My question is quite simple: how can you determine the stability of a given dynamical system? For example, what values of a make the following matrix stable:
\begin{bmatrix}
0&a \\
a&0
\end{bmatrix}
I'm just confused about how to deduce a matrix's stability from its eigenvalues. I know this is a dumb question, but help would be greatly appreciated.
Best Answer
For a continuous-time dynamical system: to find the stability of the system we first find the equilibrium solution of the system (i.e. for a continuous system $\dot x=f(x)$, by solving $f(x)=0$ ), then finding the Jacobian matrix of the system at the equilibrium solution. Now if the eigenvalues of the Jacobian matrix(at the equilibrium points) are negative or negative real part (if complex) then the equilibrium points are locally stable.
For a discrete-time dynamical system: Here equilibrium solution are solution of $f(x)=x$ and equilibrium points are locally stable if eigenvalues of the Jacobian matrix(at the equilibrium points) are of modulus less than $1$.
Here note that Jacobian matrix is nothing but a tool of local linearization in a dynamical system. Fore more details see this problems in stack exchange: 1 2