I have a system $\dot{x} = f(x)$ with a $4 \times 4$ Jacobian matrix $J$. There are two equilibria that vary with a parameter. Evaluating the eigenvalues of the Jacobian at each of the equilibria as the parameter is varied reveals that one equilibrium has two real negative eigenvalues, one zero eigenvalue, and one positive eigenvalue, so it is unstable. The other equilibrium point has three negative eigenvalues and one zero eigenvalue.
How do I examine the stability of this equilibrium? The only information I can find says that the nonlinear terms of the system of equations are what determine the stability, but I cannot find how.
Best Answer
It's complicated... Essentially you want to see what happens on the centre manifold, which in this case is also called a "slow manifold". It might help to convert the system to a "normal form" with a nonlinear transformation.