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. |