  The phase diagram of a system with two order parameters, with ${\it n_1}$ and
$n_2$ components, respectively, contains two phases, in which these order
parameters are non-zero. Experimentally and numerically, these phases are often
separated by a first-order "flop" line, which ends at a bicritical point. For
$n=n_1+n_2=3$ and $d=3$ dimensions (relevant e.g. to the uniaxial
antiferromagnet in a uniform magnetic field), this bicritical point is found to
exhibit a crossover from the isotropic $n$-component universal critical
behavior to a fluctuation-driven first-order transition, asymptotically turning
into a triple point. Using a novel expansion of the renormalization group
recursion relations near the isotropic fixed point, combined with a resummation
of the sixth-order diagrammatic expansions of the coefficients in this
expansion, we show that the above crossover is slow, explaining the apparently
observed second-order transition. However, the effective critical exponents
near that transition, which are calculated here, vary strongly as the triple
point is approached.
