Document Type


Date of Award


Degree Name

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


Mathematical Sciences

First Advisor

Casey Diekman

Second Advisor

Horacio G. Rotstein

Third Advisor

Amitabha Koshal Bose

Fourth Advisor

Yong Ick Kim

Fifth Advisor

James MacLaurin


Entrainment is a type of synchronization in which the period of an endogenous oscillator matches the period of an external forcing signal and a stable phase relationship is maintained between them. Entrainment patterns are described in terms of the number of input oscillations (N) that are phase-locked to a number of output oscillations (M), referred to as N:M patterns. Arnold tongue diagrams are used to depict the regions of N:M entrainment patterns in the input period-amplitude parameter space. Although the entrainment of self-sustained oscillators by periodic forcing are well investigated is a well-studied problem, entrainment of damped oscillators has been less explored. This thesis characterizes entrainment responses for several models of biological oscillators with the unforced system in different dynamic regimes, such as an unstable focus with large amplitude oscillations, a stable focus with weakly damped oscillations, and a stable focus with strongly damped oscillations. The main finding of this dissertation is the existence of multiple disconnected 1:1 entrainment regions when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase plane analysis and other dynamical systems tools.

Chapter 1 provides an introduction to oscillator theory and the importance of biological oscillations. Chapters 2 and 3 consider the Fitzhugh-Nagumo model of neuronal oscillations and explore its entrainment properties. Chapter 4 focuses on two models of circadian (e 24-hour) rhythms in cyanobacteria and uncovers the dynamical mechanism underlying post-translational oscillations (PTOs). Chapter 5 then analyzes entrainment of these PTO models. In Chapter 6, entrainment results are presented for other models of neuronal, circadian, and glycolytic oscillations. These investigations lead to the conclusion that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic nullcline. Concluding thoughts and future work are presented in Chapter 7.



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