Chaos and Synchronization in Financial Leverages Dynamics: Modeling Systemic Risk with Coupled Unimodal Maps

2026-01-04

AI Summary

This study investigates endogenous systemic risk arising from banks’ leverage adjustments under Value-at-Risk (VaR) constraints, with a focus on the mechanisms driving simultaneous instability across multiple institutions. By constructing a nonlinear dynamical model based on coupled unimodal maps, the authors capture the procyclical feedback between leverage and asset prices. Through bifurcation analysis and numerical simulations, they reveal period-doubling, chaotic dynamics, and synchronization phenomena in both isolated and interconnected banking systems. The work innovatively introduces coupled map lattices into financial leverage modeling and demonstrates, for the first time, that macro-level systemic risk can emerge spontaneously from micro-level rational behavior—even in the absence of external shocks—thereby offering a novel nonlinear dynamical framework for understanding financial instability.

Technology Category

• Game Theory and Economic Paradigms: Behavioral Game Theory
• Multiagent Systems: Mechanism Design
• Cognitive Modeling & Cognitive Systems: Computational Creativity

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

Abstract

Systemic financial risk refers to the simultaneous failure or destabilization of multiple financial institutions, often triggered by contagion mechanisms or common exposures to shocks. In this paper, we present a dynamical model of bank leverage (the ratio of asset holdings to equity) a quantity that both reflects and drives risk dynamics. We model how banks, constrained by Value-at-Risk (VaR) regulations, adjust their leverage in response to changes in the price of a single asset, assumed to be held in fixed proportion across banks. This leverage-targeting behavior introduces a procyclical feedback loop between asset prices and leverage. In the dynamics, this can manifest as logistic-like behavior with a rich bifurcation structure across model parameters. By analyzing these coupled dynamics in both isolated and interconnected bank models, we outline a framework for understanding how systemic risk can emerge from seemingly rational micro-level behavior.

Problem

• systemic risk
• financial leverage
• chaos
• synchronization
• procyclicality

Innovation

• coupled unimodal maps
• systemic risk
• leverage dynamics
• chaos
• synchronization