Double hopf-bifurcation in plane quadratic mass-action systems

Abstract

We consider models of open two-variable quadratic mass-action systems which may be regarded as the simplest nonlinear systems that can exhibit exotic behaviour. They occur in various branches of physics and biology such as ecology, solid, state physics and — most well-known — in chemistry and biochemistry. In this work it is shown that in spite of their simplicity these models cannot only undergo a normal but also a simultaneous Hopf-bifurcation to limit cycles at two different point of phase space. These limit cycles are found to always have opposite character of stability, thus giving rise to bistability of a limit cycle and a stationary point exterior to it. An example of a simple chemical reaction scheme with the delineated behaviour is presented.