relationship between Reproductive Ratio and Jacobian in Population Model

In class we defined the Reproductive Ratio, $R_0$ of a population modelled by SIR, SEIR,... as the average number of secondary infections caused by an average infected individual in an average ...
2
votes
0answers
33 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...