Distributional chaos in a sequence and topologically weak mixing for nonautonomous discrete dynamical systems
-
2125
Downloads
-
4297
Views
Authors
Yu Zhao
- School of Mathematics and Computer Science , Guangdong Ocean University, Zhanjiang, 524025, P. R. China.
Risong Li
- School of Mathematics and Computer Science , Guangdong Ocean University, Zhanjiang, 524025, P. R. China.
Hongqing Wang
- School of Mathematics and Computer Science , Guangdong Ocean University, Zhanjiang, 524025, P. R. China.
Haihua Liang
- School of Mathematics and Computer Science , Guangdong Ocean University, Zhanjiang, 524025, P. R. China.
Abstract
Assume that \((W, g_{1,\infty})\) is a
nonautonomous discrete dynamical system given by sequences \((g_{m})_{m=1}^{\infty}\) of continuous maps on the space \((W,d)\).
In this paper, it is proven that if \(g_{1, \infty}\) is topologically weakly mixing and satisfies that
\(g_{1}^{n}\circ g_{1}^{m}=g_{1}^{n+m}\) for any \(n,m\in\{0,1,\ldots\}\), then it is distributional chaos in a sequence.
This result extends the existing one.
Share and Cite
ISRP Style
Yu Zhao, Risong Li, Hongqing Wang, Haihua Liang, Distributional chaos in a sequence and topologically weak mixing for nonautonomous discrete dynamical systems, Journal of Mathematics and Computer Science, 20 (2020), no. 1, 14--20
AMA Style
Zhao Yu, Li Risong, Wang Hongqing, Liang Haihua, Distributional chaos in a sequence and topologically weak mixing for nonautonomous discrete dynamical systems. J Math Comput SCI-JM. (2020); 20(1):14--20
Chicago/Turabian Style
Zhao, Yu, Li, Risong, Wang, Hongqing, Liang, Haihua. "Distributional chaos in a sequence and topologically weak mixing for nonautonomous discrete dynamical systems." Journal of Mathematics and Computer Science, 20, no. 1 (2020): 14--20
Keywords
- Chaotic in the sense of Devaney
- topologically transitive
- sensitive
- nonautonomous discrete dynamical systems
- distributional chaos in a sequence
MSC
References
-
[1]
J. Banks, J. Brooks, G. Cairns, G. Davis, P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly, 99 (1992), 332--334
-
[2]
J. S. Cánovas, Li--Yorke chaos in a class of nonautonomous discrete systems, J. Differnce Equ. Appl., 17 (2011), 479--486
-
[3]
R. L. Devaney, An introduction to chaotic dynamical systems, Addison-Wesley Publishing Co., Redwood City (1989)
-
[4]
J. DvoĆáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 4649--4652
-
[5]
W. Huang, X. D. Ye, Devaney's chaos or $2$-scattering implies Li-Yorke's chaos, Topology Appl., 117 (2002), 259--272
-
[6]
S. Kolyada, M. Misiurewicz, L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval, Fund. Math., 160 (1999), 161--181
-
[7]
S. Kolyada, S. Snoha, Topological entropy of nonautononous dynamical systems, Random Comput. Dyn., 4 (1996), 205--233
-
[8]
R. S. Li, A note on the three versions of distributional chaos, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1993--1997
-
[9]
R. S. Li, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2815--2823
-
[10]
R. S. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals, 45 (2012), 753--758
-
[11]
R. S. Li, A note on uniform convergence and transitivity, Chaos Solitons Fractals, 45 (2012), 759--764
-
[12]
R. S. Li, The large deviations theorem and ergodic sensitivity, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 819--825
-
[13]
R. S. Li, A note on chaos and the shadowing property, Int. J. Gen. Syst., 45 (2016), 675--688
-
[14]
R. S. Li, Several sufficient conditions for a map and a semi-flow to be ergodically sensitive, Dyn. Syst., 33 (2018), 348--360
-
[15]
R. S. Li, T. X. Lu, Chaos in a topologically transitive semi-flow, J. Nonlinear Sci. Appl., 10 (2017), 1675--1682
-
[16]
T.-Y. Li, J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985--992
-
[17]
R. S. Li, Y. Zhao, H. Q. Wang, Furstenberg families and chaos on uniform limit maps, J. Nonlinear Sci. Appl., 10 (2017), 805--816
-
[18]
R. S. Li, Y. Zhao, H. Q. Wang, R. Jiang, H. H. Liang, $\mathcal{F}$-sensitivity and $(\mathcal{F}_{1},\mathcal{F}_{2})$-sensitivity between dynamical systems and their induced hyperspace dynamical systemsH. Liang, J. Nonlinear Sci. Appl., 10 (2017), 1640--1651
-
[19]
R. S. Li, X. L. Zhou, A note on chaos in product maps, Turkish J. Math., 37 (2013), 665--675
-
[20]
H. Liu, L. D. Wang, Z. Y. Chu, Devaney's chaos implies distributional chaos in a sequence, Nonlinear Anal., 71 (2009), 6144--6147
-
[21]
T. X. Lu, G. R. Chen, Proximal and syndetical properties in nonautonomous discrete systems, J. Appl. Anal. Comput., 7 (2017), 92--101
-
[22]
J.-H. Mai, Devaney's chaos implies existence of $s$--scrambled sets, Proc. Amer. Math. Soc., 132 (2004), 2761--2767
-
[23]
F. Martinez-Giménez, P. Oprocha, A. Peris, Distributional chaos for backward shifts, J. Math. Anal. Appl., 351 (2009), 607--615
-
[24]
B. Schweizer, J. Smital, Measures of chaos and spectral decomposition of dynamical systems of the interval, Trans. Amer. Math. Soc., 344 (1994), 737--754
-
[25]
X. Tang, G. R. Chen, T. X. Lu, Some Iterative Properties of $(\mathcal{F}_{1}, \mathcal{F}_{2})$-Chaos in Non-Autonomous Discrete Systems, Entropy, 2018 (2018), 9 pages
-
[26]
L. D. Wang, G. Huang, S. M. Huan, Distributional chaos in a sequence, Nonlinear Anal., 67 (2007), 2131--2136
-
[27]
L. D. Wang, Y. Yang, Z. Y. Chu, G. F. Liao, Weakly mixing implies distributional chaos in a sequence, Modern Phys. Lett. B, 24 (2010), 1595--1660
-
[28]
R.-S. Yang, Distribution chaos in a sequence and topologically mixing, Acta Math. Sinica (Chin. Ser.), 45 (2002), 752--758