Dynamics and Efficient Numerical Simulation of a Fractional-Order T System

In this paper, we propose and numerically investigate a fractional T system. As a fractional generalization of the classical T model, the fractional order serves as a memory parameter governing the system dynamics. By employing the fractional stability criterion, the local stability of the equilibrium points is analyzed, and the existence of Hopf bifurcation is characterized. To efficiently simulate the long-time dynamics induced by fractional memory, a linear semi-implicit numerical scheme accelerated by a sum-of-exponentials approximation of the Caputo derivative is developed. The proposed scheme is shown to be stable and enables a significant reduction in computational cost compared with classical L1 and Grünwald–Letnikov methods. Numerical experiments, including time series, phase portraits, Lyapunov exponent computations, and bifurcation diagrams, demonstrate that varying the fractional order leads to transitions among stable, periodic, and chaotic regimes. In particular, pronounced transient dynamics are observed as the fractional order approaches its critical value, highlighting the memory-induced effects inherent in fractional-order systems.