×

A three-scroll chaotic attractor. (English) Zbl 1217.37030

Summary: This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the \(z\)-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the \(z\)-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Lorenz, E.N., J. atmos. sci., 20, 130, (1963)
[2] Stewart, I., Nature, 406, 948, (2000)
[3] Ott, E., Chaos in dynamical systems, (2002), Cambridge Univ. Press Cambridge, UK · Zbl 1006.37001
[4] Chen, G.; Lü, J.H., Dynamical analysis, control and synchronization of the generalized Lorenz systems family, (2003), Science Press Beijing, (in Chinese)
[5] Chen, G.; Wang, X.F., Chaotification of dynamical systems: theory, methods and applications, (2006), Shanghai Jiao Tong Univ. Press Shanghai, (in Chinese)
[6] Gilmore, R.; Lefranc, M., The topology of chaos, (2002), Wiley New York · Zbl 1019.37016
[7] Shaw, R., Z. naturforsch. A, 36, 80, (1981)
[8] Rössler, O.E., Phys. lett. A, 57, 397, (1976)
[9] Rössler, O.E., Am. (N.Y.) acad. sci., 316, 376, (1979)
[10] Sprott, J.C., Phys. rev. E, 50, R647, (1994)
[11] Sprott, J.C., Phys. lett. A, 228, 271, (1997)
[12] Sprott, J.C.; Linz, S.J., Int. J. chaos theory appl., 5, 3, (2000)
[13] Letellier, C.; Tsankov, T.; Byrne, G.; Gilmore, R., Phys. rev. E, 72, 026212, (2005)
[14] Čelikovský, S.; Vaněcek, A., Kybernetika, 30, 4, 403, (1994)
[15] Vaněcek, A.; Čelikovský, S., Control systems: from linear analysis to synthesis of chaos, (1996), Prentice-Hall London · Zbl 0874.93006
[16] Chen, G.; Ueta, T., Int. J. bifur. chaos, 9, 1465, (1999)
[17] Lü, J.H.; Chen, G., Int. J. bifur. chaos, 12, 659, (2002)
[18] Celikovsy, S.; Chen, G., Int. J. bifur. chaos, 12, 1789, (2002)
[19] Celikovsy, S.; Chen, G., Chaos solitons fractals, 26, 1271, (2005)
[20] Rucklidge, A.M., J. fluid mech., 237, 209, (1992)
[21] Genesio, R.; Tesi, A., Automatica, 28, 3, 531, (1992)
[22] Liu, C.; Liu, T.; Liu, L.; Liu, K., Chaos solitons fractals, 22, 1031, (2004)
[23] Shilnikov, A.L., Physica D, 62, 338, (1993)
[24] Matsumoto, T.; Chua, L.O.; Tanaks, S., Phys. rev. A, 30, 1155, (1984)
[25] Baghious, E.; Jarry, P., Int. J. bifur. chaos, 3, 1, 201, (1993)
[26] Elwakil, A.; Ozoguz, S.; Kennedy, M., IEEE trans. circuits syst. I, 49, 527, (2002)
[27] Ozoguz, S.; Elwakil, A.; Kennedy, M., Int. J. bifur. chaos, 12, 1627, (2002)
[28] Elwakil, A.; Ozoguz, S.; Kennedy, M., Int. J. bifur. chaos, 13, 10, 3093, (2003)
[29] Liu, W.B.; Chen, G., Int. J. bifur. chaos, 12, 261, (2003)
[30] Lü, J.H.; Chen, G.; Cheng, D., Int. J. bifur. chaos, 14, 1507, (2004)
[31] Birman, J.S.; Williams, R.F., Topology, 22, 47, (1983)
[32] Gilmore, R., Rev. mod. phys., 70, 4, 1455, (1998)
[33] Tsankov, T.; Byrne, G.; Gilmore, R., Phys. rev. lett., 13, 134104, (2003)
[34] Letellier, C.; Roulin, E.; Rössler, O.E., Chaos solitons fractals, 28, 337, (2006) · Zbl 1084.34048
[35] Miranda, R.; Stone, E., Phys. lett. A, 178, 105, (1993)
[36] Yu, S.; Lü, J.H.; Wallace, K.; Tangb, S.; Chen, G., Chaos, 16, 3, 033126, (2006)
[37] Wang, X.F.; Chen, G.; Yu, X., Chaos, 10, 1, (2000)
[38] Takagi, T.; Sugeno, M., IEEE trans. syst. cybernet., 15, 116, (1985)
[39] Kivshar, Y.S.; Rodelsperger, F.; Benner, H., Phys. rev. E, 39, 319, (1994)
[40] Li, D.Q., Chin. phys., 15, 11, 2541, (2006)
[41] Li, D.Q., Phys. lett. A, 356, 1, 51, (2006)
[42] D.Q Li, Z. Yin, Exploring the compound structure of chaotic Chen’s attractor: A hybrid TS fuzzy approach, Chaos Solitons Fractals (2007), in press
[43] Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A., Physica D, 16, 285, (1985)
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.