Lyapunov and behavior of a linearized system

1. Apr 24, 2012

dprimedx

1. The problem statement, all variables and given/known data

Examine the behavior in the neighborhood of the origin. Derive the given Lyapunov V(x,y). Show the region $\frac{dV}{dt} = 0$ and relate it to your analysis.

2. Relevant equations

$\dot{x} = x^3 - xy$
$\dot{y} = -y + y^2 + xy - x^3$

As given by the text:
$V(x,y) = \frac{x^4}{4} + \frac{y^2}{2}$
$\dot{V}(x,y) = x^6 + yx^3 (1+x)+y^2 (1-x-y)$

3. The attempt at a solution
I've been stumped on this problem for a few days now. I spoke with my professor and he said to just "do the best you can and use what we learned in class! If it doesn't work, it doesn't work!"

For the first part of the question, I attempted to linearize the system using the Jacobian evaluated at the origin which gives the following:
{0,0}
{0,-1}

Sorry. I'm not used to LaTeX and I couldn't figure out how to enter a matrix!

To determine what sort of equilibrium that is locally, I tried two methods. The first was to check the trace (-1) and determinant (0) and check it against a chart he had given us. From this, I would guess that it would be a saddle point equilibrium. However, when I try by checking the eigenvalues (0 and -1 again), I get a line field. Am I doing something wrong here?

Part two was the difficult part. My professor instructed us to simply do as much as we can. The method he had shown us won't work for this case. Here's what I tried:

To find V(x,y), use the following:
$V(\textbf{x}) = \textbf{x}^{T}.B.\textbf{x}$
$A^{T}.B+B.A=-Identity[2]$

Typically we use Mathematica for assignments. Using this method in Mathematica, you can't solve it. Doing it by hand results in B = {{0,0},{0,1/2}} (again, sorry for the matrices!)

Now, this shouldn't be a problem. However, when you plug B into the equation for V, you get $\frac{y^2}{2}$ which isn't what the book gives. My professor said just to explain this and continue onward. I'd really like to know what it doesn't work though. Is the method correct, but the book wrong? Is there another method that would result in what the book gave?

My final issue is what is holding me back from completing this. For part c, we never covered what dV/dt = 0 represents and I'm unable to reason it myself. I know the Lyapunov, V, is the basin of attraction, but can't what the change in the basin of attraction really mean.

I'm also asked to graph everything, but it doesn't seem right to me. V doesn't fit with the global (non-linear) system and I can't find a linear system to graph. And dV/dt=0 looks rather odd and doesn't look like it relates to V at all.

Mathematica file: https://www.dropbox.com/sh/ccb10ai10kbk7ev/9Ntunu1801/p286.9.18.nb
It's apparently too large to attach!

And here's the textbook information:
Dynamics and Bifurcations, Hale and Kocak. Page 286, #9.18.

Thank you for your help and thanks for this wonderful community! I may hardly ever post, but I do lurk around almost daily to get my dose of nerdgasms.