The Effect of Lumping and Expanding on Kinetic Differential Equations

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 ...