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<p>Economic cycle change is a common nonlinear dynamical phenomenon in society. In this paper, we consider a business cycle model with distributed time delay. Firstly, the stability of the equilibrium point is discussed by utilizing the first order approximation approach. Secondly, regarding the average time delay as a bifurcation parameter, the existence of the Hopf bifurcation is studied, and an explicit algorithm for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Lastly, numerical simulations are carried out to verify the correctness of the theoretical results. This paper suggests that both stable and unstable equilibrium points as well as limit cycles can occur for this delayed business model. This study provides new ideas for policy recommendations during economic fluctuations.</p>

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Hopf Bifurcation Analysis in a Business Cycle Model with Gamma-Type Distributed Time Delay

  • 期刊论文

  • 发表日期：

    2025年5月30日

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  • Yan Lu,
  • Zechen Tian,
  • Nan Liu,
  • Haiying Liu

摘要

Economic cycle change is a common nonlinear dynamical phenomenon in society. In this paper, we consider a business cycle model with distributed time delay. Firstly, the stability of the equilibrium point is discussed by utilizing the first order approximation approach. Secondly, regarding the average time delay as a bifurcation parameter, the existence of the Hopf bifurcation is studied, and an explicit algorithm for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Lastly, numerical simulations are carried out to verify the correctness of the theoretical results. This paper suggests that both stable and unstable equilibrium points as well as limit cycles can occur for this delayed business model. This study provides new ideas for policy recommendations during economic fluctuations.