I need to determine if this system exhibits volume contraction:
$\dot x =yz-x-x^3$
$\dot y =xz-y-y^3$
$\dot z =xy-z-z^3$
My approach is to just calculate the divergence of the vector field F:
$\nabla * F$ = $\partial f_1 \over \partial x$ + $\partial f_2 \over \partial y$ + $\partial f_3 \over \partial z $
This gives $(-3x^2-1)+(-3y^2-1)+(-3z^2-1)$ which is negative for all x,y,z so system exhibits volume contraction. Is this enough to show that volume contraction is occurring or can I do more?
We are talking here about a flow $\Phi$ in ${\mathbb R}^3$, defined by an autonomous system of differential equations, i.e., by a vector field ${\bf F}$, and not about a single map $f:\>{\mathbb R}^3\to{\mathbb R}^3$.
If $B_0$ is a tiny "test body" at time $t=0$ then the flow transports and deforms $B_0$ in the time interval $[0,t]$ to a body $B_t$. The question is about the limit $$\lim_{t\to0+}{{\rm vol}(B_t)-{\rm vol}(B_0)\over t}\ .$$ Doing the computation one finds out that this limit is actually the trace of the Jacobian of the defining vector field ${\bf F}$, times ${\rm vol}(B)$. The OP has found out that this trace is $<0$ everywhere.