Hello,
I need a help with finding the equilibrium points and state of stability of the following system of ODEs:
dx/dz=x(E - Ax - y);
dy/dz=y(-1 + x);
Thank you in advance. I will be thankful for any advice.
Equilibrium occurs when dx/dz= 0 and dy/dz= 0.
y(-1+ x)= 0. Either y= 0 or x= 1.
x(E- Ax- y)= 0. If y= 0 x(E- Ax)= 0. Either x= 0 or x= E/A.
If x= 1 E- A- y= 0 so y= A- E.
Equilibrium points are (0, 0), (E/A, 0), and (1, A- E).
The "Jacobian", the matrix with the partial derivatives, is \(\displaystyle \begin{bmatrix}E- 2Ax- y & -x \\ y & -1\end{bmatrix}\).
At (0, 0) that is \(\displaystyle \begin{bmatrix}E & 0 & 0 & -1\end{bmatrix}\). That has eigen values 0 and -1. Since there are no positive eigen values, this is a stable equilibrium point.
At (E/A, 0) that is \(\displaystyle \begin{bmatrix}-E & -E/A \\ 0 & -1 \end{bmatrix}\). That has eigenvalues -E and -1. Assuming E is positive that is a stable equilibrium point.
At (1, A- E) that is \(\displaystyle \begin{bmatrix} 3A & -1 \\ A- E & -1 \end{bmatrix}\). The eigenvalue equation is more complicated so I will leave that to you!