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October  2018, 38(10): 5119-5128. doi: 10.3934/dcds.2018225

Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems

Jakub Šotola

Mathematical Institute, Silesian University in Opava, Na Rybníčku 626/1, Opava, 746 01, Czech Republic

Received  December 2017 Revised  June 2018 Published  July 2018

Fund Project: The author is supported by the grant SGS/18/2016.

We study chaotic properties of uniformly convergent nonautonomous dynamical systems. We show that, contrary to the autonomous systems on the compact interval, positivity of topological sequence entropy and occurrence of Li-Yorke chaos are not equivalent, more precisely, neither of the two possible implications is true.

Keywords: Nonautonmous dynamical systems, discrete dynamical systems, Li-Yorke chaos, topological sequence entropy, topological entropy, chaos, topological dynamics.
Mathematics Subject Classification: Primary: 54H20, 37B40, 37B55; Secondary: 37B10.
Citation: Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.  Google Scholar

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R. Bowen, Periodic points and measures for axiom A diffeomorphims, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

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A. M. Bruckner and J. Smítal, A characterization of $ω$-limit sets of maps of the interval with zero topological entropy, Ergod. Theory and Dyn. Syst., 13 (1993), 7-19.  doi: 10.1017/S0143385700007173.  Google Scholar

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M. N. Burattini, F. A. B. Coutinho, L. F. Lopez and E. Massad, Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue, Bull. Math. Biol., 68 (2006), 2263-2282.  doi: 10.1007/s11538-006-9108-6.  Google Scholar

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J. S. Cánovas, Li-Yorke chaos in a class of nonautonmous discrete systems, J. Difference Equ. Appl., 17 (2011), 479-486.  doi: 10.1080/10236190903049025.  Google Scholar

[7]

T. Downarowicz, Positive topological entropy implies chaos DC2, Proc. Amer. Math. Soc., 142 (2014), 137-149.  doi: 10.1090/S0002-9939-2013-11717-X.  Google Scholar

[8]

J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 4649-4652.  doi: 10.1016/j.cnsns.2012.06.005.  Google Scholar

[9]

G. L. Forti, L. Paganoni and J. Smítal, Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull. Austral. Math. Soc., 59 (1999), 1-20.  doi: 10.1017/S000497270003255X.  Google Scholar

[10]

N. Franzová and J. Smítal, Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc., 112 (1991), 1083-1086.  doi: 10.1090/S0002-9939-1991-1062387-3.  Google Scholar

[11]

S. Gao, Y. Liu and Y. Zhang, Analysis of a nonautonomous model for migratory birds with saturation incidence rate, Commun. Nonlinear. Sci. Numer. Simul., 17 (2012), 1659-1672.  doi: 10.1016/j.cnsns.2011.08.040.  Google Scholar

[12]

T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.  doi: 10.1112/plms/s3-29.2.331.  Google Scholar

[13]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random & Computational Dynamics, 4 (1996), 205-233.   Google Scholar

[14]

J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.  doi: 10.1090/S0002-9947-1986-0849479-9.  Google Scholar

[15]

J. Šotolová, Topological Sequence Entropy of Nonautonomous Dynamical Systems, Diploma thesis, Silesian University in Opava, 2016. Google Scholar

[16]

M. Štefánková, Inheriting of chaos in uniformly convergent nonautonmous dynamical systems on the interval, Discrete Contin. Dyn. Syst., 36 (2016), 3435-3443.  doi: 10.3934/dcds.2016.36.3435.  Google Scholar

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.  Google Scholar

[3]

R. Bowen, Periodic points and measures for axiom A diffeomorphims, Trans. Amer. Math. Soc., 154 (1971), 377-397.  doi: 10.2307/1995452.  Google Scholar

[4]

A. M. Bruckner and J. Smítal, A characterization of $ω$-limit sets of maps of the interval with zero topological entropy, Ergod. Theory and Dyn. Syst., 13 (1993), 7-19.  doi: 10.1017/S0143385700007173.  Google Scholar

[5]

M. N. Burattini, F. A. B. Coutinho, L. F. Lopez and E. Massad, Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue, Bull. Math. Biol., 68 (2006), 2263-2282.  doi: 10.1007/s11538-006-9108-6.  Google Scholar

[6]

J. S. Cánovas, Li-Yorke chaos in a class of nonautonmous discrete systems, J. Difference Equ. Appl., 17 (2011), 479-486.  doi: 10.1080/10236190903049025.  Google Scholar

[7]

T. Downarowicz, Positive topological entropy implies chaos DC2, Proc. Amer. Math. Soc., 142 (2014), 137-149.  doi: 10.1090/S0002-9939-2013-11717-X.  Google Scholar

[8]

J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlinear. Sci. Numer. Simulat., 17 (2012), 4649-4652.  doi: 10.1016/j.cnsns.2012.06.005.  Google Scholar

[9]

G. L. Forti, L. Paganoni and J. Smítal, Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull. Austral. Math. Soc., 59 (1999), 1-20.  doi: 10.1017/S000497270003255X.  Google Scholar

[10]

N. Franzová and J. Smítal, Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc., 112 (1991), 1083-1086.  doi: 10.1090/S0002-9939-1991-1062387-3.  Google Scholar

[11]

S. Gao, Y. Liu and Y. Zhang, Analysis of a nonautonomous model for migratory birds with saturation incidence rate, Commun. Nonlinear. Sci. Numer. Simul., 17 (2012), 1659-1672.  doi: 10.1016/j.cnsns.2011.08.040.  Google Scholar

[12]

T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.  doi: 10.1112/plms/s3-29.2.331.  Google Scholar

[13]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random & Computational Dynamics, 4 (1996), 205-233.   Google Scholar

[14]

J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.  doi: 10.1090/S0002-9947-1986-0849479-9.  Google Scholar

[15]

J. Šotolová, Topological Sequence Entropy of Nonautonomous Dynamical Systems, Diploma thesis, Silesian University in Opava, 2016. Google Scholar

[16]

M. Štefánková, Inheriting of chaos in uniformly convergent nonautonmous dynamical systems on the interval, Discrete Contin. Dyn. Syst., 36 (2016), 3435-3443.  doi: 10.3934/dcds.2016.36.3435.  Google Scholar

Figure 1.  Sketches of maps from proof of Lemma 3.1
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Figure 2.  Sketches of limit map f and its perturbation, dotted lines are unspecified parts of the graph
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