Stability, convergence and bifurcation of a reduced Boissonade-De Kepper model with delayed feedback

Abstract

We analyze the local stability, convergence and bifurcation properties of a reduced Boissonade-De Kepper (BD) model, in the presence of time-delayed feedback. We derive conditions to ensure stability, and also investigate the impact of model parameters on the convergence characteristics of the system. The model undergoes a Hopf bifurcation when the conditions for local stability get violated. Using Poincaré normal forms and the center manifold theory, we are able to derive explicit analytic expressions for determining the type of the Hopf bifurcation and the stability of the emerging limit cycles. We also considered the impact of either having a quadratic or a cubic term in the model. Our findings show that the quadratic term always yields a sub-critical Hopf bifurcation, whereas with a cubic term, the model can switch between a sub-critical and a super-critical bifurcation. The insights exhibit the delicate relationship between the form of the non-linear terms and the resulting system dynamics. Some of the analytical insights are corroborated with numerical computations.