We study the dynamics of macroscopically-coherent matter waves of an
ultra-cold atomic spin-one or spinor condensate on a ring lattice of six sites
and demonstrate a novel type of spatio-temporal internal Josephson effect.
Using a discrete solitary mode of uncoupled spin components as an initial
condition, the time evolution of this many-body system is found to be
characterized by two dominant frequencies leading to quasiperiodic dynamics at
various sites. The dynamics of spatially-averaged and spin-averaged degrees of
freedom, however, is periodic enabling an unique identification of the two
frequencies. By increasing the spin-dependent atom-atom interaction strength we
observe a resonance state, where the ratio of the two frequencies is a
characteristic integer multiple and the spin-and-spatial degrees of freedom
oscillate in "unison". Crucially, this resonant state is found to signal the
onset to chaotic dynamics characterized by a broad band spectrum. In a
ferromagnetic spinor condensate with attractive spin-dependent interactions,
the resonance is accompanied by a transition from oscillatory- to
rotational-type dynamics as the time evolution of the relative phase of the
matter wave of the individual spin projections changes from bounded to
unbounded.
