This paper is concerned with the development of basic concepts and constructs for a nonequilibrium theory of nonlinear control. Motivated by an example of nonstabilizability of rigid spacecraft about an equilibrium (reference attitude) but stabilizability about a revolute motion, we review recent work on the structure of those compact attractors which are Lyapunov stable. These results are illustrated and refined in a description of the asymptotic behavior of practically stabilizable systems taken form a recent work on bifurcations of the system zero dynamics. These attractors can contain periodic orbits, and necessary and sufficient condition for the existence of periodic orbits are discussed. These conditions lead to the notion of a “control one-form” and to necessary conditions for the existence of an orbitally stable periodic motion. As it turns out, even when this latter result is specialized to the equilibrium case, the criterion is new.