PaperNo Access

Sensitivity and Chaoticity on Nonautonomous Dynamical Systems

College of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, Hebei 050064, P. R. China

E-mail Address: liyufeng177@163.com

Corresponding author.

Search for more papers by this author

and

Lidong Wang

Zhuhai College of Jilin University, Zhuhai, Guangdong 519041, P. R. China

School of Science, Dalian MinZu University, Dalian, Liaoning 116600, P. R. China

E-mail Address: wld0707@dlut.edu.cn

Search for more papers by this author

https://doi.org/10.1142/S0218127420501461Cited by:5 (Source: Crossref)

Abstract

This paper discusses the sensitivity and chaotic properties of nonautonomous dynamical systems. For the first time we obtain sufficient conditions of some stronger forms of sensitivity for nonautonomous dynamical systems. Then, some chaotic relations between two nonautonomous dynamical systems are proved.

Keywords:

References

Alves, J. F., Málek, M. & Silva, L. [2015] “ Spectral invariants of periodic nonautonomous discrete dynamical systems,” J. Math. Anal. Appl. 430, 85–97. Web of Science, Google Scholar
Balibrea, F. & Oprocha, P. [2012] “ Weak mixing and chaos in non-autonomous discrete systems,” Appl. Math. Lett. 25, 1135–1141. Web of Science, Google Scholar
Bao, C., Hao, H., Li, Z. & Zhu, X. [2009] “ Time-varying system identification using a newly improved HHT algorithm,” Comput. Struct. 87, 1611–1623. Web of Science, Google Scholar
Cánovas, J. S. [2006] “ On $ω$ -limit sets of non-autonomous discrete systems,” J. Diff. Eqs. Appl. 12, 95–100. Web of Science, Google Scholar
Cánovas, J. S. [2010] “ Recent results on non-autonomous discrete systems,” SeMA J. 51, 33–40. Google Scholar
Cánovas, J. S. [2011] “ Li–Yorke chaos in a class of non-autonomous discrete systems,” J. Diff. Eqs. Appl. 17, 479–486. Web of Science, Google Scholar
Cánovas, J. S. [2013] “ On entropy of non-autonomous discrete systems,” Prog. Chall. Dyn. Syst. 54, 143–159. Google Scholar
Dvořáková, J. [2012] “ Chaos in nonautonomous discrete dynamical systems,” Commun. Nonlin. Sci. Numer. Simul. 17, 4649–4652. Web of Science, Google Scholar
Froyland, G., Lloyd, S. & Santitissadeekorn, N. [2010] “ Coherent sets for nonautonomous dynamical systems,” Physica D 239, 1527–1541. Web of Science, Google Scholar
Huang, Q., Shi, Y. & Zhang, L. [2015] “ Sensitivity of non-autonomous discrete dynamical systems,” Appl. Math. Lett. 39, 31–34. Web of Science, Google Scholar
Khan, A. & Kumar, P. [2013] “ Chaotic phenomena and non-autonomous dynamical system,” Glob. J. Theor. Appl. Math. Sci. 3, 31–39. Google Scholar
Kolyada, S. & Snoha, L. [1996] “ Topological entropy of non-autononous dynamical systems,” Fund. Math. 4, 205–233. Google Scholar
Kolyada, S., Snoha, L. & Trofimchuk, S. [2004] “ On minimality of nonautonomous dynamical systems,” Nonlin. Oscillat. 7, 83–89. Google Scholar
Li, R. [2012] “ A note on uniform convergence and transitivity,” Chaos Solit. Fract. 45, 759–764. Web of Science, Google Scholar
Li, J. & Ye, X. [2016] “ Recent development of chaos theory in topological dynamics,” Acta Math. Sci. Ser. B (Engl. Ed.) 32, 83–114. Web of Science, Google Scholar
Li, J., Oprocha, P. & Wu, X. [2017] “ Furstenberg families, sensitivity and the space of probability measures,” Nonlinearity 30, 987–1005. Web of Science, Google Scholar
Lin, X., Lan, W., Li, Z. & Li, H. [2016] “ The integral form fluid momentum theorem on time-varying systems and its application to aerodynamic force coefficient analysis,” Appl. Math. Mech. 37, 551–566. Google Scholar
Ma, C., Zhu, P. & Li, R. [2016] “ On iteration invariants for $(ℱ_{1}, ℱ_{2})$ -sensitivity and weak $(ℱ_{1}, ℱ_{2})$ -sensitivity of non-autonomous discrete systems,” J. Nonlin. Sci. Appl. 9, 5772–5779. Google Scholar
Murillo-Arcila, M. & Peris, A. [2013] “ Mixing properties for nonautonomous linear dynamics and invariant sets,” Appl. Math. Lett. 26, 215–218. Web of Science, Google Scholar
Oprocha, P. & Wilczyński, P. [2014] “ Chaos in non-autonomous dynamical systems,” An. Şt. Univ. Ovidius Constanţa 17, 209–221. Google Scholar
Oprocha, P. & Wu, X. [2017] “ On average tracing of periodic average pseudo orbits,” Discr. Contin. Dyn. Syst. 37, 4943–4957. Web of Science, Google Scholar
Pourbarat, M. & Abbasi, N. [2015] “ Nonautonomous discrete systems: Density of orbits and transitivity,” Int. J. Bifurcation and Chaos 25, 1550065-1–6. Link, Web of Science, Google Scholar
Puneet, S. & Manish, R. [2015] “ Dynamics of nonautonomous discrete dynamical systems,” Int. J. Bifurcation and Chaos 21, 3359–3371. Google Scholar
Rasuli, H. & Memarbashi, R. [2017] “ On the relation of shadowing and expansivity in non-autonomous discrete systems,” Anal. Th. Appl. 33, 11–19. Google Scholar
Shao, H., Shi, Y. & Zhu, H. [2016] “ Estimations of topological entropy for non-autonomous discrete systems,” J. Diff. Eqs. Appl. 22, 474–484. Web of Science, Google Scholar
Shi, Y. & Chen, G. [2009] “ Chaos of time-varying discrete dynamical systems,” J. Diff. Eqs. Appl. 15, 429–449. Web of Science, Google Scholar
Shi, Y. [2012] “ Chaos in non-autonomous discrete dynamical systems approached by their induced systems,” Int. J. Bifurcation and Chaos 22, 1250284-1–12. Link, Web of Science, Google Scholar
Song, X., Liu, J. & Wang, L. [2012] “ Ruelle–Takens chaos in non-autonomous dynamical systems,” Eng. Math. Lett. 1, 65–74. Google Scholar
Tian, C. & Chen, G. [2006] “ Chaos of a sequence of maps in a metric space,” Chaos Solit. Fract. 28, 1067–1075. Web of Science, Google Scholar
Wang, L., Huang, G. & Huan, S. [2007] “ Distributional chaos in a sequence,” Nonlin. Anal. 67, 2131–2136. Web of Science, Google Scholar
Wang, L., Li, Y., Gao, Y. & Liu, H. [2013] “ Distributional chaos of time-varying discrete dynamical systems,” Ann. Polon. Math. 107, 49–57. Web of Science, Google Scholar
Wu, X. & Zhu, P. [2013] “ Chaos in a class of non-autonomous discrete system,” Appl. Math. Lett. 26, 431–436. Web of Science, Google Scholar
Wu, X. [2017] “ A remark on topological sequence entropy,” Int. J. Bifurcation and Chaos 27, 1750107-1–7. Link, Web of Science, Google Scholar
Wu, X. & Chen, G. [2019] “ Answering two open problems on banks theorem for non-autonomous dynamical systems,” J. Diff. Eqs. Appl., https://doi.org/10.1080/10236198.2019.1696324. Google Scholar
Wu, X. & Zhang, X. [2019] “ Relationship among several types of sensitivity in general semi-flows,” Semigroup Forum, https://doi.org/10.1007/s00233-019-10072-7. Web of Science, Google Scholar
Wu, X., Liang, S., Luo, Y., Xin, M. & Zhang, X. [2019a] “ A remark on the limit shadowing property for iterated function systems,” U.P.B. Sci. Bull., Ser. A 81, 107–114. Google Scholar
Wu, X., Luo, Y., Ma, X. & Lu, T. [2019b] “ Rigidity and sensitivity on uniform spaces,” Topol. Appl. 252, 145–157. Web of Science, Google Scholar
Wu, X., Luo, Y., Wang, L. & Liang, J. [2019c] “ $(ℱ_{1}, ℱ_{2})$ -chaos and sensitivity for time-varying discrete systems,” U.P.B. Sci. Bull., Ser. A 81, 149–156. Google Scholar
Wu, X., Ma, X., Chen, G. & Lu, T. [2019d] “ A note on the sensitivity of semiflows,” Topol. Appl. (in Press). Web of Science, Google Scholar
Wu, X., Liang, S., Ma, X., Lu, T. & Ahmadi, S. A. [2020] “ The mean sensitivity and mean equicontinuity in uniform spaces,” Int. J. Bifurcation and Chaos 30, 2050122-1–11. Link, Web of Science, Google Scholar
Zhu, Y., Liu, Z. & Xu, X. [2012] “ Entropy of nonautonomous discrete dynamical systems,” J. Korean Math. Soc. 49, 822–826. Web of Science, Google Scholar