×
Li, Zongcheng; Shi, Yuming; Liang, Wei

Discrete chaos induced by heteroclinic cycles connecting repellers in Banach spaces. (English) Zbl 1188.37036

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 757-770 (2010).
The problem treated in the paper is reflected by its title. Discrete chaos is considered in both Banach (infinite-dimensional) and Euclidean (finite-dimensional) case. Various sorts of chaos are taken into account: in the sense of Li-Yorke, Devaney and Wiggins. An illustrative numerical example is supplied.
Reviewer: Jan Andres (Olomouc)

Cited in 10 Documents

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior

Keywords:

discrete chaos; heteroclinic cycles; repellers
PDF BibTeX XML Cite
\textit{Z. Li} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 2, 757--770 (2010; Zbl 1188.37036)
Full Text: DOI

References:

[1] Li, T. Y.; Yorke, J. A., Period three implies chaos, Amer. Math. Monthly, 82, 985-992 (1975) · Zbl 0351.92021
[2] Marotto, F. R., Snap-back repellers imply chaos in \(R^n\), J. Math. Anal. Appl., 63, 199-223 (1978) · Zbl 0381.58004
[3] Shiraiwa, K.; Kurata, M., A generalization of a theorem of Marotto, Proc. Japan. Acad., 55, 286-289 (1979) · Zbl 0451.58031
[4] Smale, S., Diffeomorphisms with many periodic points, (Differential and Combinatorial Topology (1965), Princeton University Press: Princeton University Press Princeton, NJ), 63-80 · Zbl 0142.41103
[5] Shi, Y.; Chen, G., Chaos of discrete dynamical systems in complete metric spaces, Chaos Solitons Fractals, 22, 555-571 (2004) · Zbl 1067.37047
[6] Shi, Y.; Chen, G., Discrete chaos in Banach spaces, Sci. China Ser. A: Math., 34, 595-609 (2004), Chinese version; English version: 48 (2005) 222-238
[7] Shi, Y.; Yu, P.; Chen, G., Chaotification of dynamical systems in Banach spaces, Internat. J. Bifur. Chaos, 16, 2615-2636 (2006) · Zbl 1185.37084
[8] Shi, Y.; Yu, P., Chaos induced by regular snap-back repellers, J. Math. Anal. Appl., 337, 1480-1494 (2008) · Zbl 1131.37023
[9] Kloeden, P. E., Cycles and chaos in higher dimensional difference equations, (Mitropolsky, Yu. A., Proc. 9th Int. Conference on Nonlinear Oscillators, vol. 2 (1984), Naukova Dumkapp: Naukova Dumkapp Kiev), 184-187
[10] Kennedy, J.; Yorke, J. A., Topological horseshoes, Trans, Amer. Math. Soc., 353, 2513-2530 (2001) · Zbl 0972.37011
[11] Yang, X.; Tang, Y., Horseshoes in piecewise continuous maps, Chaos Solitons Fractals, 19, 841-845 (2004) · Zbl 1053.37006
[12] Blanchard, F.; Glasner, E.; Kolyada, S.; Maass, A., On Li-Yorke pairs, J. Reine Angew. Math., 547, 51-68 (2002) · Zbl 1059.37006
[13] Block, L. S.; Coppel, W. A., (Dynamics in One Dimension. Dynamics in One Dimension, Lecture Notes in Mathematics, vol. 1513 (1992), Springer-Verlag: Springer-Verlag Berlin, Heidelberg)
[14] Shi, Y.; Yu, P., Study on chaos by turbulent maps in noncompact sets, Chaos Solitons Fractals, 28, 1165-1180 (2006) · Zbl 1106.37008
[15] Lin, W.; Chen, G., Heteroclinical repellers imply chaos, Internat. J. Bifur. Chaos, 16, 1471-1489 (2006) · Zbl 1185.37018
[16] Shi, Y.; Ju, H.; Chen, G., Coupled-expanding maps and one-sided symbolic dynamical systems, Chaos Solitons Fractals, 39, 2138-2149 (2009) · Zbl 1197.37010
[17] Li, Z.; Shi, Y.; Zhang, C., Chaos induced by heteroclinic cycles connecting repellers in complete metric spaces, Chaos Solitons Fractals, 36, 746-761 (2008) · Zbl 1142.37014
[18] Wiggins, S., Global Bifurcations and Chaos (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0661.58001
[19] Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1987), Addison-Wesley Publishing Company: Addison-Wesley Publishing Company New York
[20] Martelli, M.; Dang, M.; Steph, T., Defining chaos, Math. Magazine, 71, 2, 112-122 (1998) · Zbl 1008.37014
[21] Robinson, C., Dynamical systems: Stability, (Symbolic Dynamics and Chaos (1995), CRC Press: CRC Press Boca Raton)
[22] Touhey, P., Yet another definition of chaos, Amer. Math. Monthly, 104, 411-414 (1997) · Zbl 0879.58051
[23] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0701.58001
[24] Aulbach, B.; Kieninger, B., On three definitions of chaos, Nonlinear Dyn. Syst. Theory, 1, 1, 23-37 (2001) · Zbl 0991.37010
[25] Huang, W.; Ye, X., Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos, Topology Appl., 117, 259-272 (2002) · Zbl 0997.54061
[26] 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
[27] Rudin, W., Functional Analysis (1973), McGraw-Hill: McGraw-Hill New York · Zbl 0253.46001
[28] Lang, S., Real and Functional Analysis (1973), Springer-Verlag: Springer-Verlag New York
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.