  We investigate the dynamical evolution of globally connected Stuart-Landau
oscillators coupled through conjugate or dis-similar variables on simplicial
complexes. We report a first-order explosive phase transition from oscillatory
state to death state, with 2-simplex (triadic) interactions, as opposed to the
second-order transition with only 1-simplex (dyadic) interactions. Moreover,
the system displays four distinct homogeneous steady states in the presence of
triadic interactions, in contrast to the two homogeneous steady states observed
with dyadic interactions. We calculate the backward transition point
analytically, confirming the numerical results and providing the origin of the
dynamical states in the transition region. The study will be useful in
understanding complex systems, such as ecological and epidemiological, having
higher-order interactions and coupling through conjugate variables.
