  The bulk parameters characterizing the energy of symmetric nuclear matter and
the symmetry energy defined at normal nuclear density $\rho_0 $ provide
important information on the equation of state (EOS) of isospin asymmetric
nuclear matter. While significant progress has been made in determining some
lower order bulk characteristic parameters, such as the energy $E_0(\rho_0)$
and incompressibility $K_0$ of symmetric nuclear matter as well as the symmetry
energy $E_{sym}(\rho_0)$ and its slope parameter $L$, yet the higher order bulk
characteristic parameters are still poorly known. Here, we analyze the
correlations between the lower and higher order bulk characteristic parameters
within the framework of Skyrme Hartree-Fock energy density functional and then
estimate the values of some higher order bulk characteristic parameters. In
particular, we obtain $J_0=-355 \pm 95$ MeV and $I_0=1473 \pm 680$ MeV for the
third-order and fourth-order derivative parameters of symmetric nuclear matter
at $\rho_0 $ and $K_{sym} = -100 \pm 165$ MeV, $J_{sym} = 224 \pm 385$ MeV,
$I_{sym} = -1309 \pm 2025$ MeV for the curvature parameter, third-order and
fourth-order derivative parameters of the symmetry energy at $\rho_0 $, using
the empirical constraints on $E_0(\rho_0)$, $K_0$, $E_{sym}(\rho_0)$, $L$, and
the isoscalar and isovector nucleon effective masses. Furthermore, our results
indicate that the three parameters $E_0(\rho_0)$, $K_0$, and $J_0$ can
reasonably characterize the EOS of symmetric nuclear matter up to $2\rho_0 $
while the symmetry energy up to $2\rho_0 $ can be well described by
$E_{sym}(\rho_0)$, $L$, and $K_{sym}$.
