Isochronous rate equations describing chemical reactions

Abstract

We consider systems of ordinary differential equations in the plane featuring at most quadratic nonlinearities. It is known that, up to linear transformations of the variables, there are only four systems for which the origin is an isochronous center, that is, for which all orbits in the vicinity of the origin are periodic with the same, fixed period. On the other hand, if, after an affine transformation, the system's coefficients satisfy certain positivity requirements, these systems can be interpreted as kinetic equations for chemical reactions. Here we show that, for two of these four isochronous systems, it is possible to find an affine transformation such that the transformed system obeys all these positivity conditions. For the third we can show that this is not possible, whereas for the fourth the issue remains to some extent open. Hence for the two cases mentioned above these systems may be interpreted as kinetic equations describing isochronous chemical reactions.