Impact of Hopf and Bautin Bifurcations on an Autocatalytic Chemical Reaction System

In this paper, we discuss the influence of mathematical computations i.e. codimension one and codimension two bifurcations on an autocatalytic chemical system. In the past, it was shown that the considered dynamical system exhibits Hopf bifurcation on the positive equilibria, but in current study we have symbolically identified that the study of bifurcation in this dynamical system is not limited to Hopf bifurcation. For this purpose, a complete chart of eigenvalues for the stability of autocatalytic reaction system is provided that shows that equilibrium points E3 and P have the possibility of other type of bifurcations. Mathematically, the first Lyapunov coefficient is used to determine the type of Hopf bifurcation and is extended to second Lyapunov coefficient for the possibility of Bautin bifurcation, whereas the provided analytical results are theoretical analyzed and physical interpreted to further explore the dynamics of autocatalytic chemical reaction dynamical system in various parametric regions. It is shown that how the balancing of two reactions behave between steady and oscillatory states. Similarly, the Bautin bifurcation identify severe sensitive transition in various oscillatory regimes, where their corresponding unfolding parameters scales the transition between different oscillatory states. Finally, MATLAB is used to simulate not only the analytical results for the qualitative analysis of trajectories around equilibrium points but also to easily understand the discussed physical meaning of provided mathematical results