  We study operator growth in a bipartite kicked coupled tops (KCT) system
using out-of-time ordered correlators (OTOCs), which quantify ``information
scrambling" due to chaotic dynamics and serve as a quantum analog of classical
Lyapunov exponents. In the KCT system, chaos arises from the hyper-fine
coupling between the spins. Due to a conservation law, the system's dynamics
decompose into distinct invariant subspaces. Focusing initially on the largest
subspace, we numerically verify that the OTOC growth rate aligns well with the
classical Lyapunov exponent for fully chaotic dynamics. While previous studies
have largely focused on scrambling in fully chaotic dynamics, works on
mixed-phase space scrambling are sparse. We explore scrambling behavior in both
mixed-phase space and globally chaotic dynamics. In the mixed phase space, we
use Percival's conjecture to partition the eigenstates of the Floquet map into
``regular" and ``chaotic." Using these states as the initial states, we examine
how their mean phase space locations affect the growth and saturation of the
OTOCs. Beyond the largest subspace, we study the OTOCs across the entire
system, including all other smaller subspaces. For certain initial operators,
we analytically derive the OTOC saturation using random matrix theory (RMT).
When the initial operators are chosen randomly from the unitarily invariant
random matrix ensembles, the averaged OTOC relates to the linear entanglement
entropy of the Floquet operator, as found in earlier works. For the diagonal
Gaussian initial operators, we provide a simple expression for the OTOC.
