See this in the context of Quantum Mechanics, for the moment.
On the other hand, in general linear evolution equations (finite or infinite dimensional) I like to think of stationary states as special superpositions of states which behaves particularly well with respect to time evolution.
For example, in the case of the dynamic of two species of fishes $x,y$ given by
$$ \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix}= L \cdot \begin{pmatrix} {x}\\ {y} \end{pmatrix} $$an eigenvector $v$ of $L$ ($Lv=\lambda v$) is something like a stationary state, and for me is something like a new being made of both fish species. This kind of "cluster" behaves in a fairly simple way with respect to time: it undergoes an exponential growth
$$ v(t)=e^{\lambda t} v(0) $$See xournal 223
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Author of the notes: Antonio J. Pan-Collantes
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