A Reliable Numerical Algorithm for Stabilizing of the 2-Dimensional Logistic Hyperchaotic Trajectory
DOI:
https://doi.org/10.23851/mjs.v33i1.1048Keywords:
Chaos theory, Logistic map, Bifurcation, Lyapunov exponentAbstract
The dynamical system is the concept used to describe the behavior of several phenomena in our daily life. It comes in two types; linear and nonlinear. Two essential properties characterize the latter, stability and chaos, which in turn are classified into two categories, continuous and discrete, for the models that exhibiting chaotic behavior, which sometimes needs to be stabilized and synchronized. There are various approaches for such a purpose. In this work, the chaotic behavior of the 2D-logistic map is stabilized without adding any control parameters. This approach is considered efficient for models whose analytic solutions are challenging to find. Moreover, the modulus for the Jacobian matrix eigenvalues is greater than unity. Finally, the feasibility and effectiveness of this stabilizing method are demonstrated through some Numerical analysis.
References
K.Q. Al-Jubouri, R.M. Hussien, and N.M. Al-Saidi. Modeling of an Eco-Epidemiological system Involving Various Epidemic Diseases with Optimal Harvesting. Eurasian Journal of Mathematical and Computer Applications, 8(2),(2020).
CrossRef DOI: https://doi.org/10.32523/2306-6172-2020-8-2-4-27
K.Q. Al-Jubouri, R. M. Hussien, and N. M. Al-Saidi. The effect of harvesting policy on an eco-epidemiological model. In AIP Conference Proceedings, 2183(1), p. 070007, AIP Publishing, (2019).
CrossRef DOI: https://doi.org/10.1063/1.5136169
H. Natiq,; M.N. Al-Saidi,; M.R.M. Said. Complexity and dynamic characteristics of a new discrete-time hyperchaotic model. In Proceedings of the 2017 Second Al-Sadiq International Conference on Multidisciplinary in IT and Communication Science and Applications (AIC-MITCSA), Baghdad, Iraq, 30-31 December 2017.
CrossRef DOI: https://doi.org/10.1109/AIC-MITCSA.2017.8723012
G. Joydev, S. Banshidhar, P.Swarup. Prey-predator dynamics with prey refuge providing additional food to predator. Chaos Solitons Fractals 96, 110-119 (2017).
CrossRef DOI: https://doi.org/10.1016/j.chaos.2017.01.010
Hou, A, Guo, S: Stability and bifurcation in a state-dependent delayed predator-prey system. Int. J. Bifurc. Chaos 26(4), (2016).
CrossRef DOI: https://doi.org/10.1142/S0218127416500607
Chow, S-N., and Jack K. Hale. Methods of bifurcation theory. Vol. 251. Springer Science & Business Media, 2012.
S. N. Elaydi. Discrete Chaos: with applications in science and engineering. Chapman&Hall/CRC. (2008).
CrossRef DOI: https://doi.org/10.1201/9781420011043
E. Skinner, James, et al. "Application of chaos theory to biology and medicine." Integrative Physiological and Behavioral Science 27.1 (1992): 39-53. DOI: https://doi.org/10.1007/BF02691091
Chen, Guanrong. Controlling chaos and bifurcations in engineering systems. CRC press, 1999.
VT Pham, DS Ali, NMG Al-Saidi, K Rajagopal, FE Alsaadi, S Jafari. A novel mega-stable chaotic circuit. Radioengineering Journal. 29(1), 2020.140-146.
CrossRef DOI: https://doi.org/10.13164/re.2020.0140
A. Brock, William "Pathways to randomness in the economy: emergent nonlinearity and chaos in economics and finance." Estudios Economicos (1993): 3-55.
NMG Al-Saidi, D Younus, H Natiq, MRK Ariffin, MA Asbullah, Z Mahad. A New Hyperchaotic Map for a Secure Communication Scheme with an Experimental Realization. Symmetry. 12(11) 2020, 1881.
CrossRef DOI: https://doi.org/10.3390/sym12111881
MS Fadhil, AK Farhan, MdN Fadhil, NMG Al-Saidi. A New Lightweight AES Using a Combination of Chaotic Systems. 2020 IEEE first conference on Information Technology To Enhance e-learning and Other Application (IT-ELA). 2020. 82-88.
CrossRef DOI: https://doi.org/10.1109/IT-ELA50150.2020.9253099
AK Farhan, NMG Al-Saidi, AT Maolood, F Nazarimehr, I Hussain. Entropy analysis and image encryption application based on a new chaotic system crossing a cylinder. Entropy 21 (10), 958.
CrossRef DOI: https://doi.org/10.3390/e21100958
L. Yang, Z. R. Liu and J. M. Mao, "Controlling Hyper- chaos," Physical Review Letters, Vol. 84, No. 1, 2000, pp. 67-70. DOI: https://doi.org/10.1103/PhysRevLett.84.67
[S. L. Bu, S. Q. Wang and H. Q. Ye, Stabilizing Unstable Discrete Systems, Physical Review E, Vol. 64, 2001. DOI: https://doi.org/10.1103/PhysRevE.64.046209
Seval Işık. A study of stability and bifurcation analysis in discrete-time predator-prey system involving the Allee effect. International Journal of Biomathematics Vol. 12, No. 1 (2019).
CrossRef DOI: https://doi.org/10.1142/S1793524519500116
Xin Li, Suping Qian Controlling Unstable Discrete Chaos and Hyperchaos Systems. Applied Mathematics, 2013, 4, 1-6
CrossRef DOI: https://doi.org/10.4236/am.2013.411A2001
X. Li and Y. Chen, "Stabilizing of Two-Dimensional Discrete Lorenz Chaotic System and Three-Dimensional Discrete Rössler Hyperchaotic System," Chinese Physics Letters, Vol. 26, No. 9, 2009.
CrossRef DOI: https://doi.org/10.1088/0256-307X/26/9/090503
E. Ott, C. Grebogi and J. A. York, "Controlling Chaos," Physical Review Letters, Vol. 64, No. 11, 1990, pp. 1196- 1199. DOI: https://doi.org/10.1103/PhysRevLett.64.1196
D. Liu, L. Liu and Y. Yang, "H∞ Control of Discrete- Time Singularly Perturbed Systems via Static Output Feedback," Abstract and Applied Analysis, Vol. 2013, 2013.
CrossRef DOI: https://doi.org/10.1155/2013/528695
Gardini, Laura, et al. "A double logistic map." International Journal of Bifurcation and Chaos 4.01 (1994): 145-176.
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