Nonlinear oscillations, synchronization and chaos

The analysis of nonlinear oscillator circuits often begins with the study of a lossless circuit. When studying lossless circuits it is desirable to express their equations in Hamiltonian form. We give a method for obtaining by inspection the Hamiltonian and canonical variables for nonlinear lossless circuits composed of charge controlled capacitors, flux controlled inductors and independent voltage and current sources.; The method of averaging has been used for many years to prove the existence of oscillations in nonlinear circuits. In the past the application of averaging has tended to be ad hoc rather than systematic, and the validity of the method was not well established. We rigorize and systematize the analysis of weakly nonlinear oscillator circuits via the method of averaging. In particular we put on a rigorous foundation the concepts of averaged potential and oscillatory modes for oscillator circuits.; The first order nonuniformly sampling digital phase locked loop (DPLL) is arguably the simplest synchronization system, being described by a first order difference equation defined on the circle. We describe general first order DPLLs and give an overview of the theory of circle maps. We describe an experimental implementation of a first order DPLL and indicate how a DPLL operating in a region of parameter space where the loop exhibits chaos can be used for the generation of cryptographically secure random bits. We also consider interconnections of first order DPLLs and give an algorithm for studying their behavior in an efficient manner.; A system of more practical interest is the second order DPLL whose dynamics are more complicated than its analog counterpart. A second order digital phase locked loop may exhibit unusual behavior for some parameters due to a fractal boundary between the basin of attraction of the locked fixed point and the attracting basins of coexisting periodic orbits. The usual optimization criterion of the loop parameters using linearized analysis is insufficient, due to coexisting period orbits. We present a new optimization procedure based upon numerical bifurcation studies. We obtain numerical estimates of average lock time which can be approximated analytically using the rate of contraction of the phase space in a neighborhood of the fixed point, together with the size of the phase space that is regular for the underlying Hamiltonian system.