Existence, attractors, and fixed-/pre-assigned-time synchronization in a chaotic spatio-temporal financial model

Chaos is a fundamental feature of dynamical systems with complex nonlinear interactions. Understanding the mechanisms that generate chaos is essential for characterizing the long-term behavior of such systems. In this paper, we investigate a system of semilinear parabolic partial differential equations derived from financial markets, incorporating spatially distributed information on interest rates, investment demand, and price indices. First, by employing the Faedo–Galerkin method, the contraction mapping principle, and the regularity theory of parabolic partial differential equations, we establish the well-posedness, in the sense of Hadamard, of the initial–boundary value problem associated with the considered spatio-temporal financial system. Second, we obtain regularity results for the state trajectories and demonstrate that the concerned financial system possesses a global attractor, which characterizes its asymptotic dynamics and indicates the potential for chaotic behavior (existence of strange attractors) in the system. Third, we design control strategies for a response system corresponding to the concerned spatio-temporal financial system and prove that, under appropriate conditions, the drive and response systems achieve fixed- or preassigned-time synchronization, with explicit estimates for the settling time in the fixed-time case. Finally, numerical simulations confirm the chaotic behavior of the spatio-temporal financial system and validate the effectiveness of the proposed synchronization controls. The model provides a framework for analyzing and tracking spatio-temporal fluctuations in financial markets across spatially distributed regions, making the results both theoretically significant and practically applicable.