Find all the fixed points of the system $f(x,y)=-x+x^3, g(x,y)=-2y$ and use linearization to classify them.

Question

Find all the fixed points of the system $f(x,y)=-x+x^3, g(x,y)=-2y$ and use linearization to classify them.

I have found the solutions to be : $x = 0$ or $x = ±1$ and $y = 0 \implies 3$ fixed points $(0, 0), (1, 0) $and$ (−1, 0)$

We then calculate the Jacobian matrix, which I did for each of the above fixed points. However, my only concern is that once I calculate the Jacobian matrix at each FP how do I know whether it is a stable, saddle, or unstable node?

Answer 1

Take the Jacobian and find its eigenvalues at the fixed points.

The Jacobian is: $\begin{pmatrix} -1+3x^2 & 0 \\ 0 & -2 \end{pmatrix}$

At (0,0): $\begin{vmatrix} -1-\lambda & 0 \\ 0 & -2 - \lambda \end{vmatrix} = (\lambda+1)(\lambda+2)$

So, we have two negative eigen values, ie, stable node at (0,0).

You do this for the rest of the fixed points, here is a sheet with the correspondence of eigenvalues to type of fixed point.

http://jaffarali.org/courses/f12/map5316/stability-types.pdf

Answer 2

You have a stable node if the matrix is negative definite, unstable node if the matrix is positive definite and a saddle if it is neither of those two.

There are several conditions for positive definiteness and negative definiteness, one of the easiest being the sign of the eigenvalues. If all are positive, the matrix is positive definite, if all are negative then it is negative definite and otherwise it is neither.