Starting from an $SU(N)$ matrix quantum mechanics model with massive deformation terms and by introducing an ansatz configuration involving fuzzy four- and two-spheres with collective time dependence, we obtain a family of effective Hamiltonians, $H_n \,, (N = \frac{1}{6}(n+1)(n+2)(n+3))$ and examine their emerging chaotic dynamics. Through numerical work, we model the variation of the largest Lyapunov exponents as a function of the energy and find that they vary either as $\propto (E-(E_n)_F)^{1/4}$ or $\propto E^{1/4}$, where $(E_n)_F$ stand for the energies of the unstable fixed points of the phase space. We use our results to put upper bounds on the temperature above which the Lyapunov exponents comply with the Maldacena-Shenker-Stanford (MSS) bound, $2 \pi T$, and below which it will eventually be violated.