Global Existence and Fixed-Time Synchronization of a Hyperchaotic Financial System Governed by Semi-Linear Parabolic Partial Differential Equations Equipped with the Homogeneous Neumann Boundary Condition

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Chaotic nonlinear dynamical systems, in which the generated time series exhibit high entropy values, have been extensively used and play essential roles in tracking accurately the complex fluctuations of the real-world financial markets. We are concerned with a system of semi-linear parabolic partial differential equations supplemented by the homogeneous Neumann boundary condition, which governs a financial system comprising the labor force, the stock, the money, and the production sub-blocks distributed in a certain line segment or planar region. The system derived by removing the terms involved with partial derivatives with respect to space variables from our concerned system was demonstrated to be hyperchaotic. We firstly prove, via Galerkin's method and establishing a priori inequalities, that the initial-boundary value problem for the concerned partial differential equations is globally well posed in Hadamard's sense. Secondly, we design controls for the response system to our concerned financial system, prove under some additional conditions that our concerned system and its controlled response system achieve drive-response fixed-time synchronization, and provide an estimate on the settling time. Several modified energy functionals (i.e., Lyapunov functionals) are constructed to demonstrate the global well-posedness and the fixed-time synchronizability. Finally, we perform several numerical simulations to validate our synchronization theoretical results.