Stable cycles and tori of a system of three and four diffusive coupled oscillators

We consider chains of identical diffusive weakly coupled oscillation systems with different conditions on coupling on a chain bound. We suppose that every interacting oscillator undergoes Andronov–Hopf bifurcation, and the coefficient of coupling is proportional to the supercriticality value. For this case on a stable integral manifold of the system a normal form is constructed for which, in case of three oscillators interact, we can study the simplest stationary states and their phase transformations. As the coupling parameter changes, for the uniform stationary state corresponding to the uniform cycle of the problem there can be two cases, in the former case it loses stability with the emergence of two stable nonuniform states, and in the latter one it merges with two unstable nonuniform states and transfers to them its stability. For the stationary state corresponding to oscillations in antiphase two cases can be also distinguished. In the first one, this stationary state becomes stable due to the contraction of a stable limit cycle of the system into it (Andronov–Hopf bifurcation), and in the second case it becomes stable after branching from it an unstable limit cycle. If there are four oscillators in the chain, the system of phase difference for a small coupling coefficient was studied.