Aysomptotical stability

Prove that the system of ODEs

$y_1'=-y_1^3 + y_2^5$


is asymptotically stable in $(0,0)$.


Consider the canonical energy function $V(y_1, y_2) = \tfrac{1}{2}(y_1^2 + y_2^2)$. It is clear that this is a positive definite function. Furthermore, it is a strict Lyapunov function;

$$V' = y_1 (-y_1^3 + y_2^5) + y_2 (-y_1 y_2^4 - y_2^3) = - (y_1^4 + y_2^4),$$
which is less than $0$ everywhere except at the origin. Thus by the Lyapunov stability theorem, the system is stable at the origin.

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.