Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation

Almatroud, Othman Abdullah; Abu Hammad, Ma’mon; Dababneh, Amer; Diabi, Louiza; Ouannas, Adel; Khennaoui, Amina Aicha; Alshammari, Saleh

doi:10.3390/sym16081093

Open AccessArticle

Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation

by

Othman Abdullah Almatroud

¹,

Ma’mon Abu Hammad

²,

Amer Dababneh

²,

Louiza Diabi

^3,*,

Adel Ouannas

⁴,

Amina Aicha Khennaoui

⁵

and

Saleh Alshammari

¹

Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi Arabia

²

Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan

³

Laboratory of Dynamical Systems and Control, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria

⁴

Department of Mathematics and Computer Sciences, University of Larbi Ben M’hidi, Oum El Bouaghi 04000, Algeria

⁵

NTIC Faculty, University of Constantine 2, Constantine 25000, Algeria

^*

Author to whom correspondence should be addressed.

Symmetry 2024, 16(8), 1093; https://doi.org/10.3390/sym16081093

Submission received: 28 May 2024 / Revised: 21 June 2024 / Accepted: 28 June 2024 / Published: 22 August 2024

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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Abstract

This paper presents a new third-order symmetric difference equation transformed into a 3D discrete symmetric map. The nonlinear dynamics and symmetry of the proposed map are analyzed with two initial conditions for exploring the sensitivity of the map and highlighting the influence of the map parameters on its behaviors, thus comparing the findings. Moreover, the stability of the zero fixed point and symmetry are examined by theoretical analysis, and it is proved that the map generates diverse nonlinear traits comprising multistability, chaos, and hyperchaos, which is confirmed by phase attractors in 2D and 3D space, Lyapunov exponents (

L E_{s}

) analysis and bifurcation diagrams; also, 0-1 test and sample entropy (SampEn) are used to confirm the existence and measure the complexity of chaos. In addition, a nonlinear controller is introduced to stabilize the symmetry map and synchronize a duo of unified symmetry maps. Finally, numerical results are provided to illustrate the findings.

Keywords: symmetric map; difference equation; multistability; fixed point; chaos; control

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MDPI and ACS Style

Almatroud, O.A.; Abu Hammad, M.; Dababneh, A.; Diabi, L.; Ouannas, A.; Khennaoui, A.A.; Alshammari, S. Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation. Symmetry 2024, 16, 1093. https://doi.org/10.3390/sym16081093

AMA Style

Almatroud OA, Abu Hammad M, Dababneh A, Diabi L, Ouannas A, Khennaoui AA, Alshammari S. Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation. Symmetry. 2024; 16(8):1093. https://doi.org/10.3390/sym16081093

Chicago/Turabian Style

Almatroud, Othman Abdullah, Ma’mon Abu Hammad, Amer Dababneh, Louiza Diabi, Adel Ouannas, Amina Aicha Khennaoui, and Saleh Alshammari. 2024. "Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation" Symmetry 16, no. 8: 1093. https://doi.org/10.3390/sym16081093

APA Style

Almatroud, O. A., Abu Hammad, M., Dababneh, A., Diabi, L., Ouannas, A., Khennaoui, A. A., & Alshammari, S. (2024). Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation. Symmetry, 16(8), 1093. https://doi.org/10.3390/sym16081093

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation

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