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 $(0,0)$.
116
Answer
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.
- answered
- 2057 views
- $6.00
Related Questions
- Find the General Solution
- Leaky Buckets: Volume in a system of 2+ buckets that can be empty
- Week solution of the equation $u_t + u^2u_x = f(x,t)$
- Differential Equations- Initial Value Problem
- 2nd Order ODE IVP non homogeneous
- Two masses attached to three springs - Differential equations
- How to derive the term acting like a first derivative with respect to A that I found by accident?
- Show this initial value problem has a unique solution for initial value forall t