  We study the dynamical generation of entanglement as a signature of chaos in
a system of periodically kicked coupled-tops, where chaos and entanglement
arise from the same physical mechanism. The long-time averaged entanglement as
a function of the position of an initially localized wave packet very closely
correlates with the classical phase space surface of section -- it is nearly
uniform in the chaotic sea, and reproduces the detailed structure of the
regular islands. The uniform value in the chaotic sea is explained by the
random state conjecture. As classically chaotic dynamics take localized
distributions in phase space to random distributions, quantized versions take
localized coherent states to pseudo-random states in Hilbert space. Such random
states are highly entangled, with an average value near that of the maximally
entangled state. For a map with global chaos, we derive that value based on new
analytic results for the typical entanglement in a subspace defined by the
symmetries of the system.
  For a mixed phase space, we use the Percival conjecture to identify a
"chaotic subspace" of the Hilbert space. The typical entanglement, averaged
over the unitarily invariant Haar measure in this subspace, agrees with the
long-time averaged entanglement for initial states in the chaotic sea. In all
cases the dynamically generated entanglement is predicted by a unitary ensemble
of random states, even though the system is time-reversal invariant, and the
Floquet operator is a member of the circular orthogonal ensemble.
