Structural instability and sustained oscillations in an extended Brusselator model

When all reactions in the "Brusselator" kinetic scheme are allowed to be reversible one can demonstrate, by analyzing the focal values, that, for a range of rate constants, a unique limit cycle is created from a multiple focus. A simple modification of the kinetic scheme leads to a model that has two saddle nodes in addition to the well-known unstable focus and a stable limit cycle. In the modified system, for some parameter values, the limit cycle disappears after evolving into a saddle-node connection. For some initial conditions the modified system has unbounded solutions that describe a possible explosion. The analysis that yields these results is based on a general, constructive procedure, which can be applied to higher order physical and chemical systems.