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Abstract

The Fitzhugh-Nagumo model (FN model), which is successfully employed in modeling the function of the so-called membrane potential, exhibits various formations in neuronal networks and rich complex dynamics. This work deals with the problem of control and synchronization of the FN reaction-diffusion model. The proposed control law in this study is designed to be uni-dimensional and linear law for the purpose of reducing the cost of implementation. In order to analytically prove this assertion, Lyapunov’s second method is utilized and illustrated numerically in one- and/or two-spatial dimensions.
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[16] F. Mesdoui, N. Shawagfeh, and A. Ouannas: Global synchronization of fractional-order and integer-order N component reaction diffusion systems: Application to biochemical models. Mathematical Methods in the Applied Sciences, 44(1), (2021), 1003–1012, DOI: 10.1002/mma.6807.
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[21] A. Ouannas, M. Abdelli, Z. Odibat, X. Wang, V.-T. Pham, G. Grassi, and A. Alsaedi: Synchronization Control in Reaction-Diffusion Systems: Application to Lengyel-Epstein System. Complexity, (2019), Article ID 2832781, DOI: 10.1155/2019/2832781.
[22] A. Ouannas, Z. Odibat, N. Shawagfeh, A. Alsaedi, and B. Ahmad: Universal chaos synchronization control laws for general quadratic discrete systems. Applied Mathematical Modelling, 45 (2017), 636–641, DOI: 10.1016/j.apm.2017.01.012.
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Authors and Affiliations

Adel Ouannas
1
Fatiha Mesdoui
2
Shaher Momani
2 3
Iqbal Batiha
4 3
Giuseppe Grassi
5

  1. Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
  2. Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
  3. Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
  4. Department of Mathematics, Faculty of Science and Technology, Irbid National University, 2600 Irbid, Jordan
  5. Dipartimento Ingegneria Innovazione, Universitadel Salento, 73100 Lecce, Italy
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Abstract

Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo h-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.
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[18] F. Hadjabi, A. Ouannas,N. Shawagfeh, A.A. Khennaoui, and G. Grassi: On two-dimensional fractional chaotic maps with symmetries. Symmetry, 12(5), (2020), DOI: 10.3390/sym12050756.
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Authors and Affiliations

A. Othman Almatroud
1
Adel Ouannas
2
Giuseppe Grassi
3
Iqbal M. Batiha
4
Ahlem Gasri
5
M. Mossa Al-Sawalha
1

  1. Department of Mathematics, Faculty of Science, University of Ha'il, Ha'il 81451, Saudi Arabia
  2. Department of Mathematics and Computer Science, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria
  3. Dipartimento Ingegneria Innovazione, Universita del Salento, 73100 Lecce, Italy
  4. Department of Mathematics, Faculty of Science and Information Technology, Irbid National University, Irbid, Jordan and Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
  5. Department of Mathematics, University of Larbi Tebessi, Tebessa 12002, Algeria

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