Complex Synchronization Dynamics of Electronic Oscillators–Part I: A Time-Domain Approach via Phase-Amplitude Reduced Models

This work introduces a rigorous time-domain approach for studying the complex synchronization dynamics of periodically forced electronic oscillators, based on the well-developed theories of Phase-Amplitude reduction via the Koopman operator and dynamics of circle maps. The paper is structured in two parts. Part I presents the theoretical foundation and the numerical application of the theory. Under suitable forcing, the reduced equations simplify to a one-dimensional phase model—represented by a circle map—whose bifurcations are determined by the Phase Response Curves. This map efficiently captures the oscillator’s dynamics and enables accurate computation of resonance regions in the forcing parameter space. The influence of global isochron geometry on the map validates their critical role in phase locking, extending previous results in the theory of electronic oscillators. For more general forcing scenarios, the full Phase-Amplitude reduction effectively describes the synchronization dynamics. The developed time-domain approach demonstrates that the same limit cycle oscillator can produce periodic output with tunable spectral characteristics, operating as a frequency divider, or function as a chaotic or quasiperiodic signal generator, depending on the driving signal. As an illustrative example, the synchronization dynamics of differential LC oscillators is studied in detail. Part II is dedicated to confirming the validity, generality, and robustness of the introduced approach, which is first presented as a detailed step-by-step methodology, suitable for direct application to any oscillator. The Colpitts and ring oscillators are analyzed theoretically, and their resonance diagrams are numerically computed, following the approach established in Part I. Simulations of realistically implemented models in the Cadence IC Suite show that both synchronized and chaotic/quasiperiodic states are accurately predicted by the reduced circle map. Notably, despite the use of simplified analytical models, the theoretical framework effectively captures the qualitative behavior observed in simulation. The consistency between the theoretical and simulation results confirms both the robustness and general applicability of the proposed approach.