Computing eigenvectors for an eigenvalue of a dynamical system

Question

$\dot{x}=x\left ( 3-x-2y \right )$ $\dot{y}=y\left ( 2-x-y \right )$

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

$\begin{bmatrix} x\\ y \end{bmatrix}$

To solve for the eigenvalues, we compute the Jacobian of the matrix $A=\begin{bmatrix} \left ( 3-x-2y \right ) &0 \\ 0& \left ( 2-x-y \right ) \end{bmatrix}$

The eigenvalues are $\lambda_{1}=3,\lambda_{2}=2$

The eigenvectors associated with $\lambda_{1}$:

$A-\lambda_{1}I=\begin{bmatrix} \left ( 3-x-2y \right )-3 &0 \\ 0&\left ( 2-x-y \right )-3 \end{bmatrix}$

Then, $\left [ A-\lambda_{1} I|0 \right ]$, we get $\vec{V}=\left \langle 0,0 \right \rangle$

which is incorrect. Could anyone tell me where am I going wrong here?

Answer 1

What you're doing isn't making any sense, since it's a nonlinear system. It doesn't help that have written it "in matrix form", since the matrix depends on $x$ and $y$.

What you have to do first is to find the equilibrium points, then linearize around each of them. Then you can look at the eigenvalues of the Jacobian matrix at each equilibrium to find out whether it's stable or not.

Answer 2

You will need to express $\dot x, \dot y$ as linear functions of $x$ and $y$, for example $$\left[\begin{array}{c}\dot x\\\dot y\end{array}\right] = A\left[\begin{array}{c}x\\y\\1\end{array}\right]$$ for a matrix $A$. You can for example make a first order Taylor expansion of the original expressions (it is probably the easiest way to do). So assume you are close to the point $\left[\begin{array}{c} x_0\\ y_0\end{array}\right]$ and make a linear Taylor approximation there.

Note : The $A$ matrix should be a function of this point $\left[\begin{array}{c} x_0\\ y_0\end{array}\right]$.