  We study the dynamics of a three-mode bosonic system with mode-changing
interactions. For large mode occupations the short-time dynamics is well
described by classical mean-field equations allowing us to study chaotic
dynamics in the classical system and its signatures in the corresponding
quantum dynamics. By introducing a symmetry-breaking term we tune the classical
dynamics from integrable to strongly chaotic which we demonstrate by
calculating Poincar\'e sections and Lyapunov exponents. The corresponding
quantum system features level statistics that change from Poissonian in the
integrable to Wigner-Dyson in the chaotic case. We investigate the behavior of
out-of-time-ordered correlators (OTOCs), specifically the squared commutator,
for initial states located in regular and chaotic regions of the classical
mixed phase space and find marked differences between the two cases. The
short-time behavior is well captured by semi-classical truncated Wigner
simulations directly relating these features to properties of the underlying
classical mean field dynamics. We discuss a possible experimental realization
of this model system in a Bose-Einstein condensate of rubidium atoms which
allows reversing the sign of the Hamiltonian required for measuring OTOCs
experimentally.
