  We study the competing phases and the phase transition phenomena in an
effective spin-orbital coupled model derived for pyroxene titanium oxides
ATiSi2O6 (A=Na, Li). Using the mean-field-type analysis and the numerical
quantum transfer matrix method, we show that the model exhibits two different
ordered states, the spin-dimer and orbital-ferro state and the spin-ferro and
orbital-antiferro state. The transition between two phases is driven by the
relative strength of the Hund's-rule coupling to the onsite Coulomb repulsion
and/or by the external magnetic field. The ground-state phase diagram is
determined. There is a keen competition between orbital and spin degrees of
freedom in the multicritical regime, which causes large fluctuations and
significantly affects finite-temperature properties in the paramagnetic phase.
