Multistability, Chaos, and Synchronization in Novel Symmetric Difference Equation
Abstract
:1. Introduction
- The construction of a new third-order symmetric difference equation transformed into a 3D discrete symmetry map and its proprieties are explored.
- The diverse nonlinear traits generate the symmetry map, such as multistability, chaos, and hyperchaos, illustrated with bifurcation diagrams, Lyapunov exponents, and phase attractors.
- We use the 0-1 test and the sample entropy (SampEn) to measure and evaluate the complexity of chaos in the chaotic symmetric map.
- The stabilization and synchronization scheme of the chaotic symmetric map is realized based on the stability conditions of the discrete-time system.
2. The New Symmetric Map and Its Proprieties
- Case 1. Consider , , . The associated characteristic equation is defined by:
- Case 2. Let us assume that , , . The characteristic equation is given by:
3. Chaotic Analysis
- Case 1. In Figure 1a,b, the bifurcation diagram and the associated of the map (2) are plotted using MATLAB R2024a code. Considering to be the bifurcation parameter versus with , maps the parameters and IN1 and IN2. Subsequently, a small portion of the zone proximal to the bifurcation and the associated are plotted at , presented in detailed view as shown in Figure 1c,d. When , and , the map is chaotic with only three values being positive. Additionally, when , is hyperchaotic, where shows two positive and one displaying negative values. Moreover, the symmetric map is periodic at , and , where all values are negative. The map is quasi-periodic in , where one has zero values.
- Case 2. For versus in with , and IN1 and IN2. The symmetric map changes the dynamics between the hyperchoas, periodic, and quasi-periodic behaviors as illustrated in Figure 2a,b. For more details on , view Figure 2c,d. When , and , the map become hyperchaotic with two positive , and one negative value. It shows the periodic behavior when , and , where all three values are negative. The presence of one zero and the others is negatively expressed in the progression to quasi-periodic in the interval .
- Case 3. Similarly, take and versus in as shown in Figure 3a, Figure 3b as well as the detailed view in Figure 3c,d with . The chaotic behavior appears when , , where one of the is positive. Conversely, the periodic region exists when all the are negative at , and . When , the map has a quasi-periodic attractor characterized by one equal to zero. Thus, becomes chaotic again. According to these results, the characteristic of symmetry in the behaviors’ evolution becomes observable, reappearing over a wide field of the parameters’ systems. In contrast, symmetry breaking also occurs within a small scope.
4. Chaos Complexity
4.1. 0-1 Test
4.2. Sample Entropy
5. Chaos Control
6. Synchronization Scheme
7. Discussion and Perspectives
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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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
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 StyleAlmatroud, 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