Unstable periodic orbits embedded in a chaotic economic dynamics model

Abstract

We numerically find unstable periodic solutions embedded in a chaotic attractor in a generalized Goodwin model with an interaction between two countries and focus on a class of simple periodic orbits extracted from them. We confirm that chaotic behaviour represented by the model is qualitatively and quantitatively related to the unstable periodic solutions. The viewpoint in this article is based on recent work in physics. The result implies significance and usefulness of unstable periodic solutions embedded in chaotic economic dynamics.

Notes

¹ For simplicity the model contains some ad hoc assumptions.

² Government spending (Wolfstetter, 1982) is incorporated into Equation 5.

³ The complexity of expectation formation behind the function h _i may cause chaotic behaviour.

⁴ For some parameter settings, we have derived qualitatively the same results.

⁵ We use the fourth order Runge–Kutta method in our time integrations.

⁶ We consider solution X is periodic if ‖X(T) − X(0)‖ ≤ 10^{− 10}, where T ∈ R ₊ is the period and ‖ ·‖ is the Euclidean norm.

⁷ e.g., Cvitanović (1988)