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

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.

Citation: Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
References:
[1]

R. L. AdlerA. 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. BlanchardE. GlasnerS. 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. BurattiniF. A. B. CoutinhoL. 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. FortiL. 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. GaoY. 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. AdlerA. 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. BlanchardE. GlasnerS. 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. BurattiniF. A. B. CoutinhoL. 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. FortiL. 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. GaoY. 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
Figure 2.  Sketches of limit map f and its perturbation, dotted lines are unspecified parts of the graph
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