  We introduce a set of two-parameter models for the dark energy equation of
state (EOS) $w(z)$ to investigate time-varying dark energy. The models are
classified into two types according to their boundary behaviors at the redshift
$z=(0,\infty)$ and their local extremum properties. A joint analysis based on
four observations (SNe + BAO + CMB + $H_0$) is carried out to constrain all the
models. It is shown that all models get almost the same $\chi^2_{min}\simeq
469$ and the cosmological parameters $(\Omega_M, h, \Omega_bh^2)$ with the
best-fit results $(0.28, 0.70, 2.24)$, although the constraint results on two
parameters $(w_0, w_1)$ and the allowed regions for the EOS $w(z)$ are
sensitive to different models and a given extra model parameter. For three of
Type I models which have similar functional behaviors with the so-called CPL
model, the constrained two parameters $w_0$ and $w_1$ have negative correlation
and are compatible with the ones in CPL model, and the allowed regions of
$w(z)$ get a narrow node at $z\sim 0.2$. The best-fit results from the most
stringent constraints in Model Ia give $(w_0,w_1) = (-0.96^{+0.26}_{-0.21},
-0.12^{+0.61}_{-0.89})$ which may compare with the best-fit results $(w_0,w_1)
= (-0.97^{+0.22}_{-0.18}, -0.15^{+0.85}_{-1.33})$ in the CPL model. For four of
Type II models which have logarithmic function forms and an extremum point, the
allowed regions of $w(z)$ are found to be sensitive to different models and a
given extra parameter. It is interesting to obtain two models in which two
parameters $w_0$ and $w_1$ are strongly correlative and appropriately reduced
to one parameter by a linear relation $w_1 \propto (1+w_0)$.
