Extension of a chaotic supply chain model by incorporating distributor inventory
This article presents a novel four-dimensional chaotic supply-chain model by extending the three-dimensional framework through the inclusion of a dynamic inventory variable with decay effects. The proposed extension provides a more realistic description of distributor behaviour and its impact on production and demand oscillations in complex supply-chain systems. A qualitative analysis of the model is performed through nullcline geometry, invariant regions, equilibrium analysis, and local stability via the Jacobian matrix. The system is shown to possess multiple equilibria, and explicit expressions for the non-trivial equilibrium points are derived. The Lyapunov spectrum is computed using the Wolf QR algorithm to investigate chaotic dynamics. Parameter variations involving a1, a2, a3, a6, a7, a8, and the inventory decay parameter γ reveal transitions between stable, periodic, and chaotic states. The corresponding Lyapunov exponent diagrams confirm the presence of sustained chaos and multistability through positive maximum exponents. In addition, a synchronization and control framework is developed using a driving system and a controlled response system. Nonlinear cancellation and linear feedback controllers are designed so that the synchronization error dynamics reduce to a decoupled linear system with exponential convergence. Numerical simulations confirm the effectiveness of the proposed synchronization scheme. Overall, the proposed model enhances the theoretical understanding of chaotic supply-chain dynamics by incorporating inventory decay effects, analyzing equilibrium structures, characterizing chaos through Lyapunov spectra, and developing an effective synchronization strategy. The results provide useful insights into the stabilization and control of complex supply-chain systems.
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