At TU Dresden, I served as a lecturer for the following advanced elective course.

Advanced Elective Course

An Introductory Guide to Nonlinear Oscillations and Oscillatory Networks

Schedule: Monday, 16:40–18:10 (6th DS) Wednesday, 11:10–12:40 (3rd DS)

Lecturers: Subhendu Bhandary, Seungjae Lee, and Marc Timme

In both classical and quantum mechanics, we always begin our study with the simple harmonic oscillation. Why? Because oscillatory behavior underlies so many fundamental concepts in the physical sciences, explaining phenomena from atomic vibrations to planetary motions. For example, harmonic oscillations capture the local dynamics around a stable fixed point for any potential. Likewise, nonlinear oscillations reveal amplitude-dependent frequencies, limit-cycle behavior, synchronization effects, and even chaos—features that lie entirely beyond the scope of linear models. Yet even the simplest nonlinearity often destroys exact solvability. Regular (or naïve) perturbation methods likewise fall short. In practice, then, we need toolkits that go beyond regular perturbation to predict, control, and design the rich behaviors of real mechanical, electrical, and biological oscillators.

In this lecture, we will explore powerful methods beyond regular perturbation and apply them to nonlinear oscillatory systems. We will survey how to approximate a periodic solution for a nonlinear differential equation in terms of familiar elementary functions. Throughout, emphasis is placed on the practical implementation of asymptotic expansions, singular perturbation methods, and the interpretation of leading-order dynamics. Participants will learn various frameworks for tackling a broad class of nonlinear oscillations, laying the groundwork for applications in classical mechanics, electrodynamics, and quantum mechanics. By the end of the lecture, attendees will be equipped with both theoretical insight and hands-on techniques for deriving and validating approximate solutions to nonlinear oscillatory problems.