Chaos theory discusses the behavior of some complex systems which are sensitive to initial condition. It entails some interesting properties such as space-filling, sensitivity to initial conditions, control synchronization and dynamics which can be accessed using different control methods. This paper considers a non-linear feedback control system of underlying symmetries of chaos and bifurcation with periodic equations. Bifurcation theory plays a very vital role in the analysis of Chaos dynamics. Therefore, bifurcation process of chaotic systems involving Lyapunov exponents are studied. The work describes nonlinear systems where small changes produce notable change in the space phase with all the possible states corresponding to a unique point. A Variational Iteration Method (VIM) is adopted to determine the solution stability and bifurcation paths of the dynamic system. Thus. in the paper, state problems of the system are decomposed using Lagrange multiplier to obtain the Adomian polynomials. The polynomials generated in turn minimize the problem by providing an approximate solution that is very close to the analytic solution. A numerical illustration has been presented of a nonlinear coupled equation using VIM. Illustrations showing the graph of the phase space structure, the paths displayed in motion and the region of stability of the numerical scheme was obtained. The result shows that the chaotic nonlinear system is stable.
| Published in | Applied and Computational Mathematics (Volume 10, Issue 4) |
| DOI | 10.11648/j.acm.20211004.11 |
| Page(s) | 86-90 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2021. Published by Science Publishing Group |
Chebyshev Polynomial, Lagrange Multiplier, Variational Iterative Method, Chaos and Bifurcation
| [1] | Abbasbandy, S and Shivanian, E. Application of VIM for System of nonlinearVolterra’sIntegro-Differential Equations. Mathematical and ComputationalApplications. 14 (2), 147-158, 2009. |
| [2] | Anjum, N. and He, J. H. Laplace Transform: Making Variational Iteration Method Easier. Elsevier-Journal of Applied Mathematics Letters, 16 (1): 134-138, 2019 http://doi.org/10.1016/j.aml.2019.01.016. |
| [3] | Briggs, J. & Peat, F. D. (2000) Seven Life Lesson of Chaos: Spiritual Wisdom from the Scienceof Change. New York. |
| [4] | Butler, A. A Methodological Approach to Chaos. Federal Reserve Bank of St. Louis 72 (13), 36-48. 1990. |
| [5] | Gleick, J. Chaos: Making a New Science. New York: Viking Penguin, 1987. |
| [6] | Hayles, N. K. Chaos Bound: Orderly Disorder in Contemporary Literature and Science. Ithaca: Cornell University Press, 1990. |
| [7] | Kellert, S. H. In The Wake of Chaos Unpredictable Order in Dynamical Systems. Chicago: University of Chicago Press, 1993. |
| [8] | Levy, D. L. Chaos Theory and Strategy: Theory. Application and Managerial Implications. Strategic Management Journal, 15: 167-178, 1994. |
| [9] | Lorenz, E. N, Deterministic non periodic flow. Journal On Atmospheric Science, 20, 130–141, 1963. |
| [10] | Lorenz, E. ‘Predictability: Does the flap of a butterfly swings in Brazil set off a tornado in Texas?’ Paper presented at the meeting of the American Association for the Advancement of Science, Washington DC, 1972. |
| [11] | Mamadu, E. J and Njoseh, I. N. Variational Iteration Method for the Approximate Solution of Nonlinear Burger’s Equation, Nigerian Journal of Basic and Applied Sciences, 24 (1), 70-75, 2016. |
| [12] | Mammeri, M. Symmetry and Periodic-Chaos in 3-DSinusoid Discrete Map, Bulletin of Mathematical Analysis and Applications, 9 (1), 1-8, 2017. |
| [13] | Mendelbrot, B. Fractals, Chance and Dimensions. San Francisco: W. H. Freeman & CO, 1977. |
| [14] | Ott. E, Grebogi. C, Yorke. J. A (1990), Controlling chaos. Physical Review Letters 64: 1196–1199. |
| [15] | Porush, D. Fictions as Dissipative Structures. Prigogines Theory and Postmodernism Road show in Chaos and Order. Complex Dynamics in Literature and Science. Chicago: The University of Chicago Press, 1991. |
| [16] | Pyragas K (1992), Continuous control of chaos, by self-controlling feedback, Physics Letters 170: 421–428, 1992. |
| [17] | Radzicki, M. J. Institutional Dynamics, Deterministic Chaos and Self Organizing System. Journal of Economic Issues, 24 (1) 57-102. 1990. |
| [18] | Strogatz, S. H, Nonlinear Dynamics and Chaos. Addison-Weseley Publishing Company. United State Of America. 1994. |
APA Style
Evuiroro Edirin Judith, Ojarikre Henritta Ify. (2021). Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method. Applied and Computational Mathematics, 10(4), 86-90. https://doi.org/10.11648/j.acm.20211004.11
ACS Style
Evuiroro Edirin Judith; Ojarikre Henritta Ify. Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method. Appl. Comput. Math. 2021, 10(4), 86-90. doi: 10.11648/j.acm.20211004.11
AMA Style
Evuiroro Edirin Judith, Ojarikre Henritta Ify. Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method. Appl Comput Math. 2021;10(4):86-90. doi: 10.11648/j.acm.20211004.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.acm.20211004.11,
author = {Evuiroro Edirin Judith and Ojarikre Henritta Ify},
title = {Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method},
journal = {Applied and Computational Mathematics},
volume = {10},
number = {4},
pages = {86-90},
doi = {10.11648/j.acm.20211004.11},
url = {https://doi.org/10.11648/j.acm.20211004.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20211004.11},
abstract = {Chaos theory discusses the behavior of some complex systems which are sensitive to initial condition. It entails some interesting properties such as space-filling, sensitivity to initial conditions, control synchronization and dynamics which can be accessed using different control methods. This paper considers a non-linear feedback control system of underlying symmetries of chaos and bifurcation with periodic equations. Bifurcation theory plays a very vital role in the analysis of Chaos dynamics. Therefore, bifurcation process of chaotic systems involving Lyapunov exponents are studied. The work describes nonlinear systems where small changes produce notable change in the space phase with all the possible states corresponding to a unique point. A Variational Iteration Method (VIM) is adopted to determine the solution stability and bifurcation paths of the dynamic system. Thus. in the paper, state problems of the system are decomposed using Lagrange multiplier to obtain the Adomian polynomials. The polynomials generated in turn minimize the problem by providing an approximate solution that is very close to the analytic solution. A numerical illustration has been presented of a nonlinear coupled equation using VIM. Illustrations showing the graph of the phase space structure, the paths displayed in motion and the region of stability of the numerical scheme was obtained. The result shows that the chaotic nonlinear system is stable.},
year = {2021}
}
TY - JOUR T1 - Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method AU - Evuiroro Edirin Judith AU - Ojarikre Henritta Ify Y1 - 2021/07/09 PY - 2021 N1 - https://doi.org/10.11648/j.acm.20211004.11 DO - 10.11648/j.acm.20211004.11 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 86 EP - 90 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20211004.11 AB - Chaos theory discusses the behavior of some complex systems which are sensitive to initial condition. It entails some interesting properties such as space-filling, sensitivity to initial conditions, control synchronization and dynamics which can be accessed using different control methods. This paper considers a non-linear feedback control system of underlying symmetries of chaos and bifurcation with periodic equations. Bifurcation theory plays a very vital role in the analysis of Chaos dynamics. Therefore, bifurcation process of chaotic systems involving Lyapunov exponents are studied. The work describes nonlinear systems where small changes produce notable change in the space phase with all the possible states corresponding to a unique point. A Variational Iteration Method (VIM) is adopted to determine the solution stability and bifurcation paths of the dynamic system. Thus. in the paper, state problems of the system are decomposed using Lagrange multiplier to obtain the Adomian polynomials. The polynomials generated in turn minimize the problem by providing an approximate solution that is very close to the analytic solution. A numerical illustration has been presented of a nonlinear coupled equation using VIM. Illustrations showing the graph of the phase space structure, the paths displayed in motion and the region of stability of the numerical scheme was obtained. The result shows that the chaotic nonlinear system is stable. VL - 10 IS - 4 ER -
Email: