Finding where a function is sign definite, sign indefinite or sign semidefinite

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

Please see below

For this problem,

However, I'm confused how their got their solution. My solution is, using set builder notation,

$[ (x,y) \in \mathbf{R} : 1 - \cos x + y^4 ≥ 0 ]$ which implies that $V(0,0) = 0$ so it satisfies the first condition for being sign definite, sign semidefinite, and sign indefinite. Then from the inequality, we know that for $x \neq 0, y \neq 0$ , then $V(x,y) = 1 - \cos x + y^4 > 0$ so their Liapunov satsifes the conditions for being positive definite. However, I'm confused by how they find the domain (they call it domain of attraction) from $1 - \cos x + y^4 > 0$.

Does anybody please know how they do that?

Thanks!

 

[Office_Shredder]

What is the definition of sign definite/semi definite/indefinite?

 

[pasmith]

If you are using a Liapunov function to find a domain of attraction for (I assume) a fixed point at $(0,0)$, then you can't do that without reference to $(\dot x, \dot y)$, and since you haven't given us that information we can't really help you.