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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219 views

Stability of linear systems with singular state matrix

Given a linear time invariant system $\dot X(t) = AX(t)$ where $X \in {R^{n \times 1}}$ and $A \in {R^{n \times n}}$ is a singular matrix ($A$ has at least one zero eigenvalue). How can I study the ...
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1answer
35 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu(A\cap T^{-n}B)=\mu(A)\mu(B)$ for all $n\geq N$ for ...
0
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1answer
20 views

Kinematics Mechanics Find the length of the belt and the speed of the rack. [on hold]

For A. I know that the formula for belt is simply $L=\frac{Pi(D_a+D_b)}{2}+2C+\frac{(D_b-D_a)^2}{4C}$ Which gives me $L= 116.99"$ since C is equal to $50$ For B. However I'm stuck and can't get the ...
2
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0answers
24 views

unstable manifold in 2d dynamical systems

I have 2d dynamical non-linear system $\dot{x} = f(x)$ with $x\in \mathbb{R}^{2}$ with exactly two stable attractors $x_{1} = ( a\quad b)^{T}$ and $x_{2} = (c\quad d)^{T}$ and one saddle point (One ...
2
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1answer
40 views

Stability of a line of equilibria

I'm working with a nonlinear autonomous system $x'=f(x)$. This system stays in $\mathbb{R}^n_+$ whenever it begins there, and it has a ray of equilibria, i.e. there is a positive vector $x_0$ so that ...
3
votes
0answers
43 views

How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!
0
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0answers
23 views

Hartman-Grobman Theorem in One Dimension

Here's a (possibly) related question Dynamics question from Devaney which is the one dimensional form of Hartman-Grobman theorem on the real line. Let $p$ be a hyperbolic fixed point for $f\in C^...
1
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0answers
23 views

Linear structural stability for maps on the real line

This might be a basic question, but I think I'm completely missing the idea of this. The question comes from Devaney's An Introduction to Chaotic Dynamical Systems (p.g. 59, Ex. 11) We define the ...
2
votes
1answer
65 views

Analytical Solution to Coupled Nonlinear ODEs

I am looking to solve several coupled nonlinear ODEs like this one: $\hspace{20mm} \frac{d x(t)}{dt} = C_1 \cdot x(t) + C_2 \cdot y(t) + C_3\cdot (x(t)^2 + y(t)^2) x(t),$ $\hspace{20mm} \frac{d y(t)...
1
vote
1answer
15 views

Intersection of the domains of the inverses of a one-parameter family

Context. Let $f_c:\mathbb{C}\rightarrow\mathbb{C}$ be a holomorphic function, depending holomorphically on the parameter $c\in\mathbb{C}$. Let $\alpha_0$ be a geometrically attracting fixed point of $...
0
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0answers
17 views

Time-periodic vector field vs. time-periodic Hamiltonian

Let $X(t,x,y): \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a time-periodic Hamiltonian vector field, i.e. $X(t+1,x,y) = X(t,x,y)$ for all $t$. Let's say that $X$ is generated by a ...
0
votes
1answer
23 views

Neighborhood of inclusion in space of Lipschitz maps is 1-1

Let $B \subset \mathbb{R}^n$ be the closed unit ball. Let $i(x) = x$ denote the inclusion map. Let $\|\cdot\|$ be any norm. Given $f:B\to \mathbb{R}^n$, define the sup norm $\|f\|_\infty:=\sup_{x \in ...
0
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0answers
30 views

Complex fixed points on the bifurcation diagrams

I'm working with bifurcation diagrams, an extesion that is being made of them is the determination of complex fixed points in addition to the real fixed points, my question is: what information ...
0
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0answers
12 views

How can I show that the lag-1 autocorrelation must increase?

Assume that we have a dynamical system $$ x' = f(x) $$ and that there is a repeated disturbance in the system after each time step $\Delta T$. Between disturbances, there is a return to equilibrium $...
2
votes
2answers
369 views

invariant measure under irrational rotation on $S^1$

Prove that if $T:S^1 \to S^1$ is an irrational rotation, then the only probability measure on $S^1$ that is $T-$invariant is the lebesgue measure or a multiple or it. We are considering the lebesgue ...
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vote
1answer
54 views

Must an expanding map be weakly expansive?

Let $(X,d)$ is a metric space and $X$ has no isolated points, $T:X\rightarrow X$ is a continuous self-map. Def1. $T$ is weakly expansive if there exist $\varepsilon>0$, for any $x,y\in X$, $x\neq ...
1
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0answers
15 views

Calculate Density of Values in Cellular Automata

I am working with a special cellular automata that uses hexagonal cells rather than square cells, a hexagonal grid, rather than a square grid, and the set of complex numbers, rather than a finite set, ...
2
votes
0answers
31 views

Convergence of Discretized Geodesics?

Let $M$ be a Riemmanian manifold, $p\in M$ and $V\in T_p(M)$. Suppose $f^{-1}:U_p \mapsto \mathbb{R}^D$ is a diffeomorphism of a neighborhood of p to an open subset of $\mathbb{R}$ and define the ...
0
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0answers
48 views

classification of equilibrium points of 3d systems of ode's

I'm trying to find information about the classification of equilibrium points of 3d systems of differential equations, The qualitative analysis. I wonder if someone could refer me to some book or ...
0
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0answers
11 views

about minimal point in non- autonomous discrete system

Let $(X,d)$ be a compact metric space. In $(X,f)$, $x\in X$ is called minimal point if $N(x,U)=\{n|f^{n}(x)\in U\}$ is syndetic for every open set $U$ of $x$ i.e. there is $k\in N$ such that $\forall ...
-1
votes
1answer
23 views

Are limit cycles isolated? [closed]

Given a $C^{1}$ vector field in $\mathbb{R}^{2}$, is it true that one can find neighborhoods around each limit cycle which contain no periodic orbits other than the cycle itself?
2
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0answers
31 views

Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
2
votes
1answer
86 views

Solving $\dot{x} = -\pi\dfrac{A}{k}\sin(kx)\cos(\pi y),$ $\dot{y} = A\cos(kx)\sin(\pi y)$

The following system is given: $$\dot{x} = -\pi\dfrac{A}{k}\sin(kx)\cos(\pi y)$$ $$\dot{y} = A\cos(kx)\sin(\pi y)$$ How can I find the parametric representation $x(t)$, $y(t)$?
0
votes
1answer
779 views

Separatrix curve in dynamical systems

I have this question: What is a separatrix of a equilibrium point of a continuous dynamical system and why it is flow-invariant? Thanks Hello and thanks for the answer. I explain better. I'm ...
6
votes
1answer
66 views

Iterating a multiple of sine function makes a square wave

So, I found something curious playing around with a graphing calculator. Say we start with a function, $f_1(x) = 2\sin(x)$ and we define a constant, $C$,to be the positive fixed point for $f_1(x)$. ...
0
votes
1answer
39 views

Orbits of orthogonal vector fields in $\mathbb{R}^{2}$ [closed]

Let $f$ and $g$ be two $C^{1}$ vector fields in $\mathbb{R}^{2}$ such that $\langle f(x), g(x) \rangle = 0 \,\,\,\forall \,\, x \in \mathbb{R}^{2}$. If $f$ admits a cyclic orbit, prove that $g$ ...
1
vote
1answer
35 views

Finding bifurcation of trigonometric system

I'm really struggling to find the bifurcation(s) of the system $x'=x^2 + \cos(x+ \mu)$, $\mu \in [0,2\pi)$. I've tried substituting $y=\mu+x$, taylor expanding, and just about everything else I ...
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0answers
36 views

Katok book Exercise 2.8.3 [closed]

Prove that for any holomorphic function w wich is not a polynomial there exist a number $\lambda = exp2\pi i \alpha$, where $\alpha$ is irrational, such that the linearized equation (2.8.3) does not ...
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vote
0answers
18 views

SIR model parameter notation

I am reading on SIR models and I found this article In the article it has three groups as one without vaccination, one with only whole cell(wP) vaccination, and one with only acelluar(aP) vaccination....
37
votes
3answers
1k views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know $$\underset{j=a}{\overset{b}{\LARGE\...
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0answers
21 views

Finding attractive and repelling part of the critical manifold.

We are given the following nonlinear system , $\frac{dI}{dt} = J \\ \frac{dJ}{dt} = -0.1\left(I^{3}(C - C_{0}\right)I - F - 0.2 J\\ \frac{dC}{dt} = \epsilon\left(F + \frac{C}{\sqrt{F^2 + C^2}}\left(1-...
2
votes
2answers
29 views

Canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$

When I read some materials in topological dynamics, I met words: "canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$" without any definition. I know that $...
0
votes
1answer
39 views

Linear stability bound proof

Question: Suppose $A$ is a $2 \times 2$ real matrix with two (not necessarily distinct) real negative eigenvalues. Show that there is a constant $K$ and a real $\alpha>0$ such that $|e^{At}x| \le ...
0
votes
1answer
30 views

Can I apply Hartman-Grobman when only one eigenvalue is zero?

Consider $\begin{bmatrix} \dot x\\ \dot y \end{bmatrix} =\begin{bmatrix} xy^2 -xy\\ x-y \end{bmatrix}$. If we take the Jacobian and evaluate it at $(0,0)$, one of the eigenvalues is $-1$ and the other ...
1
vote
1answer
75 views

Hyperbolicity without ergodicity?

I have a question concerning the ergodic properties of hyperbolic Hamiltonian flows. Let $\Phi_{H}^{t}$ be a Hamiltonian flow on a symplectic manifold $\mathcal{M}$. If $\Phi_{H}^{t}$ is Anosov on a ...
0
votes
0answers
21 views

Effect of dimension increase on the domain of attraction

Consider a nonlinear system with two locally stable fixed points $s_1$ and $s_2$ which have domains of attraction $D_1$ and $D_2$ respectively. Let $d_1$ and $d_2$ be domains of attraction of one ...
4
votes
0answers
106 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 ...
1
vote
1answer
70 views

Equation of the form $\mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)$.

Let $\mathbf{\Phi}(t)$ and $\mathbf A(t)$ be matrices satisfying the differential equation $$ \mathbf{\Phi}'(t)=\mathbf A(t)\mathbf{\Phi}(t)\ . $$ If I am not mistaken, if $\mathbf A$ and its integral ...
1
vote
1answer
57 views

How to predict the incidents of synchronization for multiple oscillations.

EDIT: I changed the title of this question and made this edit based on a conversation with a friend. While I am dealing with mechanical cams the plain fact is that what I have is an oscillation in ...
0
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1answer
63 views

Prove that orbits of one system are orbits of another

Let $$\left\{\begin{align} \dot x &= x- \frac{xy}{1+\alpha x}\\ \dot y &= -y + \frac{xy}{1+\alpha x}+\delta y^2 \end{align} \right.$$ be a predator-prey model. Prove that the following ...
3
votes
1answer
208 views

Why this vector field $f$ belongs to $C^1({\bf R}^2\times {\bf R})$?

The following system is an example in a book of dynamical system(in the section about Hopf Bifurcation). $$ \begin{align} \dot{x}=\mu x- y-x\sqrt{x^2+y^2} \\ \dot{y}=x + \mu y-y\sqrt{x^2+y^2} ...
0
votes
1answer
29 views

Fixed point analysis in the Wilson-Cowan model

i guess this is a rather simple question, but given my non-mathematical background, i'm a bit stuck. i'm trying to find the jacobian matrix for the follwing dynamical system (wilson-cowan model). the ...
0
votes
1answer
43 views

Compute the fundamental matrix solution for a given system

Consider the system: $$\left \{\begin{align}\dot x &= x-y-x(x^2+y^2)\\ \dot y &= x+y-y(x^2+y^2) \end{align}\right.$$ Find the fundamental matrix solution $Y(t)$ explicitly, assuming $x(0) =...
4
votes
0answers
42 views

Quantifying Poincare map

I have a dynamical system which goes from chaos to ordered state (quasiperiodic state to be precise). I have represented this transition via a Poincare map. See the attached figures. Now, my question:...
0
votes
1answer
30 views

What are the equilibrium point of this coupled ODE?

Consider $$\dot x = x(a - bx - cy)$$ $$\dot y = y(-d + ex - fy)$$ $$a,b,c,d,e >0, f \geq 0$$ Find all the equilibrium points in the set $\mathbb{R}^2_{\geq 0}$ I can find by inspection the ...
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0answers
68 views

How to analyze ODE equilibrium stability with complex equilibria

Take this example: $y'=y^2+1$. There's no "real equilibrium", but is it right to say it has two "complex equilibria"? If so, what should be the conclusion of the derivative test? $$y'' = 2y \implies ...
5
votes
0answers
55 views

Sketching the global phase portrait for a version of the Lotka-Volterra system

I'm trying to sketch the phase portrait for a version of Lotka-Volterra given by $$\begin{cases} \dot{x} = x(3-x-2y)\\ \dot{y} = y(2-x-y) \end{cases}.$$ I can sketch this just fine except for the ...