Thank's amWhy!
I pray to some kind soul to help me on the theory of bifurcation: In the article of Tao Peng titled: "Bifurcation Behavior of a Cohen-Grossberg Neural Network of two Neurons with impulsive Effects" They are the system of differential equations: $ \begin{array}{cc} x′(t)= &ax(t)(px(t)+hf(x(t))+kf(y(t))+C_1)\\ y′(t)= &by(t)(qy(t)+uf(x(t))+vf(y(t))+C_2) \end{array} $ It states that the existence of two semi- trivial solutions; that is (x(t),0) and (0,y(t)) which is asymptotically stable, implies the existence of a nontrivial solution of the written system above.
In other word, what is the relationship between the asymptotic stability of the semi-trivial solution and the existence of a nontrivial solution?