There exist many low dimensional ODEs which describe real systems but which have not yet been studied from a mathematical perspective. These arise mostly from nonlinear systems in engineering and physics: for instance, nonlinear electronic circuits know as the varactor circuit and the saturating inductor circuit, and a nonlinear ODE describing the stability of ships in beam seas (both described by the same ODE); and the Injection Locked Frequency Divider, a circuit with significant technological applications. Rigorous results have been obtained in these cases, usually using perturbation theory. Basins of attraction and conditions for finite-time blow-up of solutions have been obtained in the former case, and conditions for frequency locking have been found for the latter.