ODELF1

From Exampleproblems

Consider the system of equations $\begin{cases} x'=y-xf(x,y) \\ y'=-x-yf(x,y)\end{cases}\,$

Find a Lyapunov function to determine the stability of the equilibrium solution $(0,0)\,$ . Consider the three cases in your calculations: a) $f(x,y)\,$ positive semidefinate, b) $f(x,y)\,$ positive definite and c) $f(x,y)\,$ negative definate.

$\begin{cases} x'=y-xf(x,y) \\ y'=-x-yf(x,y)\end{cases}\,$

Let $L=x^2+y^2\,$ .

$\frac{\partial L}{\partial t} = 2x\dot{x} + 2y\dot{y}\,$

$=2x\left[ y-xf(x,y) \right] + 2y\left[-x-yf(x,y)\right]\,$

$=-2x^2f(x,y) - 2y^2f(x,y)\,$

$=-f(x,y)[2x^2+2y^2]\,$

If $f(x,y)\,$ is positive semidefinite, then $\frac{\partial L}{\partial t} \le 0\,$ so the zero solution is uniformly stable.

If $f(x,y)\,$ is positive definite, then $\frac{\partial L}{\partial t} < 0\,$ so the zero solution is asymptotically stable.

If $f(x,y)\,$ is negative definite, then $\frac{\partial L}{\partial t} > 0\,$ so the zero solution is unstable.

Main Page : Ordinary Differential Equations : Lyapunov Functions

Retrieved from "http://www.exampleproblems.com/wiki/index.php/ODELF1"

Views

Personal tools

Create an account or log in

Search

Toolbox

Argan Oil
Natural Skin Care
Organic Skin Care