Existence of nonsingular solutions for nonlinear and homogenous linear systems

Question

I'm sure this is a simple question, but I haven't started studying ODE's yet, only the general idea.

Is it possible to have a nonsingular solution to a nonlinear system also be a solution to a homogenous linear system of differential equations?

If the answer is yes, can the solution be a saddle point?

I'm specifically interested in the applications this has to game theory, most notably the quantification of human flourishing and Nash equilibria.

reference: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3126111/

reference: https://arxiv.org/pdf/1209.5684.pdf

Edit: I don't know why this question was closed.

Answer 1

I mean yes it is defiantly possible and realistically not all that hard to construct the latter once you have the solution for the nonlinear system of equations.

However, there doesn't need to be any correlation between the graphs of what the first and second thing look like. Could they both be saddle points? of course but one could be a sink and the other a source or one a sink and one a saddle etc. its kind of like saying that 2+2=4 is the same as $x^2 -16=0 $ they agree whenever $x=\pm 4$ but if you graph the two one is a straight line at 4 and the other is a parabola; they are clearly not the same. That being said often when you solve a nonlinear system of equations you do so by approximating it with a system of linear differential equations and look at it locally. (very small littler part.) For a small enough region the linear equations will be basically the same as the nonlinear one.