  We study synchronisation between periodically driven, interacting classical
spins undergoing a Hamiltonian dynamics. In the thermodynamic limit there is a
transition between a regime where all the spins oscillate synchronously for an
infinite time with a period twice as the driving period (synchronized regime)
and a regime where the oscillations die after a finite transient (chaotic
regime). We emphasize the peculiarity of our result, having been
synchronisation observed so far only in driven-dissipative systems. We discuss
how our findings can be interpreted as a period-doubling time crystal and we
show that synchronisation can appear both for an overall regular and an overall
chaotic dynamics.

Statistical Mechanics;Chaotic Dynamics