Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: Applications to cloud-rain models

By means of Galerkin-Koornwinder (GK) approximations, an efficient reduction approach to the Stuart-Landau (SL) normal form and center manifold is presented for a broad class of nonlinear systems of delay differential equations that covers the cases of discrete as well as distributed delays. The focus is on the Hopf bifurcation as a consequence of the critical equilibrium's destabilization resulting from an eigenpair crossing the imaginary axis. The nature of the resulting Hopf bifurcation (super- or subcritical) is then characterized by the inspection of a Lyapunov coefficient easy to determine based on the model's coefficients and delay parameters. We believe that our approach, which does not rely too much on functional analysis considerations but more on analytic calculations, is suitable to concrete situations arising in physics applications. Thus, using this GK approach to the Lyapunov coefficient and the SL normal form, the occurrence of Hopf bifurcations in the cloud-rain delay models of Koren and Feingold (KF) on one hand and Koren, Tziperman, and Feingold on the other are analyzed. Noteworthy is the existence of the KF model of large regions of the parameter space for which subcritical and supercritical Hopf bifurcations coexist. These regions are determined, in particular, by the intensity of the KF model's nonlinear effects. "Islands" of supercritical Hopf bifurcations are shown to exist within a subcritical Hopf bifurcation "sea"; these islands being bordered by double-Hopf bifurcations occurring when the linearized dynamics at the critical equilibrium exhibit two pairs of purely imaginary eigenvalues.