  We analyze the quantum chaotic behavior of the Yukawa-SYK model as a function
of filling and temperature, which describes random Yukawa interactions between
$N$ complex fermions and $M$ bosons in zero spatial dimensions, for both the
non-Fermi liquid and insulating states at finite temperature and chemical
potential. We solve the ladder equations for the out-of-time-order correlator
(OTOC) for both the bosons and fermions. Despite the appearance of the chemical
potential in the Hamiltonian, which explicitly introduces an additional energy
scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn
out to be unaffected, and the Lyapunov exponents that diagnose chaos remain
maximal. As the chemical potential increases, the system is known to experience
a first-order transition from a critical phase to a gapped insulating phase. We
postulate that the boundary of the region in parameter space where each phase
is (meta)stable coincides with the curve on which the Lyapunov exponent is
maximal. By calculating the exponent in the insulating phase and comparing to
numerical results on the boundaries of stability, we show that this is
plausible.
