Chemical reactions show a separation of time scales (rapid transient decay) due to the stiffness of the ordinary differential equations (ODEs) describing their evolution. In enzyme kinetics time scale separation allows the steady-state evolution of such systems to be represented on a hierarchy of smooth, slow manifolds embedded in the full phase space of concentration variables for the complete reaction. Typically such manifolds are dynamically stable in the sense that they attract the surrounding phase flow exponentially fast; this relates to their confinement within regions of phase space bounded by the nullclines of the system. The slow manifolds also contain the true attractors of the system. Explicit formulas for manifolds of this kind can be found by iterating functional equations using a symbolic language like Maple. It has been proved that, using sufficiently smooth starting functions, e.g., the expressions for the nullclines, the nth iteration of the functional equation provides expressions for the slow manifolds is accurate to the nth power in the singular perturbation parameter(s) that appear in the ODEs. However, the iteration procedure may diverge. This can be related to the geometry of the phase flow, e.g., the phase-space region in question does not lie between system nullclines and is not locally exponentially attracting. Nevertheless, if the local phase flow has the correct properties iteration can be stabilized.
The iterative method provides global formulas for the manifolds in cases where series methods diverge. There are many advantages to such reduced descriptions: the corresponding ODEs describe the system evolution on the slow manifolds; consequently, bifurcations of the system can be analysed on the manifolds: changes in the dimensionality associated with the system evolution can be expressed as structural, geometrical changes within the slow-manifold hierarchy.