I am studying one system of nonlinear ODE's s.t. the $(x_1,x_2,x_3)=(1,0,0)$ is an equilibrium point. On this system I have: $$(1) x_1,x_2,x_3\geq 0; \\ (2) x_1+x_2+x_3=1 $$
Well, one author, analysing the stability, added a perturbation $\epsilon_1,\epsilon_2,\epsilon_3>0$ and analyses the state $(x_1,x_2,x_3)=(1+\epsilon_1,\epsilon_2,\epsilon_3)$, proving that $\epsilon_i\to0$.
However, $1+\epsilon_1+\epsilon_2+\epsilon_3>1$.
I did not understand if this state is valid to do the analysis, once it cannot be an initial condition (if $x_2,x_3$ increase, then $x_1$ might decrease).
I'd like so much to get an explanation.
Many thanks.