Until recently, deterministic non-autonomous oscillatory systems with stable amplitudes and time-varying frequencies were not recognised as such and have often been mistreated as stochastic. These systems, named chronotaxic, were introduced in \emph{Phys. Rev. Lett.} \textbf{111}, 024101 (2013). In contrast to conventional limit cycle models of self-sustained oscillators, these systems posses a time-dependent point attractor or steady state. This allows oscillations with time-varying frequencies to resist perturbations, a phenomenon which is ubiquitous in living systems. In this work a detailed theory of chronotaxic systems is presented, specifically in the case of separable amplitude and phase dynamics. The theory is extended by the introduction of chronotaxic amplitude dynamics. The wide applicability of chronotaxic systems to a range of fields from biological and condensed matter systems to robotics and control theory is discussed.