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Complex Turing patterns in chaotic dynamics of autocatalytic reactions with the Caputo fractional derivative

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Abstract

Many chemical systems exhibit a range of patterns, a noticeable and interesting class of numerical patterns that arise in autocatalytic reactions which changes with increasing spatial domains. In this paper, autocatalytic spatiotemporal patterns were demonstrated using the system of chemical species modeled with the time-fractional Caputo derivatives of subdiffusive orders. It is not new that a spectral algorithm with its entire nature is more accurate when compared with a finite-difference scheme to solve a range of integer and non-integer order time partial differential equations. This is because the Fourier spectral techniques have the upper hand on high-order spectral accuracy, and are computationally efficient. Hence, it is regarded as the best approach to existing lower-order methods for integrating the second-order partial derivatives in space. This motivates the present study to explore the usefulness of Fourier spectral methods in resolving and obtaining complex Turing patterns arising from nonlinear fractional autocatalytic reaction-diffusion problems in high dimensions. The autocatalysis model was examines for linear stability in an attempt to obtain the correct choice of parameters that are likely to lead to the formation of new complex turing-like patterns. Numerical experiments in the 2D lead to a striking range of patterns arising from catalytic reactions of fractional-order labyrinthine pattern-like structures. Analysis of pattern formation was also extended to 3D dynamics to obtain a new set of patterns like a star-, cyclic-, diamond-like, and the emergence of apple-shaped structures, which are greatly influenced by either the choice parameters that are involved or that of the initial conditions.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Federal University of Technology Akure, PMB 704, Akure, Ondo State, Nigeria

    Kolade M. Owolabi

  2. Department of Mathematics, Texas A &M University-Kingsville, Kingsville, TX, 78363-8202, USA

    Ravi P. Agarwal

  3. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 002, South Africa

    Edson Pindza

  4. Department of Mathematics and Informatics, Institute of Applied Analysis, Technische Universität Bergakademie Freiberg, 09599, Freiberg, Germany

    Swanhild Bernstein

  5. Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah, 21955, Saudi Arabia

    Mohamed S. Osman

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  1. Kolade M. Owolabi
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Owolabi, K.M., Agarwal, R.P., Pindza, E. et al. Complex Turing patterns in chaotic dynamics of autocatalytic reactions with the Caputo fractional derivative. Neural Comput & Applic 35, 11309–11335 (2023). https://doi.org/10.1007/s00521-023-08298-2

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