PapersNo Access

Emulating “Chaos $+$ Chaos $=$ Order” in Chen’s Circuit of Fractional Order by Parameter Switching

Wallace K. S. Tang

Department of Electronic Engineering, City University of Hong Kong, Hong Kong

E-mail Address: eekstang@cityu.edu.hk

Search for more papers by this author

and

Marius-F. Danca

Department of Mathematics and Computer Science, Avram Iancu University of Cluj-Napoca, Romania

Romanian Institute of Science and Technology, Cluj-Napoca, Romania

E-mail Address: danca@rist.ro

Corresponding author

Search for more papers by this author

https://doi.org/10.1142/S0218127416500966Cited by:8 (Source: Crossref)

Abstract

In this paper, the effect of the parameter switching (PS) algorithm in a fractional order chaotic circuit is investigated both in simulation and experiment. The Chen system of fractional order is focused and realized in an electronic circuit. By designing a switching circuit, the PS algorithm is implemented and it is the first time, the paradoxical “Chaos $+$ Chaos $=$ Order” is presented in an electronic circuit. Both the simulation and experimental results confirm that the obtained attractor under switching approximates the attractor of the time-averaged model. Some important design issues for the circuitry realization of the PS scheme are pointed out. Finally, our work confirms the practical usage of PS algorithm in potential applications such as attractor synthesis and chaos control.

Keywords:

References

Agrawal, S. K., Srivastava, M. & Das, S. [2012] “ Synchronization of fractional order chaotic systems using active control method,” Chaos Solit. Fract. 45, 737–752. Crossref, Web of Science, Google Scholar
Ahmad, W. M. & Sprott, J. C. [2003] “ Chaos in fractional-order autonomous nonlinear systems,” Chaos Solit. Fract. 16, 339–351. Crossref, Web of Science, Google Scholar
Asheghan, M. M., Beheshti, M. T. H. & Tavazoei, M. S. [2011] “ Robust synchronization of perturbed Chen’s fractional-order chaotic systems,” Commun. Nonlin. Sci. 16, 1044–1051. Crossref, Web of Science, Google Scholar
Caputo, M. [1967] “ Linear models of dissipation whose $Q$ is almost frequency independent — II,” Geophys. J. Roy. Astron. Soc. 13, 529–539. Crossref, Web of Science, Google Scholar
Charef, A., Sun, H. H., Tsao, Y. Y. & Onaral, B. [1992] “ Fractal system as represented by singularity function,” IEEE Trans. Automat. Contr. 37, 1465–1470. Crossref, Web of Science, Google Scholar
Danca, M. F., Tang, W. K. S. & Chen, G. [2008] “ A switching scheme for synthesizing attractors of dissipative chaotic systems,” Appl. Math. Comput. 201, 650–667. Crossref, Web of Science, Google Scholar
Danca, M. F., Romera, M., Pastor, G. & Montoya, F. [2012] “ Finding attractors of continuous-time systems by parameter switching,” Nonlin. Dyn. 67, 2317–2342. Crossref, Web of Science, Google Scholar
Danca, M. F., Bourke, P. & Romera, M. [2013] “ Graphical exploration of the connectivity sets of alternated Julia sets; M, the set of disconnected alternated Julia sets,” Nonlin. Dyn. 73, 1155–1163. Crossref, Web of Science, Google Scholar
Danca, M. F., Fečkan, M. & Romera, M. [2014] “ Generalized form of Parrondo’s paradoxical game with applications to chaos control,” Int. J. Bifurcation and Chaos 24, 1450008-1–17. Link, Web of Science, Google Scholar
Diethelm, K., Ford, N. J. & Freed, A. D. [2002] “ A predictor–corrector approach for the numerical solution of fractional differential equations,” Nonlin. Dyn. 29, 3–22. Crossref, Web of Science, Google Scholar
Dzieliński, A., Sierociuk, D. & Sarwas, G. [2010] “ Some applications of fractional order calculus,” Bull. Pol. Acad. Sci. Tech. Sci. 58, 583–592. Web of Science, Google Scholar
Foias, C. & Jolly, M. S. [1995] “ On the numerical algebraic approximation of global attractors,” Nonlinearity 8, 295–319. Crossref, Web of Science, Google Scholar
Galvao, R. K. H., Hadjiloucas, S., Kienitz, K. H. & Paiva, H. M. [2013] “ Fractional order modeling of large three-dimensional RC networks,” IEEE Trans. Circuits Syst.-I 60, 624–637. Crossref, Web of Science, Google Scholar
Gammoudi, I. E. & Feki, M. [2013] “ Synchronization of integer order and fractional order Chua’s systems using robust observer,” Commun. Nonlin. Sci. 18, 625–638. Crossref, Web of Science, Google Scholar
Hegazi, A. S., Ahmed, E. & Matouk, A. E. [2013] “ On chaos control and synchronization of the commensurate fractional order Liu system,” Commun. Nonlin. Sci. 18, 1193–1202. Crossref, Web of Science, Google Scholar
Jia, H. Y., Chen, Z. Q. & Qi, G. Y. [2014] “ Chaotic characteristics analysis and circuit implementation for a fractional-order system,” IEEE Trans. Circuits Syst.-I 61, 845–853. Crossref, Web of Science, Google Scholar
Kaslik, E. & Sivasundaram, S. [2012] “ Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions,” Nonlin. Anal.: Real. World Appl. 13, 1489–1497. Crossref, Web of Science, Google Scholar
Kingni, S. T., Nana, B., Mbouna Ngueuteu, G. S., Woafo, P. & Danckaert, J. [2015] “ Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect,” Chaos Solit. Fract. 71, 29–40. Crossref, Web of Science, Google Scholar
Lin, T. C., Lee, T. Y. & Balas, V. E. [2011] “ Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems,” Chaos Solit. Fract. 44, 791–801. Crossref, Web of Science, Google Scholar
Liu, L., Liu, C. & Zhang, Y. [2009] “ Experimental verification of a four-dimensional Chua’s system and its fractional order chaotic attractors,” Int. J. Bifurcation and Chaos 19, 2473–2486. Link, Web of Science, Google Scholar
Mao, Y., Tang, W. K. S. & Danca, M. F. [2010] “ An averaging model for chaotic system with periodic time-varying parameter,” Appl. Math. Comput. 217, 355–362. Crossref, Web of Science, Google Scholar
Munir, Y. & Canny, D. [2013] “Selecting the right CMOS analog switch,” Maxim. Appl. Note 5299. Google Scholar
Oldham, K. B. & Spanier, J. [1974] The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order (Academic Press, NY). Google Scholar
Parreño, A., Roncero-Sánchez, P., del Toro García, X., Feliu, V. & Castllo, F. [2010] “ Analysis of the fractional dynamics of an ultracapacitor and its application to a buck-boost converter,” New Trends in Nanotechnology and Fractional Calculus Applications, eds. Baleanu, D., Güvenc, Z. B. & Tenreiro Machado, J. A. (Springer Science + Business Media B.V.), pp. 97–105. Crossref, Google Scholar
Podlubny, I. [2002] “ Geometric and physical interpretation of fractional integration and fractional differentiation,” Fract. Calc. Appl. Anal. 5, 367–386. Google Scholar
Shen, Y., Yang, S. & Sui, C. [2014] “ Analysis on limit cycle of fractional-order van der Pol oscillator,” Chaos Solit. Fract. 67, 94–102. Crossref, Web of Science, Google Scholar
Sun, K., Wang, X. & Sprott, J. C. [2010] “ Bifurcation and chaos in fractional-order simplified Lorenz system,” Int. J. Bifurcation and Chaos 20, 1209–1219. Link, Web of Science, Google Scholar
Tavazoei, M. S. & Haeri, M. [2010] “ Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach,” Automatica 46, 94–100. Crossref, Web of Science, Google Scholar
Wang, X., He, Y. & Wang, M. [2009] “ Chaos control of a fractional order modified coupled dynamos system,” Nonlin. Anal.: Th. Meth. Appl. 71, 6126–6134. Crossref, Web of Science, Google Scholar
Xu, F. [2014] “ The generation of a series of multiwing chaotic attractors using integer and fractional order differential equation systems,” Int. J. Bifurcation and Chaos 24, 1450130-1–13. Web of Science, Google Scholar
Yazdani, M. & Salarieh, H. [2011] “ On the existence of periodic solutions in time-invariant fractional order systems,” Automatica 47, 1834–1837. Crossref, Web of Science, Google Scholar
Zhang, C. & Yu, S. [2011] “ Generation of multi-wing chaotic attractor in fractional order system,” Chaos Solit. Fract. 44, 845–850. Crossref, Web of Science, Google Scholar
Zhong, G. Q. & Tang, W. K. T. [2002] “ Circuitry implementation and synchronization of Chen’s attractor,” Int. J. Bifurcation and Chaos 12, 1423–1427. Link, Web of Science, Google Scholar