2
votes
0answers
42 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 ...
0
votes
0answers
18 views

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? [duplicate]

if $X$ is a vector fild in $\mathbb{R}^3$ and $h$ is a periodic orbit, then $X$ have a singularity? and in dimension $n$? I know there is singularity when $n=2$.
2
votes
0answers
15 views

If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit. [closed]

Let $X$ be a field in $\mathbb{R}^3$, $C^1$ class. If $p$ is a regular point $X$ such that $p \in \omega(p)$ then $\omega(p)$ is periodic orbit.
1
vote
0answers
17 views

Maximum intervals of a solution and singularities [closed]

Let $X$ be a vector field of $C^1$ calsse in $\Delta \subseteq \mathbb{R}^n$. Prove that if $\varphi(t)$ is a trajectory of $X$ defined maximum range $(\omega_-,\omega_+)$ with: $$\lim_{t \rightarrow ...
2
votes
0answers
20 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 ...
1
vote
1answer
60 views

Compact $\omega$-limit set $\Rightarrow$ connected

Consider the flow $\varphi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ and $L_{\omega}(x)$ the $\omega$-limit set of a point $x \in \mathbb{R}^n$. How can I show that if $L_{\omega}(x)$ is ...
1
vote
1answer
25 views

Is $ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $ dense in the rectangle $ [- A,A] \times [- B,B] $?

What conditions must $ a $ and $ b $ satisfy in order for the curve $$ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $$ to be dense in the rectangle $ [- A,A] ...
1
vote
0answers
26 views

Finding the homomorphism that links the linear part of a dynamical system to the nonlinear part.

here is a picture of my problem Basically what i have is that i was told i could find this homomorphism by doing the following ...
1
vote
1answer
75 views

Energy Transfer in a Mechanical System - Standard Pulley Scenario

I understand that this is quite a basic question, I am new to dynamics and have trouble starting off questions, I found it quite difficult to find an example question alike to the one below thus I am ...
3
votes
0answers
45 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
1answer
57 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
1
vote
1answer
33 views

What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
2
votes
1answer
86 views

Show that system is Transcritical bifurcation

In what ways can you show that transcritical bifurcation occurs? For example take the system $$\dfrac{dx}{dt}=xr+2x^2 $$
1
vote
2answers
74 views

A system of nonlinear differential equations

We have the following system in $\mathbb{R}^{2}$ $$\dot{y}_1=2-y_1y_2-y_2^2$$ $$\dot{y}_2=2-y_1^2-y_1y_2$$ i) Calculate the equilibrium points en determine their stability. ii) Draw the Phase ...
1
vote
2answers
76 views

linear differential equation problem [closed]

Consider the following system of linear differential equations: $$\begin{split} \frac{dx}{dt}&=−3x+y\\ \frac{dy}{dt}&=x−3y \end{split}$$ Find the eigenvalues and eigenvectors associated ...
1
vote
1answer
61 views

Equilibria and stability

Find all equilibria for the following system and determine their stability: $$x'=y^2-4$$ $$y'=x^2-1$$
1
vote
1answer
247 views

calculus, predator-prey system

The following system describes a predator prey system in which the prey has an Allee effect. What is the threshold of the prey to persist when alone? Find the nullclines and the steady states of the ...
1
vote
2answers
74 views

How do I Linearize

How would I solve the following problem? Linearize around the fixed points $$\left\{\begin{align}\frac{\text{d}x}{\text{d}t}&=y-x^2\\\frac{\text{d}y}{\text{d}t}&=y-x\end{align}\right.$$ I ...
0
votes
1answer
49 views

Prove that $(A,B)$ is uncontrollable $\Longleftrightarrow$ $\exists P$ $\in$ $\mathbb{R}^{nxn}$, $P \neq 0$: $PA - AP = 0$, $PB=0$

In my course advanced system Theory I had the following question: Prove the following equivalence for the pair $(A,B)$ $\in$ $\mathbb{R}^{nxn}$ x $\mathbb{R}^{nxm}$: $(A,B)$ is uncontrollable ...
1
vote
0answers
35 views

Prove that A is marginally stable iff there exist a $P$ $\in$ $S^n$, $P \succ 0$ such that $A^T + PA \leq 0$

Asymptotic stability, which means that all eigenvalues of A are in the open left half plane is easily proven. See the scan in the attachment. However, in the book the proof for the second case where ...
0
votes
1answer
34 views

Periodic points of topologically conjugated functions in dynamical systems?

I'm working on a homework problem which seems obvious, but I am having a hard time proving/completing. The problem can be stated as follows: Let $f,g:$ $\mathbb R$ $\rightarrow$ $\mathbb R$ be ...
0
votes
1answer
46 views

Topological conjugation between two flows

This is problem 1.53 from Ordinary Differential Equations by Chicone (2nd Ed). Prove that there are open intervals $U, V\subset \mathbb{R}$ both containing the origin and a differentiable map $H: U ...
0
votes
1answer
37 views

Kinematics Question Help

I'm having trouble with this question: A particle moves so that its position vector with respect to the origin $O$ of a reference frame $Oxyz$ is $$ \mathbf{r}(t)=bcos(wt) ...
1
vote
1answer
46 views

use the definition of ($\epsilon, \delta$) proof to show asymptotically stable?

Compute the solution $\phi_t \overrightarrow x_0 = e^{At} \overrightarrow x_0$ to the system $x' = -x + 4y$ and $y' = -4x - y$. i found the solution that $$\phi_t (x,y) = e^{-t}\begin{bmatrix}\cos 4t ...
1
vote
0answers
42 views

Find a nonlinear system conjugate the linear system $\overrightarrow x' = \left(\begin{array}{cc} 1 & 2\\0&-4 \end{array}\right) \overrightarrow x$.

A nonlinear system that is topologically conjugate in a neighborhood of its equilibrium point to the linear system $\overrightarrow x' = \left(\begin{array}{cc} 1 & 2\\0&-4 \end{array}\right) ...
0
votes
0answers
34 views

Find a nonhyperbolic matrix which satisfies certain conditions

A nonhyperbolic matrix A for which the planar system $\overrightarrow x' = A \overrightarrow x$ has an equilibrium point at (0,0) with the x-axis as a stable curve and the y-axis as an unstable curve. ...
0
votes
0answers
24 views

Fitting data with an AR-model

An experiment involves a discrete time dynamical system with inputs u and outputs y. $ U = \left( \begin{array}{c} -2\\ 1\\ 0\\ 0\\ -1 \end{array} \right)$ and Y = $ \left( \begin{array}{c} -1\\ ...
0
votes
2answers
33 views

finding the input sequence of a discrete-time dynamical system

I am studying Dynamical Systems, actually linear systems and I came across the following question: Consider the following discrete-time dynamical system: $x_{i+1}= \left( \begin{array}{ccc} 2 & ...
0
votes
0answers
36 views

Dependence On Initial Conditions and Parameters.

I'm having a hard time getting started with this problem. I'm not even sure if this can be done by computing some derivatives or what not or if I need to use a proof for this solution.
2
votes
1answer
82 views

Stable and unstable set of a dynamical system

Consider the following system of differential equations. $$ \dot{x} = Ax $$ with $$A=\left[ \begin{matrix}2&3\\-3&2\end{matrix} \right] $$ and with initial value $x(0)=x_0$. I want to ...
2
votes
2answers
84 views

finding the potential v(x,y)

Consider the system $\dot{x}=3x^2-1-e^{2y}, \dot{y}=-2xe^{2y}$ 1)Show that $\frac {\partial{f}}{\partial{y}}=\frac {\partial{g}}{\partial{x}}$ 2)Find the potential $V(x,y)$ 3)Show trajectories ...
0
votes
1answer
65 views

map, stability of the fixed point, cobweb

Map => $x_{n+1}=\sin x_n$ Show the stability of the fixed point $x^*=0$ is not determined by the linearizatin. Using the cobweb to show $x^*=0$ is stable. I took derivative of $\sin x_n$ and put ...
2
votes
0answers
66 views

Homoclinic orbit Hamiltonian system

The question is for which $a \in \mathbb{R}$ the system: $x'' + x - x^3 + a = 0$, has a homoclinic orbit. I let $y = x'$ so the system becomes: $x' = y$ $y' = x^3 - x - a$ and determined the ...
0
votes
1answer
81 views

Showing that a gradient flow cannot have a homoclinic solution

My problem is, given a $V: \mathbb{R}^2 \rightarrow \mathbb{R}$ that is atleast $C^2$, consider its gradient flow $$ \dot{x} = -\nabla V,\quad \text{ or }\quad \dot{x_i} = -\frac{\partial V}{\partial ...
1
vote
1answer
53 views

Show that the system $\Sigma(SAS^{-1},SB,CS^{-1},D)$ is observable/controllable iff $\Sigma(A,B,C,D)$ is observable/controllable

I am given the two linear systems: \begin{eqnarray} \Sigma_1: \dot{x}&=&Ax+Bu\\ y&=&Cx+Du \end{eqnarray} and \begin{eqnarray} \Sigma_2: \dot{x}&=&\bar{A}x+\bar{B}u\\ ...
2
votes
0answers
30 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
2
votes
2answers
159 views

Expressing an oscillator as a series of ODEs

Consider an oscillator satisfying the initial value problem $u''+w^2u=0$, where $u(0)=u_0$, $u'(0)=v_0$. Let $x_1 = u$, $x_2=u'$, and transform the equations given into the form $x' = Ax, x(0)$. Then ...
0
votes
1answer
100 views

homeomorphism of flows wrt sets

Given two flows, $\phi_t: A \to A$, and $\psi_t:B \to B$, that are topologically conjugate, and a homeomorphism, $h: A\to B$, show the following relationships to be true. In the following, $x \in A$ ...
2
votes
1answer
164 views

Show two interval homeomorphisms are topologically conjugate .

My question is the following: suppose we have two homeomorphisms $f,g:[0,1]\to[0,1]$ such that $f(0)=g(0)=0$, $f(1)=g(1)=1$ and that neither $f$ nor $g$ have a fixed point in $(0,1)$. Can we show that ...
0
votes
2answers
82 views

Controllability properties of discrete vs. continuous systems

I'm not sure what differentiates discrete from continuous systems in trying to prove certain properties of controllability. I have a chain of proofs from class that prove each other for a continuous ...
5
votes
0answers
155 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 ...
0
votes
0answers
50 views

Liapunov Function

I'm trying to find a Liapunov function for the system $$ x_1' = -2x_2+x_1^2 $$ $$ x_2' = 2x_1 + x_2^2 $$ at $(0,0)$. I tried $v(x_1,x_2) = x_1^2 +x_2^2$ first - that gave $v'(x_1,x_2) = ...
0
votes
1answer
112 views

ODE system and show infinite number of positively invariant ellipsoids

The system of ODEs is: $$ \dot{x} = -2x+yz \\ \dot{y} = x-xz \\ \dot{z} = xy $$ I found two lines of equilibria etc. but I now need to find the parameters for this "energy" or Lyapunov function, so ...
0
votes
1answer
133 views

Kepler, cartesian coordinates and ellipses

I am trying to see if I am on the right track with this. The problem: A kepler orbit (an ellipse) in Cartesian coordinates is: $$(1−\epsilon^2)x^2 + 2\alpha \epsilon x + y^2 = \alpha^2.$$ The task ...
0
votes
1answer
70 views

Showing the limit of a flow must be an equilibrium point under certain restrictions.

I'm stumped on how to approach this one: Consider the autonomous ODE $\dot{x} = f(x)$, $x \in \mathbb{R}^n$ with initial condition $x(0) = x_0$ and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ (at ...
2
votes
2answers
124 views

Prove that $x \equiv 0$ of $\dot{x}(t)=a(t)x$ is Uniformly Asymptotically Stable

I have a problem: Consider the scalar equation: $$\dot{x}(t)=a(t)x \tag{I}$$ where $a(t) \in C(\mathbb{R}^+)$. Prove that $x \equiv 0$ of $(I)$ is Uniformly Asymptotically Stable iff ...
3
votes
1answer
139 views

Construct a Liapunov function for this system

Construct a Liapunov function for the system (Determine the stability of $x \equiv 0$): I have an example:$$\begin{cases} & \mathrm { } \dot{x}= -x^3+xy^2\\ & \mathrm { } \dot{y}= ...
2
votes
1answer
326 views

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system.

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system: I have an example: $$\begin{cases} & \mathrm{ } \dot{x}= \tan(y-x)\\ & \mathrm{ } ...
0
votes
0answers
26 views

Dymamical systeme

i have this exercise : Here are graphs of the following two functions among one of the following portraits phases , which is the one corresponding to these solutions: Please help me Thank ...
0
votes
0answers
61 views

Exercise on dynamical system

using this phase portrait : a)When $t=0$ we have that $x=0$ and $y=0.25$ If $(x(t),y(t))$ is the solution satisfying the initial previous conditions what is the approximate value of $y(t)$ when ...