Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method

Evuiroro Edirin Judith; Ojarikre Henritta Ify

doi:doi:10.11648/j.acm.20211004.11

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Chaos and Bifurcation of Control Feedback System Using Variational Iteration Method

Evuiroro Edirin Judith, Ojarikre Henritta Ify

Published in Applied and Computational Mathematics (Volume 10, Issue 4)

Received: 19 May 2021 Accepted: 15 June 2021 Published: 9 July 2021

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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.

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), 2024. Published by Science Publishing Group

Keywords

Chebyshev Polynomial, Lagrange Multiplier, Variational Iterative Method, Chaos and Bifurcation

References

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[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.
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[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.
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[13]	Mendelbrot, B. Fractals, Chance and Dimensions. San Francisco: W. H. Freeman & CO, 1977.
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[18]	Strogatz, S. H, Nonlinear Dynamics and Chaos. Addison-Weseley Publishing Company. United State Of America. 1994.

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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

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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

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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

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@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}
}

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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 -

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Author Information

Evuiroro Edirin Judith

Department of Mathematics, Delta State University, Abraka, Nigeria
Ojarikre Henritta Ify

Department of Mathematics, Delta State University, Abraka, Nigeria

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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

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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

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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

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@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}
}

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