Dynamical behaviors of a chaotic system with no equilibria. (English) Zbl 1255.37013

Summary: Based on the Sprott D system, a simple three-dimensional autonomous system with no equilibria is reported. The remarkable particularity of the system is that there exists a constant controller, which can adjust the type of chaotic attractors. It is demonstrated to be chaotic in the sense of having a positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping and period-doubling route to chaos are analyzed with careful numerical simulations.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M10 Time series analysis of dynamical systems
93B05 Controllability
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
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[1] Lorenz, E. N., J. Atmos. Sci., 20, 130 (1963) · Zbl 1417.37129
[2] Rössler, O. E., Phys. Lett. A, 57, 397 (1976) · Zbl 1371.37062
[3] Chen, G. R.; Ueta, T., Int. J. Bifur. Chaos, 9, 1465 (1999) · Zbl 0962.37013
[4] Lü, J. H.; Chen, G. R., Int. J. Bifur. Chaos, 12, 659 (2002) · Zbl 1063.34510
[5] Sprott, J. C., Phys. Rev. E, 50, 647 (1994)
[6] Silva, C. P., IEEE Trans. Circ. Syst. I, 40, 657 (1993)
[7] Yang, Q. G.; Wei, Z. C.; Chen, G. R., Int. J. Bifur. Chaos, 20, 1061 (2010) · Zbl 1193.34091
[8] Yang, Q. G.; Chen, G. R., Int. J. Bifur. Chaos, 18, 1393 (2008) · Zbl 1147.34306
[9] Wang, X.; Chen, G. R., Commun. Nonlinear Sci. Numer. Simulat., 17, 1264 (2012)
[10] Falkner, V. M.; Skan, S. W., Philosophical Magazine, 7, 865 (1931)
[11] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory (1998), Springer: Springer New York · Zbl 0914.58025
[12] Hou, Z. T.; Kang, N.; Kong, X. X.; Chen, G. R.; Yan, G. J., Int. J. Bifur. Chaos, 20, 557 (2010) · Zbl 1188.34054
[13] Poincaré, H., Acta Mathematica, 13, 1 (1890)
[14] Bendixson, I., Acta Mathematica, 24, 1 (1901)
[15] Barnett, S., Polynomials and Linear Control Systems (1983), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0528.93003
[16] Feigenbaum, M. J., Physica D, 7, 16 (1983)
[17] Haniasa, M. P.; Avgerinos, Z.; Tombras, G. S., Chaos Solitons Fractals, 40, 1050 (2009)
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