  We consider dynamics of a two-component Bose-Einstein condensate where the
components correspond to different hyperfine states of the same sort of atoms.
External microwave radiation leads to resonant transitions between the states.
The condensate is loaded into the optical lattice. We invoke the tight-binding
approximation and examine the interplay of spatial and internal dynamics of the
mixture. It is shown that internal dynamics qualitatively depends on the
intra-component interaction strength and the phase configuration of the initial
state. We focus attention on two intriguing phenomena occurring for certain
parameter values. The first phenomenon is the spontaneous synchronization of
Rabi oscillations running inside neighbouring lattice sites. Another one is the
demixing of the condensate with formation of immiscible solitons when
nonlinearity becomes sufficiently strong. Demixing is preceded by the transient
regime with highly irregular behavior of the mixture.

Quantum Gases;Chaotic Dynamics;Pattern Formation and Solitons;Quantum Physics