  By means of a Floquet analysis, we study the quantum dynamics of a fully
connected Lipkin-Ising ferromagnet in a periodically driven transverse field
showing that thermalization in the steady state is intimately connected to
properties of the $N\to \infty$ classical Hamiltonian dynamics. When the
dynamics is ergodic, the Floquet spectrum obeys a Wigner-Dyson statistics and
the system satisfies the eigenstate thermalization hypothesis (ETH):
Independently of the initial state, local observables relax to the $T=\infty$
thermal value, and Floquet states are delocalized in the Hilbert space. On the
contrary, if the classical dynamics is regular no thermalization occurs. We
further discuss the relationship between ergodicity and dynamical phase
transitions, and the relevance of our results to other fully-connected
periodically driven models (like the Bose-Hubbard), and possibilities of
experimental realization in the case of two coupled BEC.
