I'm asked to study everything that is possible to know about the sytem$$\begin{cases}x'=x^2-y^2\\y'=2xy\\z'=-z\end{cases}$$
My questions here is, how much can be know about it?, how do I know I finished?
I found the equilibrium points of the system
$$x'=0\iff x^2-y^2=0\iff |x|=|y|\\y'=0\iff (x=0)\vee(y=0)\\z'=0\iff z=0$$
I conclude that the only equilibrium point if the origin. Now I try to find the equilibria of the origin, since the system is non-linear I use the Lyapunov function $V(x,y,z)=ax^2+by^2+cz^2$ which is definite positive if $a,b,c>0$.
Now $\nabla V\cdot f$ gives $$\nabla V\cdot f=2ax(x^2-y^2)+2by(2xy)+2cz(-z)=2ax^2-xy^2(4b-2a)-2cz^2.$$ I''m up to finding constants to help me to determine the equilibria of the origin, if I'm right it will be given by
$$\begin{cases}\nabla V\cdot f \leq 0 &\text{stable}\\\nabla V\cdot f <0 & \text{asymptoticalle stable}\\\nabla V \cdot f>0 & \text{unstable}\end{cases}$$
The first one seem to be the case. I thought that a valid choice would be $a=b=c=1$; then there is a boundary in which $x^2-2xy^2-2z^2\leq 0$, a ball of a radius smaller than $1$ seem to work here: if $v\in B_{1/2}(0)$ then $x,y,z<1/2$ and $x^2-2xy^2-2z^2<(1/2)^2-2(1/2)(1/2)^2-2(1/2)^2=-(1/2)^2<0$ and the origin is stable. I can't tell if the origin is asymptotically stable.
If I'm supposed to study the system, what else can I say about it?