# Asymptotic stability of semi-trivial solution and existence of a nontrivial solution

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?

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