Deterministic and stochastic theory of sustained oscillations in autocatalytic reaction systems

A special type of reaction in a uniform system with two reactants is considered which is a generalization of the first Lotka scheme. Using the Poincaré-Bendixson theorem the analytical conditions for the existence of limit cycles are derived and some examples are treated numerically in the deterministic picture. The onset of limit cycles is considered as a second order phase transition. The master equations are formulated and a general analysis of limit cycle reactions in the stochastic picture is given. The fluctuations of phase and amplitude and the correlation functions are discussed. Finally, Monte-Carlo solutions of the master equation are presented. The relations between the deterministic and the stochastic picture are discussed.