Analysis, control, synchronization, and circuit design of a novel chaotic system. (English) Zbl 1255.93076

Summary: We introduce a novel three-dimensional autonomous chaotic system with a single cubic nonlinearity. Several issues, such as the basic dynamical behaviour, equilibria, Lyapunov exponent spectrum, and bifurcations of the new chaotic system, are investigated analytically and numerically. Next, adaptive control laws are designed to stabilize the new chaotic system with unknown parameters to its unstable equilibrium point at the origin, based on adaptive control theory and Lyapunov stability theory. Then, adaptive control laws are derived to achieve global chaos synchronization of identical new chaotic systems with unknown parameters. Further to these, a novel electronic circuit realization of the proposed chaotic system is presented and examined using the Orcad-PSpice program. It is convenient to use the new chaotic system to purposefully generate chaos in chaos applications. A good qualitative agreement is shown between the simulations and the experimental results.


93C40 Adaptive control/observation systems
34D06 Synchronization of solutions to ordinary differential equations
37N35 Dynamical systems in control
94C05 Analytic circuit theory
Full Text: DOI


[1] Lorenz, E. N., Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20, 130-141 (1963) · Zbl 1417.37129
[2] Chen, G.; Ueta, T., Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[3] Ueta, T.; Chen, G., Bifurcation analysis of Chen’s attractor, International Journal of Bifurcation and Chaos, 10, 1917-1931 (2000) · Zbl 1090.37531
[4] Lü, J.; Chen, G., A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12, 659-661 (2002) · Zbl 1063.34510
[5] Sprott, J. C., Simple chaotic flows, Physical Review E, 50, 647-650 (1994)
[6] Sprott, J. C., Simplest dissipative chaotic flows, Physics Letters A, 228, 271-274 (1997) · Zbl 1043.37504
[7] Sprott, J. C., A new class of chaotic circuit, Physics Letters A, 266, 19-23 (2000)
[8] Rössler, O. E., An equation for continuous chaos, Physics Letters A, 57, 397-398 (1976) · Zbl 1371.37062
[9] van der Schrier, G.; Maas, L. R.M., The diffusionless Lorenz-equations: Shil’nikov bifurcations and reduction to an explicit map, Physica D, 141, 19-36 (2000) · Zbl 0956.37038
[10] Wei, Z. C.; Yang, Q. G., Controlling the diffusionless Lorenz equations with periodic parametric perturbation, Computers and Mathematics with Applications, 58, 1976-1987 (2009) · Zbl 1189.34118
[11] Shaw, R., Strange attractor, chaotic behavior and information flow, Zeitschrift fur Naturforschung A, 36, 80-112 (1981) · Zbl 0599.58033
[12] Yang, G.; Chen, G. R., A chaotic system with one saddle and two saddle node-foci, International Journal of Bifurcation and Chaos, 18, 1393-1414 (2008) · Zbl 1147.34306
[13] Yang, G.; Wei, Z. C.; Chen, G. R., An unusual 3D autonomous quadratic chaotic system with two stable node-foci, International Journal of Bifurcation and Chaos, 20, 1061-1083 (2010) · Zbl 1193.34091
[14] Pehlivan, I.; Uyaroglu, Y., A new chaotic attractor from general Lorenz system family and its electronic experimental implementation, Turkish Journal of Electrical Engineering and Computer Science, 18, 171-184 (2010)
[15] Cuomo, K. M.; Oppenheim, A. V., Circuit implementation of synchronized chaos with applications to communications, Physical Review Letters, 71, 65-68 (1993)
[16] Nakagawa, S.; Saito, T., An RC OTA hysteresis chaos generator, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 43, 1019-1021 (1996)
[17] Sprott, J. C., Simple chaotic systems and circuits, American Journal of Physics, 68, 758-763 (2000)
[18] Ozoguz, S.; Elwakil, A.; Kennedy, M. P., Experimental verification of the butterfly attractor in a modified Lorenz system, International Journal of Bifurcation and Chaos, 12, 1627-1632 (2002)
[19] Elwakil, A.; Kennedy, M., Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48, 289-307 (2001) · Zbl 0998.94048
[20] Pehlivan, I.; Uyaroglu, Y.; Yogun, M., Chaotic oscillator design and realizations of the Rucklidge attractor and its synchronization and masking simulations, Scientific Research and Essays, 5, 2210-2219 (2010)
[21] Yu, S.; Lü, J.; Tang, W.; Chen, G., A general multi-scroll Lorenz system family and its realization via digital signal processors, Chaos, 16, 033126 (2006) · Zbl 1151.94432
[22] Yalcin, M. E.; Suykens, J. A.K.; Vandewalle, J.; Ozoguz, S., Families of scroll grid attractors, International Journal of Bifurcation and Chaos, 12, 23-41 (2002) · Zbl 1044.37029
[23] Yalcin, M. E.; Suykens, J. A.K.; Vandewalle, J. P.L., Cellular Neural Networks, Multi-Scroll Chaos and Synchronization (2005), World Scientific: World Scientific Singapore · Zbl 1089.68117
[24] Tang, K. S.; Zhong, G. Q.; Chen, G.; Man, K. F., Generation of \(n\)-scroll attractors via sine function, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 48, 1369-1372 (2001)
[25] Lü, J.; Chen, G., Generating multi-scroll chaotic attractors: theories, methods and applications, International Journal of Bifurcation and Chaos, 16, 775-858 (2006) · Zbl 1097.94038
[26] Pehlivan, I.; Uyaroglu, Y., Rikitake attractor and its synchronization application for secure communication systems, Journal of Applied Sciences, 7, 232-236 (2007)
[27] Pehlivan, I.; Uyaroglu, Y., Simplified chaotic diffusionless Lorenz attractor and its application to secure communication systems, IET Communications, 1, 1015-1022 (2007)
[28] Uyaroglu, Y.; Pehlivan, I., Nonlinear Sprott94 Case A chaotic equation: synchronization and masking communication applications, Computers and Electrical Engineering, 36, 1093-1100 (2010)
[29] Messias, M.; Braga, D. C.; Mello, L. F., Degenerate Hopf bifurcation in Chua’s system, International Journal of Bifurcation and Chaos, 19, 497-515 (2009) · Zbl 1170.34333
[30] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physics Review Letters, 64, 821-824 (1990) · Zbl 0938.37019
[31] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Physics Review Letters, 64, 1196-1199 (1990) · Zbl 0964.37501
[32] Huang, L.; Feng, R.; Wang, M., Synchronization of chaotic systems using generalized active network, Physical Letters A, 320, 271-275 (2002)
[33] Chen, H. K., Global chaos synchronization of new chaotic systems via nonlinear control, Chaos, Solitons and Fractals, 23, 1245-1251 (2005) · Zbl 1102.37302
[34] Lu, J.; Wu, X.; Lü, J., Adaptive feedback synchronization of a new unified chaotic system, Physics Letters A, 329, 327-333 (2004) · Zbl 1209.93119
[35] Sundarapandian, V., Adaptive control and synchronization of hyperchaotic Liu system, International Journal of Computer Science, Engineering and Information Technology, 1, 29-40 (2011)
[36] Sundarapandian, V.; Sivaperumal, S., Anti-synchronization of hyperchaotic Lorenz systems by sliding mode control, International Journal on Computer Science and Engineering, 3, 2438-2439 (2011)
[37] Park, J. H.; Kwon, O. M., A novel criterion for delayed feedback control of time-delay chaotic systems, Chaos, Solitons and Fractals, 17, 709-716 (2003)
[38] Yu, Y. G.; Zhang, S. C., Adaptive backstepping synchronization of uncertain chaotic systems, Chaos, Solitons and Fractals, 27, 1369-1375 (2006)
[39] Yang, T.; Chua, L. O., Control of chaos using sampled-data feedback control, International Journal of Bifurcation and Chaos, 9, 215-219 (1999)
[40] Hahn, W., The Stability of Motion (1967), Springer Verlag: Springer Verlag New York · Zbl 0189.38503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.