# 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.

- answered
- 266 views
- $6.00

### Related Questions

- Variation of Parameter for Variable Coefficient Equation
- Differential Equations
- Calculus - 2nd order differential equations and partial derivatives
- Show that the following subset Ω of Euclidean space is open
- Solve the integral equation, (integro-differential)
- Solve the initial value problem $(\cos y )y'+(\sin y) t=2t$ with $y(0)=1$
- Ordinary Differential Equations Word Problems
- ODEs - Stability