The Effect of Lumping and Expanding on Kinetic Differential Equations

Source abstract

Let us consider the differential equation $\dot{\bf y}(t)={\bf f}({\bf y}(t))$ with an $\bf f$ from ${\bf R}^N$ to ${\bf R}^N$ and suppose that there exists a transformation $\bf h$ from ${\bf R}^N$ to ${\bf R}^{\hat N}$ ($\hat N\le N$) such that $\hat{\bf y}:={\bf h}\circ{\bf y}$ obeys a differential equation $\dot{\hat{\bf y}}(t)={\hat{\bf f}}({\hat{\bf y}}(t))$ with some function $\hat{\bf f}$; then the first equation is said to be lumpable to the second by $\bf h$. Here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction. We provided a series of necessary and sufficient conditions for the existence of such functions $\bf h$ and $\hat{\bf f}$; some of them are formulated in terms of $\bf h$ and $\bf f$ only. Beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point of view of formal reaction kinetics. Results show that each eigenvalue of the Jacobian of the nonlinear lumped system at an equilibrium is an eigenvalue of the original system at the corresponding equilibrium. (Invariant sets, equilibria, and periodic solutions are lumped into invariant sets, equilibria, and periodic solutions, respectively.) Under certain conditions a Lyapunov function of the lumped system can be used to create a Lyapunov function for the original one (to test relative stability) and vice versa---both around equilibria and far from equilibria. These general statements do not necessarily imply close qualitative resemblance of lumped and original solutions but provide criteria by which lumping schemes may be selected for this purpose. The precise meaning of the conditions in the general statements is illustrated by examples taken from formal reaction kinetics.