We theoretically study the internal Josephson effect, which is driven by spin-exchange interactions and magnetic dipole-dipole interactions, in a three-level system for spin-1 Bose--Einstein condensates, obtaining novel spin dynamics. We introduce single spatial mode approximations into the Gross--Pitaevskii equations and derive the Josephson-type equations, which are analogous to tunneling currents through three junctions between three superconductors. From an analogy with two interacting nonrigid pendulums, we identify unique varied oscillational modes, called the $0--\ensuremath{\pi}$,
0−−running,
running−−running, $2n\ensuremath{\pi} & \mathit{running}--2\ensuremath{\pi}$,
singlenonrigidpendulum, and
tworigidpendulums phase modes. These Josephson modes in the three states are expected to be found in real atomic Bose gas systems.