  The entanglement of eigenstates in two coupled, classically chaotic kicked
tops is studied in dependence of their interaction strength. The transition
from the non-interacting and unentangled system towards full random matrix
behavior is governed by a universal scaling parameter. Using suitable random
matrix transition ensembles we express this transition parameter as a function
of the subsystem sizes and the coupling strength for both unitary and
orthogonal symmetry classes. The universality is confirmed for the level
spacing statistics of the coupled kicked tops and a perturbative description is
in good agreement with numerical results. The statistics of Schmidt eigenvalues
and entanglement entropies of eigenstates is found to follow a universal
scaling as well. Remarkably this is not only the case for large subsystems of
equal size but also if one of them is much smaller. For the entanglement
entropies a perturbative description is obtained, which can be extended to
large couplings and provides very good agreement with numerical results.
Furthermore, the transition of the statistics of the entanglement spectrum
towards the random matrix limit is demonstrated for different ratios of the
subsystem sizes.
