Features in chemical kinetics. III. attracting subspaces in a hyper-spherical representation of the reactive system

In this work, we deal with general reactive systems involving N species and M elementary reactions under applicability of the mass-action law. Starting from the dynamic variables introduced in two previous works [P. Nicolini and D. Frezzato, J. Chem. Phys. 138(23), 234101 (2013); 138(23), 234102 (2013)], we turn to a new representation in which the system state is specified in a (N x M)^2- dimensional space by a point whose coordinates have physical dimension of inverse-of-time. By adopting hyper-spherical coordinates (a set of dimensionless “angular” variables and a single “radial” one with physical dimension of inverse-of-time) and by examining the properties of their evolution law both formally and numerically on model kinetic schemes, we show that the system evolves towards the equilibrium as being attracted by a sequence of fixed subspaces (one at a time) each associated with a compact domain of the concentration space. Thus, we point out that also for general non-linear kinetics there exist fixed “objects” on the global scale, although they are conceived in such an abstract and extended space. Moreover, we propose a link between the persistence of the belonging of a trajectory to such subspaces and the closeness to the slow manifold which would be perceived by looking at the bundling of the trajectories in the concentration space.