Abstract
In this paper we investigate Martelli’s chaos of inverse limit dynamical systems and hyperspace dynamical systems which are both induced from dynamical systems on a compact metric space. We give the implication of Martelli’s chaos among those systems. More precisely, we show that inverse limit dynamical system is Martelli’s chaos if and only if so is original system, and we prove that hyperspace dynamical system is Martelli’s chaos implies original system is Martelli’s chaos if the orbit of every single point set of original system is unstable in hyperspace dynamical system.
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Aulander J., Yorke J.A.: Interval maps, factors of maps, and chaos. Tohoku Math. J. 32, 177–188 (1980)
Banks J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals 25, 681–685 (2005)
Banks J., Brooks J., Cairs G., Stacey P.: On the Devaney’s definition of chaos. Am. Math. Monthly 99, 332–334 (1992)
Crannell A., Martelli M.: Dynamics of quasicontinuous systems. J. Differ. Equ. Appl. 6, 351–361 (2000)
Devaney R.L.: An Introduction to Chaotic Dynamical Systems. Addison-wesley, Redwood City (1989)
Engelking R.: General Topology. PWN, Warszawa (1977)
Fedeli A.: On chaotic set-valued discrete dynamical systems. Chaos Solitons Fractals 23, 1381–1384 (2005)
Forti G.L.: Various notions of chaos for discrete dynamical systems. A brief survey. Aequationes Math. 70, 1–13 (2005)
Kolyada S., Snoha L.: Some aspects of topological transitivity—a survey. Grazer Math. Ber, Bericht nr. 334, 3–35 (1997)
Li S.: Dynamical properties of the shift maps on the inverse limit spaces. Ergod. Th. Dynam. Sys. 12, 95–108 (1992)
Li T.Y., Yorke J.: Period three implies chaos. Am. Math. Monthly 82, 985–992 (1975)
Martelli M., Dang M., Seph T.: Defining chaos. Math. Mag. 71, 112–122 (1998)
Martelli, M.: Introduction to Dicrete Dynamical Systems and Chaos. Wiley- Interscience series in Discrete mathematics and Optimization, Wiley-Interscience, New York (1999)
Michael E.: Topologies on spaces of subsets. Trans. Am. Math. Soc. 71, 152–182 (1951)
Peris A.: Set-valued discrete chaos. Chaos Solitons Fractals 26, 19–23 (2005)
Robinson C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd edn. CRC Press Inc, Boca Raton (1999)
Robinson C.: What is a chaotic attractor?. Qual. Th. Dyn. Syst. 7, 227–236 (2008)
Roman-Flores H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals 71, 99–104 (2003)
Roman-Flores H.: Robinson’s chaos in set-valued discrete systems. Chaos Solitons Fractals 25, 33–42 (2005)
Vellekoop M., Bergund R.: On interval, transitivity=chaos. Am. Math. Monthly 101, 353–355 (1994)
Wang L., Chen Z., Liao G.: The complexity of a minimal sub-shift on symbolic spaces. J. Math. Anal. Appl. 317, 136–145 (2006)
Wang Y., Wei G.: Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. Top Appl. 155, 56–68 (2007)
Wang Y., Wei G., Campbell H.: Sensitive dependence on intial conditions between dynamical systems and their induced hyperspace dynamical systems. Top Appl. 156, 803–811 (2009)
Wiggins S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York (1990)
Zhang, Z.S.: Principles of Differential Dynamical Systems. Science Press, Beijing (1987) (in chinese)
Zhou, Z.L.: Symbolic Dynamical Systems. Shanghai Scientific and Technological Education Publishing House, Shanghai (1997) (in chinese)
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This work was supported by the Natural Science Foundation of Henan Province (092300410148), People’s Republic of China.
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Liu, L., Zhao, S. Martelli’s Chaos in Inverse Limit Dynamical Systems and Hyperspace Dynamical Systems. Results. Math. 63, 195–207 (2013). https://doi.org/10.1007/s00025-011-0188-8
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DOI: https://doi.org/10.1007/s00025-011-0188-8