Tutorials and ReviewsNo Access

A GALLERY OF CHUA ATTRACTORS: PART I

ELEONORA BILOTTA

Department of Linguistics, University of Calabria, Cubo 17B, Ponte Pietco Bucci 87036, Arcavacata di Rende, Italy

Search for more papers by this author

PIETRO PANTANO

Department of Mathematics, University of Calabria, Cubo 30B, Ponte Pietco Bucci 87036, Arcavacata di Rende, Italy

Search for more papers by this author

, and

FAUSTO STRANGES

Department of Linguistics, University of Calabria, Cubo 17B, Ponte Pietco Bucci 87036, Arcavacata di Rende, Italy

Search for more papers by this author

https://doi.org/10.1142/S0218127407017161Cited by:47 (Source: Crossref)

Abstract

Chua Oscillator exhibits a wide variety of nonlinear behavior and has become a paradigm for theoretical and experimental investigations of chaotic systems. An initial exploration of the parameter space for the circuit shows that the system and its generalizations generates a broad range of very different strange attractors. In the work described in this paper, we constructed "a gallery" of these attractors, including patterns that have never previously been observed. We identified the regions of parameter space occupied by each attractor and the initial conditions leading to production of the attractor. System behavior was characterized using time series, FFT graphs and in some cases Lyapunov exponents. In this way we created a complex picture of chaos, which we divided into six parts. The first, we publish here. The rest of our work will be published in subsequent issues of this journal. In this first paper, we describe how to build Chua Oscillator and some of its generalizations, as proposed in the recent literature. We introduce the main features that characterize the chaotic behavior of each of these systems. Finally, we offer hints on the mechanisms underlying synchronization between pairs of coupled Chua Oscillators. The investigation of chaos allows for the emergence of a complex picture, which could improve our knowledge of these challenging phenomena in contemporary science.

Keywords:

References

P. Arenaet al., IEEE Trans. Circuits Syst. I: Fund. Th. Appl. 42, 123 (1995), DOI: 10.1109/81.372854. Crossref, Web of Science, Google Scholar
P. Arenaet al., Int. J. Circuit Th. Appl. 24, 241 (1996), DOI: 10.1002/(SICI)1097-007X(199605/06)24:3<241::AID-CTA912>3.0.CO;2-J. Crossref, Web of Science, Google Scholar
M. A. Aziz-Alaoui, Multispiral chaos, Proc. 2nd Int. Conf. Control of Oscillations and Chaos1 (2000) pp. 88–91, DOI: 10.1109/COC.2000.873517. Google Scholar
E. Bilotta et al. , Sound synthesis and chaotic behaviour in Chua's oscillator , Proc. SCEE, Capo d'Orlando , Lect. Notes in Industrial Mathematics ( Springer-Verlag , Berlin , 2004 ) . Google Scholar
E. Bilotta, S. Gervasi and P. Pantano, Int. J. Bifurcation and Chaos 15, 253 (2005), DOI: 10.1142/S0218127405012156. Link, Web of Science, Google Scholar
E. Bilotta, P. Pantano and F. Stranges, Comput. Graph. 30, 359 (2006), DOI: 10.1016/j.cag.2006.02.003. Crossref, Web of Science, Google Scholar
D. Cafagna and G. Grassi, Int. J. Bifurcation and Chaos 13, 2889 (2003), DOI: 10.1142/S0218127403008284. Link, Web of Science, Google Scholar
T. L. Carrol and L. M. Pecora, IEEE Trans. Circuits Syst. 38, 453 (1991), DOI: 10.1109/31.75404. Crossref, Google Scholar
L. O. Chua, M. Komuro and T. Matsumoto, IEEE Trans. Circuits Syst. 33, 1072 (1986), DOI: 10.1109/TCS.1986.1085869. Crossref, Google Scholar
L. O. Chua and G. N. Lin, IEEE Trans. Circuits Syst. 37, 885 (1990), DOI: 10.1109/31.55064. Crossref, Google Scholar
L. O. Chua, IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E76-A, 704 (1993). Google Scholar
L. O. Chuaet al., IEEE Trans. Circuits Syst. 40, 732 (1993), DOI: 10.1109/81.246149. Crossref, Google Scholar
L. O. Chuaet al., IEEE Trans. Circuits Syst. I: Fund. Th. Appl. 40, 745 (1993), DOI: 10.1109/81.246150. Crossref, Web of Science, Google Scholar
L. O. Chua , CNN: A Paradigm for Complexity ( World Scientific , Singapore , 1998 ) . Link, Google Scholar
A. S. Elwakil, K. N. Salama and M. P. Kennedy, A system for chaos generations and its implementation in monolithic form, Proc. IEEE Int. Symp. Circuits and Systems (ISCAS 2000)V (2000) pp. 217–220. Google Scholar
R. Hilborn , Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers ( Oxford University Press , Oxford , 1994 ) . Google Scholar
T. Kapitaniak, L. O. Chua and G. Q. Zhong, IEEE Trans. Circuits Syst. 41, 499 (1994), DOI: 10.1109/81.298367. Crossref, Google Scholar
J. Kaplan and J. Yorke, Lecture Notes in Mathematics 730 (Springer-Verlag, 1979) p. 219. Google Scholar
M. P. Kennedy, Frequenz 46, 66 (1992). Web of Science, Google Scholar
M. Komuroet al., Int. J. Bifurcation and Chaos 1, 139 (1991), DOI: 10.1142/S0218127491000105. Link, Web of Science, Google Scholar
R. N. Madan, Observing and learning chaotic phenomena from Chua's circuit, Proc. 35th Midwest Symp. Circuits and Systems (1992) pp. 736–745, DOI: 10.1109/MWSCAS.1992.271218. Google Scholar
R. N. Madan , Chua's Circuit: A Paradigm for Chaos ( World Scientific , Singapore , 1993 ) . Link, Google Scholar
S. Mascolo and S. Grassi, Observers for hyperchaos synchronization with application to secure communications, IEEE Proc. IEEE Int. Conf. Control Applications2 (1998) pp. 1016–1020, DOI: 10.1109/CCA.1998.721611. Google Scholar
T. Matsumoto, IEEE Trans. Circuits Syst. 31, 1055 (1984), DOI: 10.1109/TCS.1984.1085459. Crossref, Google Scholar
T. Matsumoto, L. O. Chua and M. Komuro, IEEE Trans. Circuits Syst. 32, 797 (1985), DOI: 10.1109/TCS.1985.1085791. Crossref, Google Scholar
T. Matsumoto, L. O. Chua and K. Kobayashi, IEEE Trans. Circuits Syst. 33, 1143 (1986), DOI: 10.1109/TCS.1986.1085862. Crossref, Google Scholar
T. Matsumoto, L. O. Chua and K. Ayaki, IEEE Trans. Circuits Syst. 35, 909 (1988), DOI: 10.1109/31.1836. Crossref, Google Scholar
S. Özoğuz, A. S. Elwakil and K. N. Salama, IEEE Electron. Lett. 38, 685 (2002), DOI: 10.1049/el:20020524. Crossref, Web of Science, Google Scholar
T. O. Parker and L. O. Chua , Practical Numerical Algorithms for Chaotic Systems ( Springer-Verlag , NY , 1988 ) . Google Scholar
O. E. Rössler, Phys. Lett. A 71, 155 (1979), DOI: 10.1016/0375-9601(79)90150-6. Crossref, Web of Science, Google Scholar
K. N. Salama, S. Özoğuz and A. S. Elwakil, IEEE Trans. Circuits Syst. 3, 176 (2003). Google Scholar
L. P. Shil'nikov, IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 40, 784 (1993), DOI: 10.1109/81.246153. Crossref, Web of Science, Google Scholar
S. Strogatz , Nonlinear Dynamics and Chaos ( Addison-Wesley , MA , 1994 ) . Google Scholar
J. A. K. Suykens and J. Vandewalle, IEEE Proc.-G 138, 595 (1991). Google Scholar
J. A. K. Suykens and J. Vandewalle, IEEE Trans. Circuits Syst. 40, 861 (1993), DOI: 10.1109/81.251829. Crossref, Google Scholar
J. A. K. Suykens, A. Huang and L. O. Chua, Archiv für Elektronik und Ubertragung-stechnik 51, 131 (1997). Google Scholar
W. K. Tanget al., IEEE Trans. Circuits Syst. 48, 1369 (2001), DOI: 10.1109/81.964432. Crossref, Google Scholar
K. Thamilmaran, M. Lakshmanan and A. Venkatesan, Int. J. Bifurcation and Chaos 14, 221 (2004), DOI: 10.1142/S0218127404009119. Link, Web of Science, Google Scholar
A. Tsuneda, Int. J. Bifurcation and Chaos 15, 1 (2005), DOI: 10.1142/S0218127405011990. Link, Web of Science, Google Scholar
M. E. Yalçin, J. A. K. Suykens and J. Vandewalle, IEEE Trans. Circuits Syst. 47, 425 (2000), DOI: 10.1109/81.841929. Crossref, Google Scholar
M. E. Yalçinet al., IEEE Electron. Lett. 37, 147 (2001), DOI: 10.1049/el:20010114. Crossref, Web of Science, Google Scholar
M. E. Yalçin , J. A. K. Suykens and J. Vandewalle , Cellular Neural Networks, Multi-Scroll Chaos and Synchronization ( World Scientific , Singapore , 2005 ) . Link, Google Scholar
T. Yang and L. O. Chua, Int. J. Bifurcation and Chaos 10, 2015 (2000), DOI: 10.1142/S0218127400001286. Link, Web of Science, Google Scholar
C. W. Wu and L. O. Chua, Int. J. Bifurcation and Chaos 3, 1619 (1993), DOI: 10.1142/S0218127493001288. Link, Web of Science, Google Scholar
C. W. Wu , Synchronization in Coupled Circuits and Systems ( World Scientific , Singapore , 2002 ) . Link, Google Scholar
G. Q. Zhong and F. Ayrom, IEEE Trans. Circuits Syst. 32, 501 (1985), DOI: 10.1109/TCS.1985.1085728. Crossref, Google Scholar
G. Q. Zhong and F. Ayrom, IEEE Trans. Circuits Syst. 13, 93 (1985). Google Scholar
G. Q. Zhong, IEEE Trans. Circuits Syst. 41, 934 (1994), DOI: 10.1109/81.340866. Crossref, Google Scholar
G. Q. Zhong, C. Wu and L. O. Chua, IEEE Trans. Circuits Syst. 45, 186 (1998), DOI: 10.1109/81.661689. Crossref, Google Scholar