Coupled Oscillator Model of the Business Cycle with Fluctuating Goods Markets

Abstract

The sectoral synchronization observed for the Japanese business cycle in the Indices of Industrial Production data is an example of synchronization. The stability of this synchronization under a shock, e.g., fluctuation of supply or demand, is a matter of interest in physics and economics. We consider an economic system made up of industry sectors and goods markets in order to analyze the sectoral synchronization observed for the Japanese business cycle. A coupled oscillator model that exhibits synchronization is developed based on the Kuramoto model with inertia by adding goods markets, and analytic solutions of the stationary state and the coupling strength are obtained. We simulate the effects on synchronization of a sectoral shock for systems with different price elasticities and the coupling strengths. Synchronization is reproduced as an equilibrium solution in a nearest neighbor graph. Analysis of the order parameters shows that the synchronization is stable for a finite elasticity, whereas the synchronization is broken and the oscillators behave like a giant oscillator with a certain frequency additional to the common frequency for zero elasticity.

Direct Evidence for Synchronization in Japanese Business Cycles

“…The observed oscillation with the same period is induced by the interaction between two pendulum clocks mediated by the beam. This phenomenon is called synchronization, or entrainment [16,17].…”

Entrainment in forced Winfree systems

“…In engineering, synchronization is applied to power grids [57], microwave oscillations [200], beam steering [41,80] and antenna technology [115]. In economy, synchronization is used for analysis of goods markets [90] and the global economic activity [71], measuring market movements [58], and stock arrangement [110].…”

Synchronization of coupled second-order Kuramoto oscillators

“…Except for the interest of exploring the synchronization of coupled units, the second-order oscillator has various applications. These second-order oscillators can also be obtained with minor changes and applied to various systems, such as the Josephson junction arrays (Levi et al, 1978;Watanabe and Strogatz, 1994;Trees et al, 2005), goods markets (Ikeda et al, 2012), dendritic neurons (Sakyte and Ragulskis, 2011), and power generators (Filatrella et al, 2008;Rohden et al, 2012Lozano et al, 2012;Bergen and Hill, 1981;Hill and Chen, 2006).…”

“…The Kuramoto model is not only simple and amenable to analytical considerations, but it is also easy to generalize in different directions. By adding frequency adaptations (inertias), the second-order oscillators model has been proposed and developed to describe the dynamics of several systems: tropical Asian species of fireflies (Ermentrout, 1991); Josephson junction arrays (Levi et al, 1978;Watanabe and Strogatz, 1994;Trees et al, 2005); goods markets (Ikeda et al, 2012); dendritic neurons (Sakyte and Ragulskis, 2011); and power grids (Filatrella et al, 2008). Many important conclusions about the stability of power grids have been obtained through analysis of this model (Rohden et al, 2012Lozano et al, 2012;Hellmann et al, 2016;Kim et al, 2015;Gambuzza et al, 2017;Fortuna et al, 2011;Grzybowski et al, 2016;Maïzi et al, 2016;Manik et al, 2017a;Pinto and Saa, 2016;Rohden et al, 2017;Witthaut et al, 2016).…”

“…Among the different models of oscillator dynamics, the Kuramoto model Kuramoto and Nishikawa (1987), and its various generalizations Rodrigues et al (2016), are some of the most popular models. Within this class of generalized Kuramoto models, second-order oscillator models, that is, oscillators with inertias, have been used for describing the dynamics of fireflies Ermentrout (1991), Josephson junction arrays Levi et al (1978); Watanabe and Strogatz (1994); Trees et al (2005), goods markets Ikeda et al (2012), dendritic neurons Sakyte and Ragulskis (2011), and power grids Filatrella et al (2008). Due to the effect of inertias, phenomena such as hysteresis, bi-stability and abrupt transitions are found for these second-order oscillators Tanaka et al (1997a,b); Gao and Efstathiou (2018).…”