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

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This study introduces an efficient Galerkin-Koornwinder (GK) approximation for analyzing Hopf bifurcations in nonlinear delay differential equations. The method simplifies determining bifurcation types in physics models, like cloud-rain dynamics.

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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.