In this course, we shall explore a few interesting physics phenomena as well as some useful mathematical techniques in differential equations. 1>First, one of the main characteristics we are concerned is the time periodicity. On the one hand, we study different scenarios that give the periodic dynamics. On the other hand, we consider the synchronization problem for the system consisting of many oscillators which have strong coupling relation. This is a universal phenomenon including circadian rhythms, electrical generators, Josephson junction arrays, heart, intestinal muscles, menstrual cycles, and fireflies. We will give a concise account of the recent development of this research direction including the most updated results obtained by my research group. 2> Secondly, we shall introduce some recent results on the regularity issue of the stationary solutions for the linearized Boltzmann equations. Overall, we shall introduce at least three type physics equations : fluid equations, gas dynamics and the Kuramoto oscillator systems.