## Complex eigenvalues in linearized equilibria

Hi all,

Suppose a dynamical system defined on $[0,1] \rightarrow [0,1]$. For example, it can be dynamic movement of probabilities or a share in the whole (in my case, it's replicator dynamics).

Suppose this system to have multiple equilibria and several of them are found on the boundary, e.g. in 2-dimensional case : $x=1, y=0$.

My question is theoretical - if you linearize around these boundary equilibria and check the eigenvalues, these are always real and never complex. According to my intuition, it cannot be complex because we couldn't have oscillations near or on the boundary. But is there a formal way of showing this?