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Stability and Lyapunov functions

Following is the problem that I solved the first part:

Consider the system x' = f(x), where f: R^2 into R^2, is defined by:

f(x) = [ (x1)^3 + (x1) (x2)^2 ]
[ (x1)^2 (x2) + (x2)^3 ]

a- Find all equilibrium points of the system.
b- Use an appropriate Lyapunov function to determine the stability of the equilibrium points. If an equilibrium point is stable, is it asymptotically stable ?

- For question a: II found that (xo) = (0,0)^T is the only equilibrium point with (x1)=0 and (x2)= 0., and since the eigenvalues are 0 implies that it is a nonhyperbolic point

For question b: I tried the lyapunov function V(x) = a(x1)^2+ b(x2)^2, and I got
1/2 V'(x) = a {(x1)^4 + (x1)^3 (x2)} + b {(x2)^4 + (x1) (x2)^3} .

I also tried another Lyapunov with power 4 that gave me almost the same trend.

I am stuck there. I don't know how to continue since V'(x) is not 0. Could you help me conclude this problem ?

Thank you, Professor

Solution Preview

Here is the solution (attached below)

Note that you can write f(x) as:

dx1/dt = x1 * [x1^2 ...

Solution Summary

The following posting helps with problems involving stability and Lyapunov functions.