  We consider a truncation of the BMN matrix model to a configuration of two
fuzzy spheres, described by two coupled non-linear oscillators dependent on the
mass parameter $\mu$. The classical phase diagram of the system generically
($\mu \neq 0$) contains three equilibrium points: two centers and a
center-saddle; as $\mu \to 0$ the system exhibits a pitchfork bifurcation. We
demonstrate that the system is exactly integrable in quadratures for $\mu=0$,
while for very large values of $\mu$, it approaches another integrable point
characterized by two harmonic oscillators. The classical phase space is mixed,
containing both integrable islands and chaotic regions, as evidenced by the
classical Lyapunov spectrum. At the quantum level, we explore indicators of
early and late time chaos. The eigenvalue spacing is best described by a Brody
distribution, which interpolates between Poisson and Wigner distributions; it
dovetails, at the quantum level, the classical results and reemphasizes the
notion that the quantum system is mixed. We also study the spectral form factor
and the quantum Lyapunov exponent, as defined by out-of-time-ordered
correlators. These two indicators of quantum chaos exhibit weak correlations
with the Brody distribution. We speculate that the behavior of the system as
$\mu \to 0$ dominates the spectral form factor and the quantum Lyapunov
exponent, making these indicators of quantum chaos less effective in the
context of a mixed phase space.
