An Arrow of Time in Stable Autonomous Generation

Abstract

Does stable autonomous generation select a direction of time? This question matters both for learning models that autonomously generate prescribed dynamics and for designing controllers that make an invariant set attracting without changing its dynamics. Periodic motion and quasiperiodic torus rotations admit stable generation in either direction. For a chaotic attractor, however, reversing the equations reproduces its backward trajectories while turning attraction into repulsion. Can additional latent variables or autonomous feedback restore attraction without changing the reversed motion? We prove an obstruction to such exact generation by finite-dimensional autonomous systems with $C^2$ vector fields, compact attracting sets, and continuous readouts. The result applies to the classical Lorenz attractor and, more generally, to dynamics with an ergodic invariant measure of positive metric entropy whose intrinsic local stable fibers are totally disconnected on a set of positive measure. The proof connects this geometry to an entropy contradiction: a hypothetical attracting realization would force the output entropy to vanish, contrary to the prescribed chaotic dynamics. We use reservoir-computing examples to illustrate the difficulty of stable autonomous generation for time-reversed chaos: both time directions are closely tracked while the networks are driven by data, but only the forward example retains Lorenz geometry during autonomous generation. The theorem identifies a class of dynamics that cannot be learned for exact, stable autonomous generation by finite-dimensional smooth models satisfying the stated assumptions, regardless of the learning procedure or the number of latent variables.