For linear continuous dynamical systems, is the only possible equilibrium point 0? In all the examples I have seen, only the 0 equilibrium point is considered. I know this is not true for nonlinear continuous dynamical systems.
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2Consider the example $\dot x=0$. What are the equilibria? – Artem Nov 17 '15 at 16:33
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It is not. All the points in the null space of $A$ are equilibrium points. The null space of $A$ is $\{0\}$ if and only if $A$ is invertible. For example take
$$A = \begin{bmatrix}-1 & 1 \\ 0 & 0\end{bmatrix}$$
So, the null space of $A$ is the space spanned by the vector $[0 ~~ 1]^T$, so all the points $[0 ~~ \alpha]^T$ for all $\alpha \in \mathbb{R}$ are equilibrium points.
obareey
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Yes, it is for every hyperbolic (nonzero eigenvalues) linear system $\dot{x} = Ax$. Just solve the equation $A x = 0$.
demitau
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