March 3rd, 2012, 08:05 AM  #1 
Newbie Joined: Mar 2012 Posts: 2 Thanks: 0  When is this Lyapunov?
I went to my prof for help on this and he couldn't figure it out, although he assured me he has figured it out in the past. I am trying to determine the values of k for which V(x,y)=x^2+ky^2 is Lyapunov for the following first order system: x'=x+yx^2y^2+(x)(y^2) ; y'=y+xyy^2(x^2)(y) I'd also like to like to be able to say something about the domain of stability for the case k=1. So far I've intuitively determined that there is a neighborhood around (0,0) for which the Lyanpunov's(k=1) derivative is always positive, thus unstable, thus there is no domain of stability for the case k=1. I'm having some trouble showing it in a rigorous manner, though. Any help would be greatly appreciated. 
March 6th, 2012, 04:56 PM  #2 
Newbie Joined: Mar 2012 Posts: 2 Thanks: 0  Re: When is this Lyapunov?
I have actually figured this one out. I made a mistake for the k=1 case. I think I misplaced a negative or something. It turns out that when k=1, the derivate of V(x,y) can be factored as 2(x+y+1)(x^2xy+y^2) and since (x^2xy+y^2) is always positive in any deleted neighborhood of the origin, we need for x+y+1 to be greater than 0. This gives y>x1. So we have asymptotic stability at the origin and the domain of stability could be a ball of radius sqrt(2)/2 centered at the origin. Sqrt(2)/2 is the infimum of the distances from (0,0) to the line y=x1 and it is in the region where y>x1. As for finding the values of k for which V(x,y)=x^2+ky^2 is Lyapunov for the system in question, here is what I did: As (x,y) gets close to the origin, the only terms of the gradient derivative that aren't negligible are the 2 degree terms, meaning V'(x,y)~2x^2+2xy2ky^2. So I'm saying that in a small enough neighborhood of the origin, all of the terms of 3rd and 4th degree are small. Okay, so then we need 2x^2+2xy2ky^2<0 or equivalently, k>(x/y)(x/y)^2. Then I substitute cot(a)=(x/y) and maximimize the right side giving k > 1/4. Sorry, I am skipping some steps. If something doesn't make sense let me know. 

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