Fractional-Order Financial System and Fixed-Time Synchronization

Abstract

This study is concerned with the dynamic investigation and fixed-time synchronization of a fractional-order financial system with the Caputo derivative. The rich dynamic behaviors of the fractional-order financial system with variations of fractional orders and parameters are discussed analytically and numerically. Through using phase portraits, bifurcation diagrams, maximum Lyapunov exponent diagrams, 0–1 testing and time series, it is found that chaos exists in the proposed fractional-order financial system. Additionally, a complexity analysis is carried out utilizing approximation entropy SE and C0 complexity to detect whether chaos exists. Furthermore, a synchronization controller and an adaptive parameter update law are designed to synchronize two fractional-order chaotic financial systems and identify the unknown parameters in fixed time simultaneously. The estimate of the setting time of synchronization depends on the parameters of the designed controller and adaptive parameter update law, rather than on the initial conditions. Numerical simulations show the effectiveness of the theoretical results obtained.