Feature ArticlesNo Access

A Simple Chaotic Flow with a Continuously Adjustable Attractor Dimension

Buncha Munmuangsaen

Fabrinet Co., Ltd, Klongluang, Patumthani, 12120, Thailand

E-mail Address: nopnop99@hotmail.com

Search for more papers by this author

Julien Clinton Sprott

Department of Physics, University of Wisconsin – Madison, Madison, WI 53706, USA

E-mail Address: sprott@physics.wisc.edu

Search for more papers by this author

Wesley Joo-Chen Thio

Department of Electrical and Computer Engineering, The Ohio State University, Columbus OH, 43210, USA

E-mail Address: wesley.thio@gmail.com

E-mail Address: thio.7@osu.edu

Search for more papers by this author

Arturo Buscarino

Dipartimento di Ingegneria Elettrica Elettronica ed Informatica, Università degli Studi di Catania, viale A. Doria 6, 95125 Catania, Italy

E-mail Address: arturo.buscarino@dieei.unict.it

Search for more papers by this author

, and

Luigi Fortuna

Dipartimento di Ingegneria Elettrica Elettronica ed Informatica, Università degli Studi di Catania, viale A. Doria 6, 95125 Catania, Italy

E-mail Address: luigi.fortuna@dieei.unict.it

Search for more papers by this author

https://doi.org/10.1142/S0218127415300360Cited by:36 (Source: Crossref)

Abstract

This paper describes two simple three-dimensional autonomous chaotic flows whose attractor dimensions can be adjusted continuously from $2.0$ to $3.0$ by a single control parameter. Such a parameter provides a means to explore the route through limit cycles, period-doubling, dissipative chaos, and eventually conservative chaos. With an absolute-value nonlinearity and certain choices of parameters, the systems have a vast and smooth continual transition path from dissipative chaos to conservative chaos. One system is analyzed in detail by means of the largest Lyapunov exponent, Kaplan–Yorke dimension, bifurcations, coexisting attractors and eigenvalues of the Jacobian matrix. An electronic version of the system has been constructed and shown to perform in accordance with expectations.

Keywords:

References

1. Blakely, J. N., Eskridge, M. B. & Corron, N. J. [2007] “ A simple Lorenz circuit and its radio frequency implementation,” Chaos 17, 023112-1–5. Crossref, Web of Science, Google Scholar
2. Buscarino, A., Fortuna, L., Frasca, M. & Sciuto, G. [2014] A Concise Guide to Chaotic Electronic Circuits, Springer Briefs in Applied Sciences and Technology (Springer Intl. Pub.). Crossref, Google Scholar
3. Carroll, T. L. & Pecora, L. M. [1991] “ Synchronizing chaotic circuits,” IEEE Trans. Circuits Syst. 38, 453–456. Crossref, Google Scholar
4. Chlouverakis, K. E. & Sprott, J. C. [2004] “ A comparison of correlation and Lyapunov dimensions,” Physica D 200, 156–164. Crossref, Web of Science, Google Scholar
5. Cuomo, K. M. & Oppenheim, A. V. [1993] “ Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65–68. Crossref, Web of Science, Google Scholar
6. Fortuna, L., Frasca, M. & Xibilia, M. G. [2009] Chua’s Circuit Implementations, Yesterday, Today and Tomorrow (World Scientific, Singapore). Link, Google Scholar
7. Hoover, W. G. [1995] “ Remark on some simple chaotic flows,” Phys. Rev. E 51, 759–760. Crossref, Web of Science, Google Scholar
8. Hoover, W. G., Sprott, J. C. & Hoover, C. G. [2016] “ Ergodicity of a singly-thermostated harmonic oscillator,” Commun. Nonlin. Sci. Numer. Simul. 32, 234–240. Crossref, Web of Science, Google Scholar
9. Kaplan, J. L. & Yorke, J. A. [1979] “ Functional differential equations and approximations of fixed points,” in Lecture Notes in Mathematics, Vol. 730, eds. Peitgen, H.-O. & Walther, H.-O. (Springer, Berlin), pp. 204–227. Google Scholar
10. Kilias, T., Kelber, K., Mogel, A. & Schwarz, W. [1995] “ Electronic chaos generators: Design and applications,” Int. J. Electron. 79, 737–753. Crossref, Web of Science, Google Scholar
11. Kocarev, L., Halle, K. S., Eckert, K. & Chua, L. O. [1992] “ Experimental demonstration of secure communications via chaotic synchronization,” Int. J. Bifurcation and Chaos 2, 709–713. Link, Google Scholar
12. Lorenz, E. N. [1963] “ Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141. Crossref, Web of Science, Google Scholar
13. Munmuangsaen, B. & Srisuchinwong, B. [2011] “ Elementary chaotic snap flows,” Chaos Solit. Fract. 44, 995–1003. Crossref, Web of Science, Google Scholar
14. Posh, H. A., Hoover, W. G. & Vesely, F. J. [1986] “ Canonical dynamics of the Nosé oscillator: Stability, order, and chaos,” Phys. Rev. A 33, 4253–4265. Crossref, Web of Science, Google Scholar
15. Qi, G., van Wyk, M. A., van Wyk, B. J. & Chen, G. [2008] “ On a new hyperchaotic system,” Phys. Lett. A 372, 124–136. Crossref, Web of Science, Google Scholar
16. Rosenstein, M. T., Collins, J. J. & De Luca, C. J. [1993] “ A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D 65, 117–134. Crossref, Web of Science, Google Scholar
17. Rössler, O. E. [1976] “ An equation for continuous chaos,” Phys. Lett. A 57, 397–398. Crossref, Web of Science, Google Scholar
18. Sprott, J. C. [1994] “ Some simple chaotic flows,” Phys. Rev. E 50, R647–R650. Crossref, Web of Science, Google Scholar
19. Sprott, J. C. [1997] “ Some simple chaotic jerk functions,” Am. J. Phys. 65, 537–543. Crossref, Web of Science, Google Scholar
20. Sprott, J. C. [2003] Chaos and Time-Series Analysis (Oxford University Press, Oxford). Crossref, Google Scholar
21. Sprott, J. C. [2007] “ Maximally complex simple attractors,” Chaos 17, 033124-1–6. Crossref, Web of Science, Google Scholar
22. Sprott, J. C. [2010] Elegant Chaos: Algebraically Simple Chaotic Flows (World Scientific, Singapore). Link, Google Scholar
23. Srisuchinwong, B. & Munmuangsaen, B. [2011] “ A highly chaotic attractor for a dual-channel single-attractor, private communication system,” in Chaos Theory: Modeling, Simulation and Applications, eds. Skiadas, C. H., Dimotikalis, I. & Skiadas, C. (World Scientific, Singapore), pp. 399–405. Link, Google Scholar
24. Zaslavsky, G. M. [2007] The Physics of Chaos in Hamiltonian Systems (World Scientific, Singapore). Link, Google Scholar