Study of the Complex Synchronization Dynamics of Periodically Forced Electronic Oscillators via Reduction to Generalized Action-Angle Variables

Abstract

Over the last years, spectral theory of dynamical systems has attracted attention complementing and expanding classical techniques. In particular, the spectrum of the Koopman operator contains important information regarding the geometry of the phase space and provides an appropriate conjugacy that globally linearizes the system. Spectral expansions of appropriate observables generalize notions like the isochrons and the isostables providing efficient numerical algorithms for their global computation. Moreover the operator’s spectrum can be connected with known notions like the stable and unstable manifolds (of limit points or limit cycles) and through the definition of appropriate functional spaces, in which spectral expansions are valid, the Laplace domain analysis is extended to nonlinear systems. In this way, generalized action-angle coordinates can be defined. The application of this reduction on periodic systems provides a rigorous and general framework for the study of periodically perturbed oscillators, generalizing approximate first order models that are usually utilized in the field of electronics. These models rely essentially on frequency-domain arguments, interpreting synchronization as the the result of harmonic mixing, and suffer from the limitations of weak forcing and nearly harmonic oscillations. Under certain conditions for the periodic forcing, the study is equivalent to that of a one dimensional circle map based on which the synchronization regions, referred to as Arnold tongues, can be derived. Considering two classical electronic oscillators (Colpitts and differential LC) we study analytically and numerically the nonlinear dynamics associated with each of them determining the bifurcation curves that result in qualitatively different phase portraits. Using Shilnikov theory we prove the qualitative structure of the bifurcation diagram of the Colpitts oscillator and show the existence of chaotic behavior. Regarding the differential LC oscillators, combining normal form theory and the Melnikov method, we analytically prove the existence of each bifurcation curve and the topological equivalence, at least in a neighborhood of the origin, of all systems belonging to this family. Periodically stimulating the above systems, we determine, based on the developed theory, the synchronization regions and extend the study, using the complete description with action-angle coordinates, to cases of general periodic forcing. The numerical results and the conducted simulations verify the validity and the robustness of the proposed methods. The study is concluded with the investigation of the chaotic synchronization of the Colpitts oscillator. The rigorous developed methods introduce a new approach and generalize the approximate techniques used in the electronics literature, allow the study of the synchronization dynamics of periodically perturbed oscillators in a general context, provide capabilities regarding the design of accurate timing systems and demonstrate that a single limit cycle oscillator can function as a divider, multiplier or a chaotic generator, depending on the driving signal.