The onset of new time domain surveys and the imminent increase in astronomical data exposes the shortcomings in traditional time series analysis (such as power spectra analysis) in characterizing the abundantly varied, complex and stochastic light curves of both X-ray Binaries (XRBs) and Active Galactic Nuclei (AGN). Recent applications of alternative methods to Fourier analysis have shown promise in characterizing higher modes of variability and timescales in AGN. Methods of analysis developed to study the recurrences of dynamical trajectories in phase space, which can be constructed from one-dimensional time series such as light curves, provide complementary information about not only characteristic timescales revealed by other methods, but also direct indications of the nature of the underlying physics in these objects. Studying a light curve in an embedded phase space provides an alternative perspective of the dynamics that generate the light curve, which can be leveraged to complement and compensate for other methods. We apply methods from nonlinear dynamics to a sample of optical light curves of Kepler-monitored AGN, X-ray light curves of XRBs monitored by RXTE and MAXI, and hard X-ray light curves of XRBs and AGN observed by Swift/BAT. We detect regions in the light curves that deviate from regularity, indicate deterministic mechanisms, and reveal evidence of nonlinearity and chaos in some XRBs. We provide evidence for the types of possible underlying processes driving the dynamics of the light curves and their diverse classes of variability. We propose future application of novel methods of analysis to a larger sample and other time-varying systems in preparation for future time domain missions.