Symbolic dynamics in the belousov–zhabotinskii reaction: from rössler’s intuition to experimental evidence for shil’nikov’s homoclinic chaos

The Belousov-Zhabotinskii reaction has revealed most of the well-known scenarios to chaos including period-doubling, intermittency, quasiperiodicity, frequency locking, fractal torus …. However, although the data have been shown to display unambiguous features of deterministic chaos, the understanding of the nature and the origin of the observed behavior has been incomplete. In 1976, Rössler suggested an intuitive interpretation to explain chemical chaos. His feeling was that nonperiodic wandering trajectories might arise in chemical systems from a pleated slow manifold (Fig. 1a), if the flow on the lower surface of the pleat had the property of returning trajectories to a small neighborhood of an unstable focus lying on the upper surface. In this communication, we intend to revisit the terminology introduced by Rössler of “spiral-type”, “screw-type” and “funnel-type” strange attractors in terms of chaotic orbits that occur in nearly homoclinic conditions. According to a theorem by Shil’nikov, there exist uncountably many nonperiodic trajectories in systems which display a homoclinic orbit biasymptotic to a saddle-focus O, providing the following condition is fulfilled: ρ/λ < 1, where the eigenvalue of O are (−λ, ρ ± iω). This subset of chaotic trajectories is actually in one to one correspondance with a shift automorphism with an infinite number of symbols. Since homoclinic orbits are structurally unstable objects which lie on codimension-one hypersurfaces in the constraint space, one can reasonably hope to cross these hypersurfaces when following a one-parameter path. The bifurcation structure encountered near homoclinicity involves infinite sequences of saddle-node and period-doubling bifurcations. The aim of this paper is to provide numerical and experimental evidences for Shil’nikov homoclinic chaos in nonequilibrium chemical systems.