Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization

Nestor Tsafack; Jacques Kengne

doi:doi:10.11648/j.wjap.20190402.12

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Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization

Nestor Tsafack, Jacques Kengne

Published in World Journal of Applied Physics (Volume 4, Issue 2)

Received: 25 July 2019 Accepted: 13 August 2019 Published: 11 September 2019

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Abstract

Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.

Published in	World Journal of Applied Physics (Volume 4, Issue 2) This article belongs to the Special Issue Symmetry and Multi-Stability in Simple Chaotic Systems and Circuits
DOI	10.11648/j.wjap.20190402.12
Page(s)	24-34
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

Chua’s Circuit System, Adjustable Symmetry, Coexisting Bifurcations, Coexisting Attractors, Pspice Circuit Simulations

References

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Nestor Tsafack, Jacques Kengne. (2019). Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization. World Journal of Applied Physics, 4(2), 24-34. https://doi.org/10.11648/j.wjap.20190402.12

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ACS Style

Nestor Tsafack; Jacques Kengne. Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization. World J. Appl. Phys. 2019, 4(2), 24-34. doi: 10.11648/j.wjap.20190402.12

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AMA Style

Nestor Tsafack, Jacques Kengne. Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization. World J Appl Phys. 2019;4(2):24-34. doi: 10.11648/j.wjap.20190402.12

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@article{10.11648/j.wjap.20190402.12,
  author = {Nestor Tsafack and Jacques Kengne},
  title = {Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization},
  journal = {World Journal of Applied Physics},
  volume = {4},
  number = {2},
  pages = {24-34},
  doi = {10.11648/j.wjap.20190402.12},
  url = {https://doi.org/10.11648/j.wjap.20190402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20190402.12},
  abstract = {Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.},
 year = {2019}
}

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TY  - JOUR
T1  - Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization
AU  - Nestor Tsafack
AU  - Jacques Kengne
Y1  - 2019/09/11
PY  - 2019
N1  - https://doi.org/10.11648/j.wjap.20190402.12
DO  - 10.11648/j.wjap.20190402.12
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 24
EP  - 34
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20190402.12
AB  - Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.
VL  - 4
IS  - 2
ER  -

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Author Information

Nestor Tsafack

Electrical Engineering Department, Institute of Technology Fotso Victor, University of Dschang, Bandjoun, Cameroon; Department of Physics, Faculty of Sciences, University of Dschang, Dschang, Cameroon
Jacques Kengne

Electrical Engineering Department, Institute of Technology Fotso Victor, University of Dschang, Bandjoun, Cameroon

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APA Style

Nestor Tsafack, Jacques Kengne. (2019). Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization. World Journal of Applied Physics, 4(2), 24-34. https://doi.org/10.11648/j.wjap.20190402.12

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ACS Style

Nestor Tsafack; Jacques Kengne. Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization. World J. Appl. Phys. 2019, 4(2), 24-34. doi: 10.11648/j.wjap.20190402.12

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AMA Style

Nestor Tsafack, Jacques Kengne. Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization. World J Appl Phys. 2019;4(2):24-34. doi: 10.11648/j.wjap.20190402.12

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@article{10.11648/j.wjap.20190402.12,
  author = {Nestor Tsafack and Jacques Kengne},
  title = {Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization},
  journal = {World Journal of Applied Physics},
  volume = {4},
  number = {2},
  pages = {24-34},
  doi = {10.11648/j.wjap.20190402.12},
  url = {https://doi.org/10.11648/j.wjap.20190402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wjap.20190402.12},
  abstract = {Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization
AU  - Nestor Tsafack
AU  - Jacques Kengne
Y1  - 2019/09/11
PY  - 2019
N1  - https://doi.org/10.11648/j.wjap.20190402.12
DO  - 10.11648/j.wjap.20190402.12
T2  - World Journal of Applied Physics
JF  - World Journal of Applied Physics
JO  - World Journal of Applied Physics
SP  - 24
EP  - 34
PB  - Science Publishing Group
SN  - 2637-6008
UR  - https://doi.org/10.11648/j.wjap.20190402.12
AB  - Background: Since the invention of Chua’s circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua’s circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system’s coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua’s circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.
VL  - 4
IS  - 2
ER  -

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