Let $x(t),v(t),f(x(t))\in\mathbb{R}^3$ then consider the following system of ODEs \begin{align} \dot{v} &= f(x)-v \\ \dot{x} &= v \\ \end{align} with initial conditions $x(0) = x_0$ and $v(0) = v_0$. I would like to study how the Jacobian determinant $\left|\frac{\partial x}{\partial x_0}\right|$ evolves in time. This is essentially how a volume in the coordinates $x$ changes w.r.t time.
This is what I have done so far: Letting $X = \frac{\partial x}{\partial x_0}$, then by Jacobi's formula we have \begin{align} \frac{d}{dt}|X| &= |X| \, \text{tr}\left(X^{-1}\frac{d}{dt}X\right). \end{align} Differentiating once more with respect to $t$ gives \begin{align} \frac{d^2}{dt^2}|X| &= \frac{d}{dt}|X| \, \text{tr}\left(X^{-1}\frac{d}{dt}X\right)+|X| \,\text{tr}\left(\frac{d}{dt}X^{-1}\frac{d}{dt}X\right)+|X| \,\text{tr}\left(X^{-1}\frac{d^2}{dt^2}X\right)\\ &= |X|\left( \text{tr}\left(X^{-1}\frac{d}{dt}X\right)^2- \,\text{tr}\left(\left(X^{-1}\frac{d}{dt}X\right)^2\right)+ \,\text{tr}\left(X^{-1}\frac{d^2}{dt^2}X\right)\right) \end{align} Now here is a step I am not entirely sure about: use the fact that $$X^{-1}\frac{d}{dt}X = \left(\frac{\partial x}{\partial x_0}\right)^{-1}\frac{\partial \dot{x}}{\partial x_0} = \frac{\partial v}{\partial x}$$ and similarly for the term with the double time derivative. Then we have \begin{align} \frac{d^2}{dt^2}|X| &= |X|\left( \text{tr}\left(\frac{\partial v}{\partial x}\right)^2- \,\text{tr}\left(\left(\frac{\partial v}{\partial x}\right)^2\right)+ \,\text{tr}\left(\frac{\partial f(x)}{\partial x}\right) - \text{tr}\left(\frac{\partial v}{\partial x}\right)\right) \end{align} Finally, another step that I'm not entirely sure about: aren't $v$ and $x$ independent of each other? I.e., $\frac{\partial v}{\partial x}=0$? If so does this then just reduce down to the following second order ODE in $|X|$ $$\frac{d^2}{dt^2}|X| = |X| \nabla\cdot f(x).$$ So if $f(x)$ is divergence-free, then $|X|$ depends linearly on $t$? And if the divergence of $f(x)$ is constant, then $|X|$ is oscillatory? This feels wrong! Please help!