Generation of quasiperiodic oscillations in pairs of coupled maps

We investigate analytically the emergence of quasiperiodic and mode-locked states of arbitrary period in pairs of coupled maps. Quasiperiodic and mode-locked states arise from Naimark–Sacker bifurcations and exhibit very rich local dynamics. We determine analytically both states for pairs of maps under symmetric and asymmetric couplings. Lyapunov spectra, portraits in parameter-space and in phase-space are used to show the transition from quasiperiodic and from mode-locked states into chaotic states and to check the range of validity of the approximations implicit in the standard normal forms.