Naturalistic cognition in human performance is defined by dynamical responses to stimuli. Allostasis Machines (AMs) are characterized by an internal model and corresponding output trajectory characterizing a generalized response to stresses and sudden changes. The effects of the environment on the internal model are collectively known as perturbations, with a generalized response analogous to allostatic load. AMs consist of a sensory input, an internal model, a source of environmental perturbation, and an dynamical output that represents the response to perturbation over time. These dynamical output trajectories characterize this response either by recovering from perturbation (well-matched, ergodic), or drifting to a new stable state (accommodative, non-ergodic). We construct a quantitative model of AMs and consider their behaviors in a variety of scenarios, including isolated, serial, and new state perturbations. Control-theoretic strategies and multi-scale information processing can also be employed to provide AM models with more sophisticated feedback and control mechanisms. Understanding the difference between well-matched responses (stably matching environmental states) and allostatic drift (hysteretic responses to perturbation) clarifies how nonlinear responses produce continuous stability.