  The scrambling time and its delay are calculated using holography in an
asymptotically AdS black hole solution of the gauged
Einstein-Maxwell-Dilaton-Axion (EMDA) theory, the dyonic Kerr-Sen-AdS$_4$ black
hole, perturbed by rotating and charged shock waves along the equator. The
leading term of the scrambling time for a black hole with large entropy is
logarithmic in the entropy and hence supports the fast scrambling conjecture
for this black hole solution, which implies that the system under consideration
is chaotic. We also find that the instantaneous minimal Lyapunov index is
bounded by $\kappa=2\pi T_H/(1-\mu\mathcal{L})$, which is analogous to the
surface gravity but for the rotating shock waves, and becomes closer to
equality for the near extremal black hole. For a small value of the AdS scale,
we found that the Lyapunov exponent can exceed the bound for a large value of
$\mathcal{L}$. Due to the presence of the electric and magnetic charge of the
shock waves, we also show that the scrambling process of this holographic
system is delayed by a time scale that depends on the charges of the shock
waves. The calculations also hold for the ultra-spinning version of this black
hole. The result of this paper generalizes the holographic calculations of
chaotic systems which are described by an EMDA theory in the bulk.
