Canards, Black Swans and Control of Chemical Reactions

The work is devoted to the investigation of critical phenomena using the geometric theory of singular perturbations, namely, the black swans and canards techniques. The interest to critical phenomena is occasioned by not only of safety reason, in many cases namely the critical regime is the most effective in technological processes. The sense of criticality here is as follows. The critical regime corresponds to chemical reaction separating the domains of self-accelerating reactions and domains of slow reactions. Recall that a canard (or French duck) is a trajectory of a singularly perturbed system of differential equations if it, at first, follows a stable slow integral manifold, and then an unstable one. In both cases the distances travelled are more than infinitesimally small. The slow integral manifold is defined as an invariant surface of slow motions. A canard trajectory may be considered as the result of gluing stable (attractive) and unstable (repulsive) slow integral manifolds at one point of the breakdown surface, due to the availability of an additional scalar parameter in the differential system. If we take an additional function of a vector variable parameterizing the breakdown surface, we can glue the stable and unstable slow integral manifolds at all points of the breakdown surface simultaneously. As a result we obtain the continuous stable/unstable integral surface or black swan. It is possible to consider the gluing function as a special kind of incomplete feedback control. This guarantees the safety of chemical regimes, even with perturbations, during a chemical process.