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

Area of Science:

• Nonlinear Dynamics
• Mathematical Physics
• Differential Equations

Background:

• Nonlinear systems with delays often exhibit complex behaviors like Hopf bifurcations.
• Analyzing these bifurcations typically requires advanced functional analysis.
• Existing methods can be computationally intensive for practical applications.

Purpose of the Study:

• To present an efficient reduction approach for nonlinear delay differential equations using Galerkin-Koornwinder (GK) approximations.
• To focus on analyzing Hopf bifurcations and characterizing their nature (supercritical or subcritical).
• To apply the method to concrete physics problems, specifically cloud-rain delay models.

Main Methods:

• Utilizing Galerkin-Koornwinder (GK) approximations for reduction to Stuart-Landau (SL) normal form and center manifold.
• Employing Lyapunov coefficient calculations based on model coefficients and delay parameters.
• Analyzing Hopf bifurcations in Koren and Feingold (KF) and Koren, Tziperman, and Feingold cloud-rain models.

Main Results:

• The GK approach provides an efficient method for analyzing Hopf bifurcations in systems with discrete and distributed delays.
• Lyapunov coefficients are determined analytically, simplifying the characterization of bifurcation types.
• Coexistence of supercritical and subcritical Hopf bifurcations is identified in the KF model, influenced by nonlinear effects.
• Regions of supercritical Hopf bifurcations exist within subcritical regions, bordered by double-Hopf bifurcations.

Conclusions:

• The GK approximation offers a computationally tractable and analytic method for studying Hopf bifurcations in nonlinear delay systems.
• The approach is well-suited for practical physics applications, demonstrating its utility in climate modeling.
• Complex bifurcation phenomena, including coexistence and islands of different bifurcation types, are revealed in the analyzed cloud-rain models.