  Recently, a complete set of differential equations for the effective neutrino
masses and mixing parameters in matter have been derived to characterize their
evolution with respect to the ordinary matter term $a \equiv 2\sqrt{2}G^{}_{\rm
F} N^{}_e E$, in analogy with the renormalization-group equations (RGEs) for
running parameters. Via series expansion in terms of the small ratio
$\alpha^{}_{\rm c} \equiv \Delta^{}_{21}/\Delta^{}_{\rm c}$, we obtain
approximate analytical solutions to the RGEs of the effective neutrino
parameters and make several interesting observations. First, at the leading
order, $\widetilde{\theta}^{}_{12}$ and $\widetilde{\theta}^{}_{13}$ are given
by the simple formulas in the two-flavor mixing limit, while
$\widetilde{\theta}^{}_{23} \approx \theta^{}_{23}$ and $\widetilde{\delta}
\approx \delta$ are not changed by matter effects. Second, the ratio of the
matter-corrected Jarlskog invariant $\widetilde{\cal J}$ to its counterpart in
vacuum ${\cal J}$ approximates to $\widetilde{\cal J}/{\cal J} \approx
1/(\widehat{C}^{}_{12} \widehat{C}^{}_{13})$, where $\widehat{C}^{}_{12} \equiv
\sqrt{1 - 2 A^{}_* \cos 2\theta^{}_{12} + A^2_*}$ with $A^{}_* \equiv
a/\Delta^{}_{21}$ and $\widehat{C}^{}_{13} \equiv \sqrt{1 - 2 A^{}_{\rm c} \cos
2\theta^{}_{13} + A^2_{\rm c}}$ with $A^{}_{\rm c} \equiv a/\Delta^{}_{\rm c}$.
Finally, after taking higher-order corrections into account, we find compact
and simple expressions of all the effective parameters.
