Figure 1.
Sketches of maps from proof of Lemma 3.1
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Lyapunov exponents of cocycles over non-uniformly hyperbolic systems
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 |
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 and Continuous Dynamical Systems,
2018, 38
(10)
: 5119-5128.
doi: 10.3934/dcds.2018225
![]()
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Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc., 112 (1991), 1083-1086.
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T. N. T. Goodman,
Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.
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S. Kolyada and L. Snoha,
Topological entropy of nonautonomous dynamical systems, Random & Computational Dynamics, 4 (1996), 205-233.
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J. Smítal,
Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.
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J. Šotolová,
Topological Sequence Entropy of Nonautonomous Dynamical Systems, Diploma thesis, Silesian University in Opava, 2016. |
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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. |
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. |
| [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. |
| [3] |
R. Bowen,
Periodic points and measures for axiom A diffeomorphims, Trans. Amer. Math. Soc., 154 (1971), 377-397.
doi: 10.2307/1995452. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [12] |
T. N. T. Goodman,
Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350.
doi: 10.1112/plms/s3-29.2.331. |
| [13] |
S. Kolyada and L. Snoha,
Topological entropy of nonautonomous dynamical systems, Random & Computational Dynamics, 4 (1996), 205-233.
|
| [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. |
| [15] |
J. Šotolová,
Topological Sequence Entropy of Nonautonomous Dynamical Systems, Diploma thesis, Silesian University in Opava, 2016. |
| [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. |
Figure 2.
Sketches of limit map f and its perturbation, dotted lines are unspecified parts of the graph
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