Yu-Lan Wang

This paper investigates a time-fractional reaction–diffusion system, and develops an efficient numerical method for the system. The fractional orders serve as key bifurcation parameters, capable of inducing instabilities and rich pattern structures. We analyze and and simulate the stability conditions, bifurcation behavior, and pattern formation mechanisms of the system, and explore how variations in the fractional orders modulate the underlying dynamics.

This paper investigates the chaotic and pattern dynamics of the time-fractional Ginzburg–Landau equation. First, we propose a high-precision numerical method that combines finite difference schemes with an improved Grünwald–Letnikov fractional derivative approximation. Subsequently, the effectiveness of the proposed method is validated through systematic comparisons with classical numerical approaches. Finally, numerical simulations based on this method reveal rich dynamical phenomena in the fractional Ginzburg–Landau equation: the system exhibits complex behaviors including chaotic oscillations and novel two- and three-dimensional pattern structures. This study not only advances the theoretical development of numerical solutions for fractional GLE but also provides a reliable computational tool for deeper understanding of its complex dynamical mechanisms.

This paper focuses on the dynamical behavior of the fractional-order Hastings–Powell (HP) model. First, a high-precision numerical method is given and the limitations of the method are analyzed. Then, the stability of the equilibrium point of the model is analyzed, numerical simulations of the model with different orders and initial values are carried out, and a comparison is made between the high-precision numerical method and other methods. Finally, the Turing instability analysis and the weak nonlinear analysis of the HP model with diffusion terms are carried out and numerical simulations are performed in a nine-point difference format. The consistency between the numerical results and the numerical simulations is also verified.

In this paper, we investigated the nonlinear dynamics and pattern formation of a fractional-order three-variable Oregonator model. We first performed a linear stability analysis of the model without diffusion, deriving equilibrium points and Jacobian eigenvalues, and verified Matignon's stability conditions. A high-precision numerical scheme was developed, and simulations revealed that even tiny variations in fractional order produce significant changes in long-term trajectories. For the reaction-diffusion model, we analyzed Turing instability under integer-order diffusion and derived the critical wave-number conditions via Routh-Hurwitz criteria. Weakly nonlinear analysis near the Turing threshold yielded coupled amplitude equations whose coefficients predicted stripe, hexagon, and mixed patterns. Extensive two-dimensional numerical experiments confirmed the theoretical predictions: Depending on diffusion coefficients and other parameters, the model evolved into bullseye, spiral, labyrinthine, or spot-stripe mixtures.

It is always a difficult and hot topic to effectively explore the dynamic behavior of phytoplankton-zooplankton models. Thus, we investigated dynamic properties and a numerical solution of a fractional-order phytoplankton-zooplankton ecological model (PZEM) that incorporates the effects of toxic substances and additional food transmission in the environment. First, stability, Turing instability, Hopf bifurcation, and weakly nonlinear analysis were analyzed for the PZEM. Second, a new high-precision numerical method was developed for the fractional PZEM without diffusion terms. We compared the method with other methods to determine the effectiveness of the present method. A discretization method was established for the PZEM with diffusion term. Finally, numerical simulation verified the feasibility of the theory. Numerical simulations showed the chaotic attractor and some novel pattern dynamical behaviors of the PZEM.

Phytoplankton-zooplankton ecological models (PZEM) are crucial for comprehending the dynamics and interactions within marine ecosystems. Leveraging the memory effect of fractional-order derivatives , the fractional-order PZEM can more effectively capture biological significance. This paper investigates a PZEM that incorporates the effects of toxic substances and additional food transmission in the environment. Initially, we examine the stability of the fractional-order diffusionless model solution and the Hopf bifurcation with respect to the fractional order parameter. Subsequently, using weak nonlinear analysis, we obtained the amplitude equation close to the Turing instability threshold, which helped us classify different modal selections. Finally, an Euler dispersion scheme is constructed for the model, which effectively reveals the mixed pattern dynamic behavior observed in numerical simulations. Our findings uncover new dynamical behaviors of the model, contributing to a deeper understanding of PZEM in biological contexts. Mathematics Subject Classification: 92D40; 65M06; 34C23; 35B20; 35K55; 91B55.

Effective exploration of the pattern dynamic behaviors of reaction–diffusion models is a popular but difficult topic. The Schnakenberg model is a famous reaction–diffusion system that has been widely used in many fields, such as physics, chemistry, and biology. Herein, we explore the stability, Turing instability, and weakly non-linear analysis of the Schnakenberg model; further, the pattern dynamics of the fractional-in-space Schnakenberg model was simulated numerically based on the Fourier spectral method. The patterns under different parameters, initial conditions, and perturbations are shown, including the target, bar, and dot patterns. It was found that the pattern not only splits and spreads from the bar to spot pattern but also forms a bar pattern from the broken connections of the dot pattern. The effects of the fractional Laplacian operator on the pattern are also shown. In most cases, the diffusion rate of the fractional model was higher than that of the integer model. By comparing with different methods in literature, it was found that the simulated patterns were consistent with the results obtained with other numerical methods in literature, indicating that the Fourier spectral method can be used to effectively explore the dynamic behaviors of the fractional Schnakenberg model. Some novel pattern dynamics behaviors of the fractional-in-space Schnakenberg model are also demonstrated.

The two-mass nonlinear vocal cord vibration system (VCVS) serves as a mechanical representation of the fundamental vocalization process. Traditional models of the VCVS, which are based on integer-order dynamics, often overlook the impact of memory effects. To address this limitation and enhance the accuracy of simulations, this study incorporates the memory effects of vocal cord vibrations by integrating the Grunwald–Letnikov fractional derivative into the two-mass nonlinear VCVS framework. Initially, a high-precision computational scheme is formulated for the two-mass nonlinear fractional-order VCVS. Subsequently, the model undergoes a comprehensive series of numerical simulations to investigate its dynamic characteristics. The findings reveal that the dynamics of the fractional-order VCVS exhibit a significantly higher complexity compared to the conventional integer-order models, with the emergence of novel chaotic behaviors that were previously unobserved.

The establishment of a financial system should not only consider the current situation, but also need to refer to the past. Due to the memory of the fractional derivative, a fractional-order system can more effectively describe the historical significance of the financial system. Most scholars use the prediction–correction scheme to study fractional-order systems. This paper provides a higher-precision numerical method for the financial system, which more effectively simulate the system. Based on the definition of the Grünwald–Letnikov fractional derivative, the integer-order system with nonconstant demand elasticity is extended to the fractional-order setting, and its dynamic behavior is studied, with some novel chaotic attractors found. The research results are helpful for improving the understanding of the financial system and the financial market and for predicting financial risks.