Chaotic Dynamics, Complexity Analysis and Control Schemes in Fractional Discrete Market System

The study of economic maps has consistently attracted researchers due to their rich dynamics and practical relevance. A deeper understanding of these systems enables the development of more effective control strategies. In this work, we examine the influence of the fractional order υ with the Caputo fractional difference on an economic market map. The primary contribution is the comprehensive analysis of how both commensurate and incommensurate fractional orders affect the stability and complexity of the map. Numerical investigations, including phase portraits, largest Lyapunov exponents, and bifurcation analysis, reveal that the system undergoes a cascade of period-doubling bifurcations before transitioning into chaos. To further characterize the dynamics, complexity is evaluated using the 0–1 test and C0 complexity, both confirming chaotic behavior. Furthermore, two-dimensional control schemes are introduced and theoretically validated to both stabilize the chaotic economic market map and achieve synchronization with a combined response map. The theoretical and numerical results are validated through MATLAB 2016 simulations.