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Chaotic solutions found in the gravity model of network transport

Hajime Koike*

Hideki Takayasu and Misako Takayasu†

  • *Contact author: hajimekoike@outlook.jp
  • †Contact author: takayasu@comp.isct.ac.jp
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Phys. Rev. Research 7, 023232 – Published 5 June, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.023232

Abstract

The gravity model is a mathematical model that applies Newton's universal law of gravitation to socioeconomic transport phenomena and has been widely used to describe world trade, intercity traffic flows, and business transactions for several decades. However, its strong nonlinearity and diverse network topology make a theoretical analysis difficult, and only a short history of studies on its stability exists. In this study, the stability of gravity models defined on networks with few nodes is analyzed in detail using numerical simulations. It was found that, other than the previously known transition of stationary solutions from a unique diffusion solution to multiple localized solutions, parameter regions exist where periodic solutions with the same repeated motions and chaotic solutions with no periods are realized. The smallest network with chaotic solutions was found to be a ring with seven nodes, which produced a new type of chaotic solution in the form of a mixture of right and left periodic solutions.

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Phys. Rev. Research 7, 023232– Published 5 June, 2025
  • Received 24 October 2024
  • Accepted 20 April 2025
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