A physical system with qualitatively uncertain dynamics

Abstract

THE notion of determinism in classical dynamics has been eroded since Poincaré's work¹ led to the recognition that dynamical systems can exhibit chaotic behaviour, in which small perturbations grow exponentially fast. For a chaotic system, ubiquitous measure-ment errors, noise and computer round-off severely limit the time over which, given a precisely defined initial state, one can predict the detailed subsequent evolution. Practically speaking, the behaviour of such systems is quantitatively non-deterministic. Nevertheless, as the state of the system tends to be confined to an 'attractor' in phase space, at least its qualitative behaviour is predictable. Another challenge to determinism arises, however, when a system has competing attractors towards which an initial state may be drawn. Perturbations make it difficult to determine the fate of the system near the boundary between sets of initial conditions (basins) drawn toward different attractors, particularly if the boundary is geometrically convoluted². Recently, mathematical mappings were found³ for which the entire basin of a given attractor is riddled with 'holes' leading to a competing attractor. Here we present the first example of a physical system with this property. Perturbations in such a system render uncertain even the qualitative fate of a given initial state: experiments lose their reproducibility. We suggest that 'riddled' systems of this kind may be by no means uncommon.