Aysomptotical stability

Prove that the system of ODEs

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

$y_2'=-y_1y_2^4-y_2^3$

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.