Generating 3-D multi-scroll chaotic attractors: a hysteresis series switching method. (English) Zbl 1162.93353

Lü, Jinhu; Han, Fengling; Yu, Xinghuo; Chen, Guanrong

Automatica 40, No. 10, 1677-1687 (2004).

Summary: This paper introduces a systematic method-a hysteresis series switching approach-for generating multi-scroll chaotic attractors from a three-dimensional linear autonomous system, including 1-D \(n\)-scroll, 2-D \(n-m\)-grid scroll, and 3-D \(n-m-l\)-grid scroll chaotic attractors. The chaos generation mechanism is studied by analyzing the system trajectories and the hysteresis switching dynamics of the controlled chaotic systems are explored. Moreover, a two-dimensional Poincaré return map is rigorously derived. This map and its maximum Lyapunov exponent are applied to verifying the chaotic behaviors of the generated 3-D multi-scroll chaotic attractors.

MSC:

93C10	Nonlinear systems in control theory
34C55	Hysteresis for ordinary differential equations
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

Chaos generation; Multi-scroll chaotic attractor; Hysteresis series; Switching control

References:

[1]	Cafagna, D.; Grassi, G., New 3D-scroll attractors in hyperchaotic Chua’s circuit forming a ring, International journal of bifurcation and chaos, 13, 10, 2889-2903, (2003) · Zbl 1057.37026
[2]	Chen, G. (Ed.) (1999). Controlling chaos and bifurcations in engineering systems. Boca Raton: CRC Press.
[3]	Chen, G.; Dong, X., From chaos to order: methodologies, perspectives and applications, (1998), World Scientific Singapore
[4]	Chen, G.; Lü, J., Dynamics of the Lorenz system family: analysis, control and synchronization, (2003), Science Press Beijing, (in Chinese)
[5]	Elwakil, A.S.; Kennedy, M.P., Systematic realization of a class of hysteresis chaotic oscillators, International journal of circuit theory and applications, 28, 4, 319-334, (2000) · Zbl 1031.34044
[6]	Elwakil, A.S.; Kennedy, M.P., Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices, IEEE transactions on circuits and systems, part I, 48, 3, 289-307, (2001) · Zbl 0998.94048
[7]	Elwakil, A. S., Salama, K. N., & Kennedy, M. P. (2000). A system for chaos generation and its implementation in monolithic form. Proceedings of the IEEE symposium on circuits and systems, Geneva (pp. 217-220).
[8]	Han, F., Lü, J., Yu, X., Chen, G., & Feng, Y. (2004). A new systematic method for generating multi-scroll chaotic attractors from a linear second-order system with hysteresis. Accepted by Dynamics of Continuous, Discrete and Impulsive Systems, to appear.
[9]	Kataoka, M.; Saito, T., A two-port VCCS chaotic oscillator and quad screw attractor, IEEE transactions on circuits and systems, part I, 48, 2, 221-225, (2001)
[10]	Kennedy, A.S.; Kennedy, M.P., Chaotic oscillators derived from Saito’s double-screw hysteresis oscillator, IEICE transactions on fundamentals, E82, 1769-1775, (1999)
[11]	Lü, J.; Lu, J.; Chen, S., Chaotic time series analysis and its applications, (2002), Wuhan University Press China, (in Chinese)
[12]	Lü, J.; Zhou, T.; Chen, G.; Yang, X., Generating chaos with a switching piecewise-linear controller, Chaos, 12, 2, 344-349, (2002)
[13]	Lü, J.; Yu, X.; Chen, G., Generating chaotic attractors with multiple merged basins of attractiona switching piecewise-linear control approach, IEEE transactions on circuits and systems, part I, 50, 2, 198-207, (2003) · Zbl 1368.37041
[14]	Nakagawa, S.; Saito, T., An RC OTA hysteresis chaos generator, IEEE transactions on circuits and systems, part I, 43, 12, 1019-1021, (1996)
[15]	Newcomb, R.W.; El-Leithy, N., Chaos generation using binary hysteresis, Circuit system, signal processing, 5, 3, 321-341, (1986) · Zbl 0608.58033
[16]	Ozoguz, S.; Elwakil, A.S.; Salama, K.N., N-scroll chaos generator using nonlinear transconductor, Electronics letters, 38, 14, 685-686, (2002)
[17]	Saito, T., An approach toward higher dimensional hysteresis chaos generators, IEEE transactions on circuits and systems, part I, 37, 3, 399-409, (1990) · Zbl 0704.94028
[18]	Saito, T.; Nakagawa, S., Chaos from a hysteresis and switched circuit, Philosophical transactions of the royal society of London series A, 353, 47-57, (1995) · Zbl 0870.58097
[19]	Storace, M.; Parodi, M.; Robatto, D., A hysteresis-based chaotic circuitdynamics and applications, International journal of circuit theory and applications, 27, 6, 527-542, (1999) · Zbl 0990.94041
[20]	Suykens, J.A.K.; Vandewalle, J., Generation of n-double scrolls (n = 1,2,3,4,…), IEEE transactions on circuits and systems, part I, 40, 11, 861-867, (1993) · Zbl 0844.58063
[21]	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, part I, 48, 11, 1369-1372, (2001)
[22]	Yalcin, M.E.; Ozoguz, S.; Suykens, J.A.K.; Vandewalle, J., N-scroll chaos generatorsa simple circuit model, Electronics letters, 37, 3, 147-148, (2001)
[23]	Yalcin, M.E.; Suykens, J.A.K.; Vandewalle, J.; Ozoguz, S., Families of scroll grid attractors, International journal of bifurcation and chaos, 12, 1, 23-41, (2002) · Zbl 1044.37029
[24]	Zhong, G.Q.; Man, K.F.; Chen, G., A systematic approach to generating n-scroll attractors, International journal of bifurcation and chaos, 12, 12, 2907-2915, (2002)