Chaos of discrete dynamical systems in complete metric spaces. (English) Zbl 1067.37047

Summary: This paper is concerned with chaos of discrete dynamical systems in complete metric spaces. Discrete dynamical systems governed by continuous maps in general complete metric spaces are first discussed, and two criteria of chaos are then established. As a special case, two corresponding criteria of chaos for discrete dynamical systems in compact subsets of metric spaces are obtained. These results extend and improve the existing relevant results of chaos in finite-dimensional Euclidean spaces.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
54E40 Special maps on metric spaces
Full Text: DOI


[1] Arkhangel’skii, A. V.; Pontryagin, L. S., General topology, (Encyclopaedia of mathematical sciences, vol. 17 (1990), Springer-Verlag: Springer-Verlag New York)
[2] Banks, J.; Brooks, J.; Cairns, G.; Davis, G.; Stacey, P., On Devaney’s definition of chaos, Amer. Math. Monthly, 99, 332-334 (1992) · Zbl 0758.58019
[3] Chen, G., Chaotification via feedback: the discrete case, (Chen, G.; Yu, X., Chaos control: theory and applications (2003), Springer-Verlag: Springer-Verlag Heidelberg), 159-177 · Zbl 1330.93107
[4] Chen, G.; Hsu, S.; Zhou, J., Snap-back repellers as a cause of chaotic vibration of the wave equation with a Van der Pol boundary condition and energy injection at the middle of the span, J. Math. Phys., 39, 12, 6459-6489 (1998) · Zbl 0959.37027
[5] Devaney, R. L., An introduction to chaotic dynamical systems (1987), Addison-Wesley Publishing Company: Addison-Wesley Publishing Company New York
[6] Keener, J. P., Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. Soc., 261, 589-604 (1980) · Zbl 0458.58016
[7] Kennedy, J.; Yorke, J. A., Topological horseshoes, Trans. Amer. Math. Soc., 353, 2513-2530 (2001) · Zbl 0972.37011
[8] Li, T.; Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82, 985-992 (1975) · Zbl 0351.92021
[9] On the Marotto-Li-Chen theorem and its application to chaotification of multi-dimensional discrete dynamical systems, Chaos, Solitons & Fractals, 18, 807-817 (2003), See also · Zbl 1069.37024
[10] Marotto, F. R., Snap-back repellers imply chaos in \(R^n\), J. Math. Anal. Appl., 63, 199-223 (1978) · Zbl 0381.58004
[11] Martelli, M.; Dang, M.; Seph, T., Defining chaos, Math. Mag., 71, 2, 112-122 (1998) · Zbl 1008.37014
[12] Robinson, C., Dynamical systems: stability, symbolic dynamics and chaos (1995), CRC Press: CRC Press Boca Raton · Zbl 0853.58001
[16] Wang, X. F.; Chen, G., Chaotifying a stable map via smooth small-amplitude high-frequency feedback control, Int. J. Circ. Theory Appl., 28, 305-312 (2000) · Zbl 1043.93026
[17] Wiggins, S., Global bifurcations and chaos (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0661.58001
[18] Yang, X.; Tang, Y., Horseshoes in piecewise continuous maps, Chaos, Solitons & Fractals, 19, 841-845 (2004) · Zbl 1053.37006
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.