3
votes
0answers
31 views

Sharkovskii's theorem in two dimensions?

Question A weak form of Sharkovskii's theorem in $1$D dynamical systems states that, if a continuous function $f:I\to I$ does not include a periodic point of least period $2$ on $I$, then ...
6
votes
2answers
115 views

Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as: $$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, ...
1
vote
0answers
67 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
1
vote
1answer
23 views

Unique solution to a arbitrary non-linear system under monotonicity assumptions

I have a map $f:\mathbb{R}^n\times\mathbb{R}^m \to \mathbb{R}^n$ of two arguments $x, y$, which has a following properties: The jacobian matrix of $f$ wrt to the first argument $\frac{\partial ...
2
votes
0answers
32 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
0
votes
1answer
77 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
5
votes
2answers
78 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
2
votes
1answer
72 views

a system of finite difference equations

Let $a,b>0$ such that $ab<1$ consider the system$$x_{t+1}=x_ty_t+ay_t$$ $$y_{t+1}=x_ty_t+bx_t$$ I would like you to help me answer the following: find values $a$ and $b$ ​​for which the ...
1
vote
2answers
39 views

Finding a Lyapunov function for a given system of equations

I've got the following system of equations: $$ \begin{cases} x_1'=-8x_1^3-x_2 \\x_2'=-4x_2-4x_1^3 \end{cases} $$ I'm trying to check, if the equilibrium point in $(0,0)$ is stable or not. I am ...
1
vote
2answers
71 views

In search of periodic solutions of a system of ODEs by means of Fourier series

Consider the following non-linear system of ODEs : \begin{cases} x' = y \\ y' = x^2-\lambda x. \end{cases} In search of a solution such that $y(0) = y(2 \pi) = 0$, I am being told to seek $x$ and $y$ ...
0
votes
1answer
56 views

FermiPasta-Ulam problem

Consider $H(q,p) = \frac{1}{2} \sum\limits_{j=1}^{n+1} {(p_j^2 + (q_{j}-q_{j-1})^2)}$ $H(q,p) $ is the Hamiltonian considered in the FermiPasta-Ulam problem. Consider canonical transformation $Q = ...
1
vote
1answer
41 views

bifurcation with more than parameter

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$ $x˙ = \alpha_1 + \alpha_2 x − x^2$. Find a change of coordinates $y = \phi(x)$ such that ...
0
votes
1answer
53 views

near identity change of coordinates

Problem: Consider the scalar differential equation $$x' = \frac{4x – 24x^2 – 16x^3}{1 – 12x – 12x^2}.$$ which has a fixed point at $x^* = 0 $. For $x$ close to $x^* = 0 $ find a near identity ...
2
votes
2answers
90 views

the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
1
vote
2answers
84 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
78 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
71 views

Why was this change of variables made in this system of differential equations?

I'm trying to understand an example from my notes. I'm given a system of linear differential equations as follows $$x'=2x-y$$ $$y'=2x-2$$ The notes solve these by making the change of variables ...
1
vote
1answer
57 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 ...
6
votes
2answers
273 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
3
votes
1answer
44 views

How to divide solutions of system of ODE

If I have the following system of ODEs: \begin{align} F'_1=aF_1+bF_2 \nonumber\\ F'_2=-bF_1+cF_2 \nonumber \end{align}for which I know the solutions of $F_1$ and $F_2$ as sums of exponential of ...
2
votes
2answers
96 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 ...
1
vote
1answer
109 views

Find the index of the equilibrium points of the system (Question on solution)

I have the following system: $$\dot{x} = 2xy$$ $$\dot{y} = 3x^2-y^2$$ I have the following solution: The system has one equilibrium point at the origin. Let the curve $\Gamma$ surrounding the origin ...
3
votes
1answer
49 views

Dynamical system equilibrium point increment

I am reading through the dynamical systems theory and there is an example of a Mass-Spring system. The state equations are given by $\displaystyle \frac{d x_1}{dx}(t) = x_2(t)$ $\displaystyle ...
2
votes
1answer
2k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
0
votes
1answer
74 views

Question on dynamical system

i have this exercise : we consider the following model : $$ \begin{cases} x'& = x(4-x-y)\\ y'&=y(2+2\alpha-y-\alpha x) \end{cases} $$ a) Find the critical point $P$ does not depend on ...
0
votes
2answers
116 views

Existence of invariant set in dynamical system generated by ODE

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations? $x'=x+\sin{(xy+2)}-7$ $y'=-y+\arctan{(x^2+y^3-6)}$ My idea is to use this fact: Not ...