Driven nonlinear dynamical systems are everywhere in biology. Powerful mathematical tools like Fourier analysis or the principal of superposition are not available in nonlinear systems. This thesis investigates three different dynamical systems drawn from biology and machine learning as a function of the ratio of the system’s response timescale to the signal’s timescale. We focus our studies on regimes where the timescale of the external driving signal is comparable to the timescale of internal responses. The methods here may not solve every driven nonlinear problem because no universal method likely exists for strongly driven non-linear systems. Nevertheless, the results will help solve future similar challenges.