Constructing Lyapunov functions using eigenvectors

[1670frank]

Homework Statement

How can I produce a Lyapunov function using the eigenvalues and vectors
x'=-x+y
y'=-x

Homework Equations

The Attempt at a Solution

So I got the matrix using jacobian and I got the matrix
-1 1
-1 0
then i found the eigenvalues to be λ_1= (-1+sqrt3 i)/2 and λ2= (-1-sqrt3 i)/2 and i found the eigenvectors v1= (1 1/2 +sqrt3i /2) v2= (1 1/2 -sqrt3i/2) I am not exactly sure how I can proceed further though. I am not sure how to construct Lyapunov function using this information...

 

[mighty2000]

Hello, great question! To construct a Lyapunov function using the eigenvalues and eigenvectors, you can use the following steps:

1. Start by defining a function V(x,y) that is positive definite and continuously differentiable. This function will serve as your Lyapunov function.

2. Next, you will need to calculate the derivative of V(x,y) with respect to time, which can be written as V'(x,y). This will give you an expression that includes the derivatives of x and y.

3. Now, substitute the given system of equations for x' and y' into the expression for V'(x,y). This will give you an expression that includes x, y, x', and y'.

4. Simplify the expression using the given system of equations to eliminate x' and y'. This will leave you with an expression that only includes x and y.

5. Finally, to construct the Lyapunov function, you will need to choose the coefficients of x and y in such a way that the resulting expression is negative definite, meaning that it is always negative except at the equilibrium point (0,0). This will ensure that the Lyapunov function is decreasing along the trajectories of the given system, which is a key requirement for stability.

I hope this helps! Please let me know if you have any further questions or need clarification on any of the steps. Good luck with your calculations!