  The critical behavior of many physical systems involves two competing
$n^{}_1-$ and $n^{}_2-$component order-parameters, ${\bf S}^{}_1$ and ${\bf
S}^{}_2$, respectively, with $n=n^{}_1+n^{}_2$. Varying an external control
parameter $g$, %(e.g. uniaxial stress or magnetic field), one encounters
ordering of ${\bf S}^{}_1$ below a critical (second-order) line for $g<0$ and
of ${\bf S}^{}_2$ below another critical line for $g>0$. These two ordered
phases are separated by a first-order line, which meets the above critical
lines at a bicritical point, or by an intermediate (mixed) phase, bounded by
two critical lines, which meet the above critical lines at a tetracritical
point. For $n=1+2=3$, the critical behavior around the (bi- or tetra-)
multicritical point either belongs to the universality class of a
non-rotationally invariant (cubic or biconical) fixed point, or it has a
fluctuation driven first-order transition. These asymptotic behaviors arise
only very close to the transitions. We present accurate renormalization-group
flow trajectories yielding the effective crossover exponents near
multicriticality.
