This paper investigates the dynamic behavior of a fractional-order reaction-diffusion system for vegetation pattern formation, incorporating interactions between plant biomass, soil water, and salt concentration. The model utilizes Grünwald–Letnikov fractional derivatives to capture memory effects and anomalous diffusion. A novel high-order numerical scheme is developed, featuring a high-accuracy, short-memory time discretization with a nine-point finite difference method in space to enhance stability. Bifurcation analysis is performed to determine equilibrium stability and identify Turing instability thresholds. Numerical simulations illustrate the emergence of diverse vegetation patterns, highlighting how different fractional orders influence spatiotemporal dynamics. Overall, the proposed framework provides an effective computational tool for analyzing complex fractional ecological systems.

This paper presents a new numerical method for simulating the dynamic behavior of the fractional-in-time Gierer-Meinhardt reaction-diffusion model with periodic boundary conditions. A recursive algorithm for binomial coefficients is introduced, avoiding numerical instabilities associated with Gamma functions. High-precision polynomial expansions and the short-memory principle are employed to enhance efficiency and accuracy. Numerical simulations reveal diverse pattern formation.

This paper investigates the chaotic dynamics and control of fractional-order chaotic systems. A high-precision numerical method based on the Grünwald–Letnikov definition is developed to explore the system's dynamics. We also design linear feedback and adaptive control strategies to achieve chaotic synchronization and system stabilization. Numerical simulations validate the effectiveness of our methods, showing successful synchronization and control of the chaotic system.

This paper investigates the chaotic dynamics in a fractional-order vocal fold vibration (VCV) model based on the Grünwald-Letnikov fractional derivative (GLFD). Studying the characteristics of vocal fold vibration is of great significance for revealing its vibration mechanism, the etiology of abnormal vibrations, and natural speech synthesis. Traditional vocal fold vibration models are based on integer-order systems and are unable to describe the memory effects present in real physical systems. To overcome this limitation, this paper introduces fractional derivatives and develops a high-precision numerical method to simulate the fractional-order VCV model. By incorporating nonlinear elastic and damping forces, the model can more accurately describe the complex dynamic characteristics of vocal fold vibrations, including memory effects and non-locality. The numerical simulation results reveal novel chaotic behaviors in the fractional-order VCV model, which have not been observed in integer-order models. These findings provide new insights into the possible dynamic states of vocal fold vibrations and lay the foundation for further theoretical and experimental studies on the vocal cord vibration mechanism.