Synchronization has been observed and applied in various disciplines, including circadian rhythms, firefly flashes, and biological oscillators. To demonstrate such phenomena, the Kuramoto model has been proposed and investigated in diverse applications. In this talk, I shall briefly give an introduction and introduce the Kuramoto model involving the amplitude dynamics. Then I shall present the main results and numerical simulations. Finally, I will give a summary and present some ongoing projects.

We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations, which consists of a bifurcation without parameters and a classical bifurcation. Our main result classifies the hybrid bifurcation when a line of equilibria with an exchange point of normal stability vanishes. We showcase the efficacy of our approach by proving stable periodic coexistent solutions in an ecosystem of two competing predators with Holling’s type II functional response. This is a joint work with Alejandro López Nieto, Phillipo Lappicy, Nicola Vassena, and Hannes Stuke.

Over the past two decades, various infectious diseases such as influenza, norovirus, and coronaviruses including SARS-CoV, MERS-CoV, and SARS-CoV-2 have consistently posed significant global threats. Alongside newly identified variant strains and seasonal outbreaks, these pathogens continue to challenge public health systems worldwide. Furthermore, the rapid dissemination of misinformation during outbreaks has amplified global concerns. In response, the Republic of Korea has implemented targeted quarantine and intervention strategies rather than broad-scale lockdowns, effectively managing disease spread while adapting policies according to the severity and characteristics of each epidemic. To provide accurate scientific insights and guide public health policies, we have developed and applied mathematical and statistical models for forecasting and analyzing the transmission dynamics of various infectious diseases, including COVID-19, influenza, and norovirus. These models enable assessment of vaccination strategies, evaluation of intervention effectiveness, and prediction of seasonal outbreak patterns. In this talk, I’d like to share our latest findings on forecasting infectious disease transmission and identifying effective policy measures for long term prevention and control.

It is well known that the boundary dynamics of vortex patches is locally well-posed in the Hölder space $C^{1,\alpha}$ for $0<\alpha<1$, while the well-posedness in $C^1$ remains an open problem. In this paper, we establish the local well-posedness of vortex patches in the class $C^{1,\varphi}$, where $\varphi$ is a modulus of continuity satisfying certain structural assumptions. It has been observed that the velocity operator in the contour dynamics equation is closely related to the Hilbert transform. Motivated by this, we solve both the contour dynamics equation and the equation for its associated Hilbert transform simultaneously. In doing so, we derive several properties of the Hilbert transform and its variant in critical spaces which are essential for controlling the velocity operator and its Hilbert transform.

The Zakharov-Kuznetsov equation was formally derived from the 3D Euler-Poisson system with a constant external magnetic field to describe the behavior of ions in the direction of the magnetic field. We discuss planar traveling solitary waves of the 3D Euler-Poisson system in the direction oblique to the magnetic field and compare it in the context of the Zakharov-Kuznetsov limit.