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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44 views

Foliations vs Laminations

What's the big difference/similarity between foliations and laminations? What kind of theorems hold for both of them? Is there something that makes them essentially the same/different?
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47 views

proving asymptotic stability dynamical system

I want to show the origin of the dynamical system \begin{align} \dot{x}_1 &= -2x_1+x_2+x_1^3x_2^2\\ \dot{x}_2 &= -x_1-2x_2+x_1^2x_2^3 \end{align} is asymptotically stable over an invariant ...
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1answer
27 views

Differentiation of unit force vector

I was reading a paper and don't know how the following was derived. Given that $f = \begin{bmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} + g \\ \end{bmatrix}$ and ...
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1answer
25 views

Two ODE planar systems that are orthogonal to each other

Given two planar systems X'=F(X) and X'=G(X) (so F and G are both $C^1$). Assume the dot product of F(X) and G(X) is always zero on $R^2$. Now if F has a closed orbit, prove that G has a zero. My ...
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32 views

Computing eigenvectors for an eigenvalue of a dynamical system

$\dot{x}=x\left ( 3-x-2y \right )$ $\dot{y}=y\left ( 2-x-y \right )$ In matrix form: $\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix}$ $=\begin{bmatrix} \left ( 3-x-2y \right ) &0 \\ 0& ...
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3answers
22 views

solving coupled equation

$\dot{x}=x\left ( 3-x-2y \right )$ $\dot{y}=y\left ( 2-x-y \right )$ The above is a coupled equation. The fixed point condition requires all x,y for which $x\ast =0$ and$ y\ast=0$ solving, I arrive ...
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2answers
25 views

Time shifted function in differential equation from a control systems problem

Suppose we had the following differential equation with zero initial conditions: $$\ddot{x}\left(t\right)+2\dot{x}\left(t\right)+x\left(t\right)=0.5u\left(t\right)$$ where $u\left(t\right)$ is given ...
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0answers
27 views

'Fast' and 'slow' Eigendirection?

Can someone give an intuition and a definition of what a "fast" and "slow" eigendirection means? A reasonable google search reveals nothing that would help. Thanks in advance.
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1answer
46 views

Proving solutions to the anisotropic kepler system that meet certain constraints lie on the position axes of configuration space

The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{3/2}} \end{equation*} \begin{equation*} y''=\frac{-y}{(\mu x^2 + y^2)^{3/2}} \end{equation*} With $\mu>1$ a constant ...
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1answer
36 views

Fixed points of dynamical systems

I haven't been able to find an answer to this question anywhere, so I thought I should post it here. In doing a fixed-points/stability analysis, one is required to find the fixed points of a dynamical ...
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1answer
9 views

Discrete-time variation of constant formula

The difference equation is given \begin{align*} x(k + 1) = Ax(k) + Bu(k) \end{align*} with an initial condition $x(0)= x_0$. Inductively, I derived the solution $$x(k) = A^kx_0+\sum_{j=0}^{k-1} ...
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2answers
33 views

Finding a circle in polar coordinates

I have converted the system of ODEs, $$x'=x-y-x(x^2+5y^2)$$ $$y'=x+y-y(x^2+y^2),$$ to polar coordinates and got this: $$ r' = r-r^3(1+4\sin^2(\theta)\cos^2(\theta))$$ ...
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2answers
71 views

Phase portrait of ODE in polar coordinates

Given the system of ODEs in polar coordinates, $$r' = r(1-r^2)(4-r^2)$$ $$\theta'=2-r^2,$$ one can determine its equilibrium points and limit cycles as follows: $\gamma_1:= \begin{cases} r = 0,\\ ...
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57 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 ...
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1answer
94 views

If the orbit of a point $x$ is a closed set, then either $x$ has a periodic iterated or its omega-limit is empty.

QUESTION: Let $X$ be a topologically complete metric space and $T:X\to X$ a continuous map. Let $x\in X$ be a point whose orbit is a closed set. Show that either $x$ has an iterated that is periodic ...
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0answers
26 views

Is there a way to directly recover differentials from an integral; anti-separation of variables?

I have an expression that relates the tension in a string, moving in two dimensions, to its acceleration. I'm sure that it's a solved problem, but in my exploration I saw that twice the tension is the ...
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1answer
35 views

Approximating Borel Measure with Atomic Measures

I see some posts that are related to this one, e.g. Borel Measures: Atoms (Summary) I have a sort of particular question: I have one professor saying the following is true, while another says it's ...
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1answer
20 views

About Duffing equation

Is there a relation between Duffing equation and Van der Pol equation? My second question is what is the application(s) of stochastic Duffing equn. in practice ?
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1answer
79 views

Is every basin of attraction completely invariant?

I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
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1answer
29 views

Fixed points of $u'= u(1-u+ \alpha v)$, $v' = \rho v(1-v + \beta u)$

Hopefully a simple question however it has me stumped... I have the following system: $\frac{dN_1}{dt} = r_1N_1(1- \frac{N_1}{k_1} + b_{12} \frac{N_2}{k_1})$ $\frac{dN_2}{dt} = r_2N_2(1- ...
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1answer
23 views

Proof involving Poisson bracket

Not being able to understand how each term has been simplified to get from the third step to the fourth step. So how did 1/2m become 1/m and {qj,plpl}pk become {qj,pl}plpk and how did k/4 become ...
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0answers
23 views

How to do this Poisson bracket proof

For the proof of the above equation, I understand the first step which has been obtained from the definition but in the second step I don't understand why they are summing over $j$ first ...
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1answer
19 views

Poisson bracket proof

For this question I understand the first line of the solution which they have obtained from the definition but how have they simplified each term to get to the second line from the first line? The ...
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0answers
23 views

Parameter estimation - Holt's Two parameter Linear Exponential Smoothing

The reference for the below equations can be found in the Link . Note that $k$ is the timestamp and $i$ is the $i^{th}$ entry of a vector or $(i,i)^{th}$ entry of a matrix, $F$ in this case Equation 1 ...
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1answer
24 views

Is a gradient system considered an ODE or PDE?

When you have a system of the type $$\dfrac{dx(t)}{dt} = \nabla V(x)$$ Is this considered an ODE or a PDE? Because you have a single derivative with respect to $t$ on the lefthand side, whereas on ...
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68 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, ...
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1answer
55 views

Finding periodic points of diffeomorphism of the circle

I want to find all the periodic points of the following diffeomorphism of the circle: $f(x) = x + \frac{1}{4} + \frac{1}{10} \sin(8 \pi x) \mod 1$ Where a periodic point is $p$ such that $f^n(p) = ...
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1answer
32 views

Poisson bracket proofs

I understand the first sentence you wrote for the need of a different summation index. However, i'm still not able to understand the individual steps. Like how in the first line we have four ...
0
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2answers
29 views

Sketching a dynamical system

Sketch the dynamical system \begin{align} \dot x_1 = x_2 \\ \dot x_2 = 1 \end{align} Firstly we may integrate this to find $$x_2(t) = t + A$$ $$x_1(t) = \frac12t^2+At + B$$ How do I then change ...
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1answer
27 views

How to model this recurrence?

I'm having some problems on how to model this situation correctly, using difference equations. Say there's a medicine that has a half-life of 12 hours (every 12 hours, the amount of it on your blood ...
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0answers
20 views

How does Melnikov function for a Hamiltonian change if one considers an augmented symplectic manifold?

Suppose we have a nonautonomous nearly integrable Hamiltonian system, periodic in $t$ with period $2\pi / \omega$ $$H_{\epsilon}(x,y,t)=H_{0}(x,y) + \epsilon H_{1}(x,y,t)$$ with $(x,y,t) \in ...
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0answers
29 views

How to discretize state space with uniform grid

Let us consider a general continuous time stochastic differential equation represented by *dx* = A(x)dt + B(x)udt + $\sigma$ dw where A(x) represent the ...
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1answer
35 views

On “bounded” in intuition for a theorem involving autonomous 1st-order ODEs

Suppose we have an autonomous first-order ordinary differential equation $$\frac{dx}{dt} = f(x) \tag{*}$$ where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE ...
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2answers
275 views

Prov that function is eventually periodic to origin.

Let $f:\mathbb{Z}^4 \rightarrow \mathbb{Z}^4$ by $f(w,x,y,z) = (\mid w-x \mid,\mid x-y \mid,\mid y-z \mid,\mid z-w \mid)$ Prove that for any $(w,x,y,z) \in \mathbb{Z}^4$ there is $n>0$ such that ...
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0answers
30 views

Two trajectories $x_1(t)$ and $x_2(t)$ of the six dimensional system

Two trajectories $x_1(t)$ and $x_2(t)$ of the six dimensional system $\dot{x}=A_{6x6}x$ are given by $$x_1(t)=t^2e^{-2t}v_1+te^{-2t}v_2+e^{-2t}v_3+v_4 ,\> \> x_2(t)=v_5+e^{-t}v_6 \> \> ...
1
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1answer
45 views

Index of circle containing fixed point

Given $\dot x = x^2-y^4 $ and $ \dot y = y^2 -x^4 $ Find the index of the circle $x^2 + y^2 = a^2$ with $a < 1$ Attempt: I employed linear analysis by finding all the fixed points. There are ...
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1answer
47 views

On corollary and theorem involving autonomous 1st-order ODEs

Suppose we have an autonomous first-order ordinary differential equation $$\frac{dx}{dt} = f(x) \tag{*}$$ where $f$ is continuously differentiable for all $x \in D \subseteq \mathbb R$ s.t. the ODE ...
2
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2answers
26 views

Invariant Subspaces Dynamical Interpretation

Consider the linear system $\dot{x}=A_{nx n}x$ ; $x(0)=x_0$. Recall that a subspace $\mathcal{U}$ s (dynamically) invariant to the flow if for all initial conditions $x_0 \in \mathcal{U}$ the ...
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1answer
29 views

Nonhomogenous Linear System

You are given the time dependent linear system in $\mathbb{R^n}$ $\bf{\dot{x}}$=$A$$\bf{x}$ where $A$ is a fixed $n\times n$ matrix and $b : \mathbb{R} \rightarrow \mathbb{R^n}$ is continuous. $i)$ ...
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1answer
28 views

stability of the equilibrium at the $\bf{x}=0$

Consider the system $\bf{\dot{x}}$=$A$$\bf{x}$ in $\mathbb{R^6}$. You are given that the null-space of $A$ has dimension 2 and the stable subspace of $A$ has dimension 2 and that is an ...
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1answer
45 views

Root mean square function (RMS) in the ODEs of a dynamical system

I am trying to obtain the differential equations of a dynamic system in which one of the blocks calculates the RMS of a signal. When arranging the ODEs I see that the differential version of the RMS ...
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31 views

Stability of the equilibrium at the origin

Consider the system $\bf{\dot{x}}$=$A_{3x3}$$\bf{x}$ You are given that $A$ is nonsingular and the unstable subspace of $A$ is trivial. Is this information sufficient to infer the stability of ...
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1answer
44 views

Dynamical System in Polar Coordinates

I have a dynamical system defined by : $ \dot x = {(x+iy)^n + (x-iy)^n \over2}$ and $\dot y = {(x+iy)^n - (x-iy)^n \over2i}$ Converting the system to polar coordinates gives the system: $\dot r = ...
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2answers
51 views

System of ODEs - from Cartesian to polar

Given the system of ODEs, $$\dot{x}=x^2+3y^2-1$$ $$\dot{y}=-2xy$$ How does one transform it into polar coordinates $(\rho, \theta)$? Here's my line of reasoning: let $x=\rho \cos(\theta)$, $y=\rho ...
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3answers
50 views

modal truncation of state space system while preserving certain eigenvalues

Given a state space system $(A,B,C)$: $$\dot{x}=Ax+Bu\\y=Cx$$ Is there any method to obtain a reduced system $(A_r,B_r,C_r)$, where $$\dot{x}=A_rx+B_ru\\y=C_rx,$$ such that the eigenvalues of $A_r$ ...
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1answer
43 views

How to determine whether dynamical systems are dissipative, conservative or expanding?

I have the dynamical systems: $dx/dt=y$, $dy/dt=-w^2x$ $dx/dt=y$, $dy/dt=-by-w^2x$ $dx/dt=a(y-x)$, $dy/dt=x(b-z)-y$, $dz/dt=xy-cz$ with $a,b,c,w\in\mathbb{R}$ How do I ...
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1answer
98 views

Show that $\bf{x^*}$ is an equilibrium point.

Let $\dot{\bf{x}}=f(\bf{x})$ be a dynamical system with $f: \mathbb{R}^n\rightarrow \mathbb{R}^n$ being continuous and locally Lipschitz. Suppose that a particular solution trajectory satisfies ...
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1answer
69 views

How many completely invariant domains can there be for a rational function?

I am considering rational functions $R:\overline{\mathbb{C}}\rightarrow\overline{\mathbb{C}}$ of degree $d\geq 2$. A completely invariant set $U$ is a set for which it and its complement are ...
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0answers
31 views

Systems of second order differential equation

i'm following a course in Hamiltonian systems and regarding the part of linear systems I found this exercise from a book and need to solve it. My ideas are just after the test of the exercise. ...
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1answer
72 views

Arnold's proof of Liouville's Theorem on integrable systems

My question happens to be almost identical to the one left unanswered/closed here, which gives a bit of background information - it may not be necessary. I hope the reason it was closed on ...