# 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 $(0,0)$.

Ichbinanonym

93

## 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.

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

- answered
- 1046 views
- $6.00

### Related Questions

- Differentai equations, question 2.
- Solve the Riemann Problem
- Week solution of the equation $u_t + u^2u_x = f(x,t)$
- ODEs - Stability
- A linear ODE
- Laplace transforms and initial value problems.
- Ordinary Differential Equations Integrating Factors Assignment
- Find the general solution of the system of ODE $X'=\begin{bmatrix} 1 & 3 \\ -3 & 1 \end{bmatrix} X$