Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer †
Abstract
:1. Introduction
2. The Proposed Dynamical System
2.1. System Description
- The system has only five terms. Till now only few chaotic 3D dynamical systems with five terms have been reported in literature. Most of them belong to the class of jerk systems and they have been introduced by Professor Sprott and his colleagues [47,48]. However, there are few others, which are written in the general 3D dynamical system’s form [49,50,51,52,53]. In 1997, Fu and Heidel rigorously proved that there can be no simpler system with a quadratic nonlinearity [54].
- By solvingIt is easy to verify that (1) has a line of equilibrium points . Therefore, system’s attractor is hidden.
- The amplitude is easily controllable.
- The system is symmetric through the transformation .This symmetry is observed in Figure 3, where two symmetric system’s periodic attractors, in the three different planes, for and initial conditions , with red color, and , with blue color, has been captured.
2.2. Dynamical Analysis
3. Realization of the System
4. Application to Secure Communications
- (A.1)
- , where n is the number of columns of E. So, by considering the form of E, one can see that this is equivalent to D having full column rank.
- (A.2)
- The nonlinear part satisfies the Lipschitz property, that is, there exists a positive scalar such that
- Compute the matrix R, following the procedure given in ([67] Appendix A.)
- Compute the matrices , that correspond to the system
5. Simulation Results
6. Conclusions and Future Aspects
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Moysis, L.; Volos, C.; Pham, V.-T.; Goudos, S.; Stouboulos, I.; Gupta, M.K.; Mishra, V.K. Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer. Technologies 2019, 7, 76. https://doi.org/10.3390/technologies7040076
Moysis L, Volos C, Pham V-T, Goudos S, Stouboulos I, Gupta MK, Mishra VK. Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer. Technologies. 2019; 7(4):76. https://doi.org/10.3390/technologies7040076
Chicago/Turabian StyleMoysis, Lazaros, Christos Volos, Viet-Thanh Pham, Sotirios Goudos, Ioannis Stouboulos, Mahendra Kumar Gupta, and Vikas Kumar Mishra. 2019. "Analysis of a Chaotic System with Line Equilibrium and Its Application to Secure Communications Using a Descriptor Observer" Technologies 7, no. 4: 76. https://doi.org/10.3390/technologies7040076