Dynamic Behavior of the Fractional-Order Ananthakrishna Model for Repeated Yielding

This paper introduces and analyzes a novel fractional-order Ananthakrishna model. The stability of its equilibrium points is first investigated using fractional-order stability criteria, particularly in regions where the corresponding integer-order model exhibits instability. A linear finite difference scheme is then developed, incorporating an accelerated L1 method for the fractional derivative. This enables a detailed exploration of the model’s dynamic behavior in both the time domain and phase plane. Numerical simulations, including Lyapunov exponents, bifurcation diagrams, phase and time diagrams, demonstrate that the fractional model exhibits stable and periodic behaviors across various fractional orders. Notably, as the fractional order approaches a critical threshold, the time required to reach stability increases significantly, highlighting complex stability-transition dynamics. The computational efficiency of the proposed scheme is also validated, showing linear CPU time scaling with respect to the number of time steps, compared to the nearly quadratic growth of the classical L1 and Grünwald-Letnikow schemes, making it more suitable for long-term simulations of complex fractional-order models. Finally, four types of stress-time curves are simulated based on the fractional Ananthakrishna model, corresponding to both stable and unstable domains, effectively capturing and interpreting experimentally observed repeated yielding phenomena.