# 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 $(0,0)$.

## Answer

$$V' = y_1 (-y_1^3 + y_2^5) + y_2 (-y_1 y_2^4 - y_2^3) = - (y_1^4 + y_2^4),$$

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.

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