In this doctoral dissertation we consider three problems in nonlinear dynamics -- a nonlinear second order delay differential equation (DDE), a nonlinear third order microelectromechanical (MEMS) system and a nonlinear first order DDE governing the spread of infectious diseases, in particular COVID-19. In the first problem, we consider the delayed Duffing equation $\ddot{x}+x_{d}+x^{3}=0$ where $x_d$ indicates delayed $x$. We find that as the delay is increased from zero, an infinite number of limit cycles of ever-increasing amplitude are born in a remarkable bifurcation. In the second problem, we construct a model of a MEMS cantilever which is heated by a laser. The model shows limit cycles. When two oscillators are coupled together, we get a very rich bifurcation sequence. The third problem relates to the spread of the pandemic COVID-19. We develop a model which is realistic and accounts for various features of the disease such as asymptomatic and latent (pre-symptomatic) transmission as well as interventions like contact tracing and mass testing.