Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents

TL;DR

This paper investigates irregular and chaotic economic dynamics using Lyapunov exponents and attractor localization in a firm model, proposing control methods to improve long-term forecasting accuracy.

Contribution

It introduces an analytical approach for attractor localization and Lyapunov dimension calculation, and demonstrates control techniques to transition from chaos to regularity in economic models.

Findings

Analytical formulas for Lyapunov exponents and dimensions of the attractor.

Effective use of delayed feedback control to stabilize dynamics.

Complexity in numerical Lyapunov dimension estimation across trajectories.

Abstract

Cyclicity and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run. We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy. Using of a mid-size firm model as an example, we demonstrate the use of effective analytical and numerical procedures for calculating the quantitative characteristics of its irregular limiting dynamics based on Lyapunov exponents, such as dimension and entropy. We use an analytical approach for localization of a global attractor and study limiting dynamics of the model. We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically. With the help of delayed feedback control (DFC), the possibility of transition from irregular limiting dynamics to regular periodic dynamics is shown to solve the problem of reliable forecasting. At the same time, we demonstrate the complexity and ambiguity of applying numerical procedures to calculate the Lyapunov dimension along different trajectories of the global attractor, including unstable periodic orbits (UPOs).