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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Finding a parametrization

I am trying desperately to find a parametrization for the following: $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$ $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$ I tried to devide the equation ...
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1answer
52 views

$\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1]??

Let $\alpha$ be an irrational real number. I wonder whether $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1] in which $\{x\}$ means the fractional part of x. This is equivalent to the ...
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22 views

Projection of periodic trajectories

Let $(\bar x(t),\bar u(t)),\, t\in [0,1]$ solution of $$ \left \{ \begin{array}{l} \dot x_1 (t) = u(t)\, f(x(t)) \\ \dot x_2 (t) = u(t)\\ x_1(0) = 0, x_1(1)=1 \\ x_2(0) = x_2(1) \end{array} \right. $$ ...
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23 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
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1answer
49 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 $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
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1answer
24 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 ...
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27 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 ...
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113 views
+100

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!
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29 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^...
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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 ...
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25 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 ...
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1answer
67 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)...
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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 ...
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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 ...
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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 ...
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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 $...
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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 $...
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20 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, ...
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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 ...
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13 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 ...
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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?
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36 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 ...
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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 ...
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1answer
87 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 $x(t)$, $y(t)$?
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1answer
67 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 $f_1(x)$. ...
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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 $g$ ...
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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....
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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-...
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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 $...
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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 ...
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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 ...
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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 ...
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1answer
76 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 ...
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1answer
71 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 ...
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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 ...
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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 ...
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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) =...
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43 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:...
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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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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 ...
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71 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 ...
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1answer
33 views

Proving that an equation has a unique stable limit cycle

I'm preparing for my exam and I stumbled upon a question and I am a bit lost on how to write the correct solution. The question goes as follows: Prove that the equation $\ddot{x} + \mu(x^{4}-1)\...
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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 ...
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1answer
41 views

About minimal group actions?

Let $G$ be infinite group and $G$ act on compact metric space $(X, d)$, $\varphi:G\times X\rightarrow X$. $\varphi:G\times X\rightarrow X$ is called minimal action, whenever there is not proper ...
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18 views

Finding braches of equilibria

Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$ The point $x^* = (0,0,0)^...
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2answers
36 views

Finding the stable and unstable manifold of this system

Consider the system $$\begin{cases}\dot{x} = x \\ \dot{y} = -y + x^2\end{cases}$$ This has fixed point $\overline{X} = (0,0)$, which is a saddle point. The aim is to find the equation of the stable ...