Date of Award


Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Amitabha Koshal Bose

Second Advisor

Casey Diekman

Third Advisor

Horacio G. Rotstein

Fourth Advisor

Denis L. Blackmore

Fifth Advisor

Victor Victorovich Matveev


The circadian rhythm refers to an internal body process that regulates many body processes including the sleep-wake cycle, digestion and hormone release. The ability of a circadian system to entrain to the 24-hour light-dark cycle is one of the most important properties. There are several scenarios in which circadian oscillators do not directly receive light-dark forcing. Instead they are part of hierarchical systems in which, as \peripheral" oscillators, they are periodically forced by other \central" circadian oscillators that do directly receive light input. Such dynamics are modeled as hierarchical coupled limit cycle systems. Those models usually have a large population, and are non-autonomous. In this dissertation, a coupled Kuramoto model and a coupled Novak-Tyson model are developed to study the entrainment of hierarchical coupled circadian oscillators. Direct simulations usually are incapable of revealing the full dynamics of such models. One goal of this dissertation is to apply proper mathematical methods to simplify the original systems. A phase reduction method is applied for reducing the original system to phase model. A parameterization method is introduced for simplifying such systems, and it is also applied for computing invariant manifolds of some biological oscillators. A novel tool, entrainment map, is developed and extended to a higher dimensional situation. Compared with direct simulations, the map has the advantages of describing the conditions for existence and stability of the limit-cycle solutions, as well as studying forcing and coupling strength dependent bifurcations. It is also more practical to calculate the entrainment times by just iterating the map rather than by direct simulations.