Dr. Noah Corbett fulfilled all of the requirement for the Ph.D. in Mathematics and defended his dissertation on Wednesday, April 1, 2026.
Advisors:
Jason Mireles-James, Ph,D, (University of Texas, Austin, 2009)
Vincent Naudot, Ph.D. (Dijon-France, 1996)
Title: A Computational Approach to Periodic Orbits of State-Dependent Delay Differential Equations
Abstract: The field of delay differential equations (DDEs) concerns the study of systems whose evolution depends on certain past states of the system. Of particular interest are the state-dependent DDEs, whose delay terms are non-constant and depend on the current state itself. In this thesis, we provide rigorous solution-finding techniques for a certain class of one-dimensional state-dependent DDEs, as well as a state-dependent delayed Van der Pol equation. This technique is inspired by the classical Picard-Lindelöf theorem and is successful in proving the existence and uniqueness of orbits in such systems under certain reasonable restrictions. We then employ the Lagrange-Chebyshev interpolating operator to frame our results in the computational setting, allowing us to algorithmically obtain periodic orbits of the systems in question. These algorithms are based on the Newton-Kantorovich theorem and the resulting illustrations and numerics are discussed in detail.
