Multistationarity is neither necessary nor sufficient to oscillations

The theory of reaction kinetics is full with tradition, beliefs, conjectures and refutations. Let us mention just a few of them. Reversible reactions have a positive stationary concentration (true [28]), they only have one (false [7]). “Closed sytems tend to equilibrium” – if this statement is meant to mean that reversible detailed balanced reactions have a single, locally relatively asymptotically stable stationary point, then it is true [17,31,35] and has also been generalized to the case of complex balanced reactions [13,14,19,20]. Autocatalytic, autoinhibitory, or enzymatic steps are needed to obtain oscillatory reactions [2,23], false [22], cf. also [29]. Feed-back is needed to oscillation [5]. (It was even possible to produce chaos with monoand bimolecular steps without autocatalysis [24].) A more detailed account of history from this aspect can be found in [11, section 4.2]. In this short note we consider the statement (or, rather, conjecture) according to which multistationarity is a necessary or sufficient condition of oscillation in complex chemical reactions [3,6,10]. More precisely, the assertion is that in order to have periodic solutions in a complex chemical reaction at some values of the parameters, the reaction must have more than one stationary point for some values of the parameters. The majority of model reactions (as, e.g., the Autocatalator, the Ivanova reaction, the Lotka–Volterra reaction, the Oregonator) have this property. What is more, real chemical systems with oscillatory behavior also share this property and they have partly been designed using this conjecture [9,30]. (Although the reaction of chlorite with thiocyanate exhibits oscillations without known bistability so far [1], and the