Date of Award

Spring 1-1-2010

Document Type


Degree Name

Master of Science (MS)


Applied Mathematics

First Advisor

James Meiss

Second Advisor

James Curry


The goal of this research is to explore criteria sufficient to produce oscillations, sample some dynamical systems that oscillate, and investigate synchronization. A discussion on linear oscillators attempts to demonstrate why autonomous oscillators are inherently nonlinear in nature. After describing some criteria on second-order dynamics that ensure periodic orbits, we explore the dynamics of two second-order oscillators in both autonomous and periodically driven fashion. Finally, we investigate the phenomena of synchronization with the nonlinear phase-locked loop. Methods of analysis are exemplified as they become relevant including Poincaré; maps and the Zero-One test for chaos.

The Poincaré-Bendixson theorem is used to demonstrate the existence of periodic orbits in R2 under extraordinarily general conditions. Liénard's equation and theorem are introduced, which provide an intuitive parameterization for a class of oscillators. Liénard's equation is a second- order, ordinary differential equation that characterizes an oscillator with respect a state dependent damping function and a restoring force function. Liénard's theorem establishes sufficient criteria under which the Liénard's equation has a unique, stable, limit cycle.

The Duffing equation conforms with the Liénard equation, yet produces limit cycles without satisfying Liénard's theorem. Our Duffing dynamics are explained in the context of a nonlinear spring model. We survey the parameter space, which form both pitchfork and hyperbolic potential wells with respect to the displacement. These two wells characterize the bifurcations between the four fundamental undamped dynamical modes. One interesting result is that chaotic trajectories of the Duffing equation are able to quickly shed light on a multitude of quasi-periodic trajectories at the boundaries of the Poincaré map.

Next we introduce an oscillator that is similar to many engineered oscillators. The Van der Pol (VDP) oscillator model is presented in the context of a nonlinear current source in parallel with an inductor, a capacitor, and a resistor. It provides a net negative conductance destabilizing the equilibrium, and is tamed into global stability by increasing damping by the square of the voltage. The VDP oscillator is the opposite of the Duffing equation in that its nonlinearity is in the damping function, with a linear restoring force function. Like the VDP oscillator, many engineered oscillators are self-excited, autonomous systems that produce limit cycles.

Finally, we investigate the process of synchronization with the phase-locked loop (PLL). Synchronization is a nonlinear process in which systems entrain their frequencies to external signals or other systems. Naturally occurring PLLs lie at the foundation of synchronization. We describe the basic topology of the PLL. Interestingly, the phase model introduced conforms with Liénard's equation and is similar to the model used for the Josephson junction and the driven pendulum. Perhaps explaining the prevalence of synchronization, we show that almost any nonlinear functional can serve as a phase detector. We briefly demonstrate a phase-lock of two oscillators with phase- noise analysis. Finally, we report on the nonlinear behavior of the PLL when subjected to a modulated input.