Synchronization and multi-frequency oscillations in the low-dimensional chain of the self-oscillators

The problem of growing complexity of the dynamics of the coupled phase oscillators as the number of oscillators in the chain increases is considered. The organization of the parameter space (parameter of the frequency detuning between the second and the first oscillators versus parameter of dissipative coupling) is discussed. The regions of complete synchronization, quasi-periodic regimes of different dimensions and chaos are identified. We discuss transformation of the domains of different dynamics as the number of oscillators grows. We use the method of charts of Lyapunov exponents and modification of the method of the chart of dynamical regimes to visualize two-frequency regimes of different type. Limits of applicability of the quasi-harmonic approximation and the features of the dynamics of the original system which are not described by the approximate phase equations are discussed for the case of three coupled oscillators.