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<p>Given the complex interplay between foreign investment, Gross Domestic Product (GDP) growth, and National Debt (ND), as well as the negative correlation between national debt and nominal interest rate, this paper constructs a GDP-IR-ND dynamics model incorporating foreign investment as a key factor, building upon the conventional IS-LM framework. To investigate the lag effect inherent in foreign investment, we introduce a time delay into the model, highlighting the delayed influence of foreign investment on the domestic economy. We analyze the system's stability and Hopf bifurcation characteristics. The non-zero equilibrium point is calculated, and the impact of the time delay in foreign investment on the domestic money and product markets is examined. Results reveal that the equilibrium characteristics of the domestic product market and the currency market are significantly influenced by the time delay associated with the debt level and foreign investment. Specifically, when the time delay of foreign investment is less than the critical threshold of 2.6&#xa0;years, the economic system experiences short-term fluctuations but ultimately returns to stability. However, when the time delay reaches this critical value, the system undergoes a Hopf bifurcation, entering a state of cyclical fluctuation. If the time delay exceeds 2.6&#xa0;years, the economic system may face the risk of a systemic crisis.</p>

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Equilibrium Characteristics and Hopf bifurcation Analysis of GDP-IR-ND Dynamic With Time Delay of Foreign Investment

  • 期刊论文

  • 发表日期：

    2026年3月2日

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  • Yuanyuan Ma,
  • Siying Wang,
  • Shu Liu,
  • Qiannan Zhang

摘要

Given the complex interplay between foreign investment, Gross Domestic Product (GDP) growth, and National Debt (ND), as well as the negative correlation between national debt and nominal interest rate, this paper constructs a GDP-IR-ND dynamics model incorporating foreign investment as a key factor, building upon the conventional IS-LM framework. To investigate the lag effect inherent in foreign investment, we introduce a time delay into the model, highlighting the delayed influence of foreign investment on the domestic economy. We analyze the system's stability and Hopf bifurcation characteristics. The non-zero equilibrium point is calculated, and the impact of the time delay in foreign investment on the domestic money and product markets is examined. Results reveal that the equilibrium characteristics of the domestic product market and the currency market are significantly influenced by the time delay associated with the debt level and foreign investment. Specifically, when the time delay of foreign investment is less than the critical threshold of 2.6 years, the economic system experiences short-term fluctuations but ultimately returns to stability. However, when the time delay reaches this critical value, the system undergoes a Hopf bifurcation, entering a state of cyclical fluctuation. If the time delay exceeds 2.6 years, the economic system may face the risk of a systemic crisis.