2021/06/18 by Hatim Khudhair, Hatim K. Khudhair, Yanzhi Zhang +4 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Theoretical and Computational Physics #math.DS #nlin.PS
paper · pdf · doi:10.48550/arxiv.2106.10263
18 pages, 13 figures
arxiv created 2021/06/18 · openalex publication_date 2021/06/18 · arxiv updated 2021/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Pattern formation in the classical and fractional Schnakenberg equations is studied to understand the nonlocal effects of anomalous diffusion. Starting with linear stability analysis, we find that if the activator and inhibitor have the same diffusion power, the Turing instability space depends only on the ratio of diffusion coefficients κ1/κ2. However, the smaller diffusive powers might introduce larger unstable wave numbers with wider band, implying that the patterns may be more chaotic in the fractional cases. We then apply a weakly nonlinear analysis to predict the parameter regimes for spot, stripe, and mixed patterns in the Turing space. Our numerical simulations confirm the analytical results and demonstrate the differences of normal and anomalous diffusion on pattern formation. We find that in the presence of superdiffusion the patterns exhibit multiscale structures. The smaller the diffusion powers, the larger the unstable wave numbers and the smaller the pattern scales.