Topological characterisation of a chaotic attractor with an additional branch generated from economic data

TL;DR

The paper demonstrates that chaotic dynamics can be extracted from macroeconomic time series by constructing a deterministic three-variable system with GPoM, revealing a chaotic attractor in $\mathbb{R}^3$ for $(x,y,z)$ corresponding to unemployment, inflation, and exchange rate. It develops a topological characterization using Poincaré sections, first return maps, UPOs, linking numbers, and templates, illustrated with the Rössler system and applied to the economic model. A key finding is the presence of an additional branch in the first return map, which, after appropriate encoding, does not change the attractor’s topological class but reflects Poincaré-section geometry; economically, this points to a nontrivial role for the exchange rate in the inflation–unemployment dynamics and stagflation episodes. The work underscores the value of combining global modelling with topological analysis to interpret nonlinear economic time series and suggests future bifurcation analyses to map routes to chaos across parameter regimes.

Abstract

There are insights of chaotic properties in economic systems and data. To prove the existence of chaotic dynamics, the establishment of a deterministic model is mandatory. A global modelling tool (GPoM) is used to search for mathematical models of equations from economic data: unemployment, inflation and nominal exchange rate over 30 years. A system of three differential equations is chosen as a model, whose solution is a chaotic attractor in $\mathbb{R}^3$. The model extracted from the data is not able to fit them, but it provides equations linking those multiple economic variables and reveals significant impact of exchange rate on unemployment and inflation evolution. The topological characterisation of the chaotic attractor solution exhibits an additional branch in its first return map to the Poincaré section. Consequences of this particular structure are analysed and interpreted economically.

Figures (11)

• Figure 1: Topological characterisation of a chaotic attractor. A. $\mathcal{R}$ is the chaotic attractor solution of the Rössler system \ref{['eq:rossler']}. B. Bounding torus of the attractor indication the required number of components to create the appropriate Poincaré section. C. Position of the Poincaré section \ref{['eq:rossler_01_section_rho']} in the phase space with a flow evolving clockwise. D. First return map with periodic points indicating the position of the Unstable Periodic Orbits (UPOs). E. Numerical representation of the UPOs. F. Table containing numerical linking numbers between UPOs. G. Template describing the topological properties of the attractor. H. Linking matrix detailing torsions and permutations for each branch in the template with the validation when theoretical linking numbers between orbits correspond to numerical linking numbers.
• Figure 2: Original time series (first column) and the normalized and resampled time series (second column) used to perform global modelling with GPoM.
• Figure 3: Attractor $\mathcal{E}$ solution of the model \ref{['eq:economic']}.
• Figure 4: Differential equations system and the original values.$x$ corresponds to the unemployment rate and $y$ to the inflation rate. The original time series projected in the $(x, y)$ phase space mainly evolves clockwise. The solution of the model \ref{['eq:economic']} in grey also evolves clockwise.
• Figure 5: A. Three-dimensional representation of the chaotic attractor solution $\mathcal{E}$ and the Poincaré section $\mathcal{P}$. B. Computed points of $\mathcal{P}$.