Early models of chemical oscillations failed to provide bounded solutions

Before the development of the Brusselator, the Selkov and Turing models were considered as possible prototype models of chemical oscillations. We first analyse their Hopf bifurcation branches and show that they become vertical past a critical value of the control parameter. We explain this phenomenon by the emergence of canard orbits. Second, we analyse all solutions in the phase plane and show that some initial conditions lead to unbounded trajectories even if there exists a locally stable attractor. Our findings mathematically explain why these too simple twovariable models fail to account for the emergence of chemical oscillations. They support the conclusion that the Brusselator is the first minimal two-variable model explaining the onset of stable oscillations in a way fully compatible with thermodynamics and the law of mass action. © 2018 The Author(s) Published by the Royal Society. All rights reserved.