Taiwanese Journal of Mathematics

On Stronger Forms of Sensitivity in Non-autonomous Systems

Radhika Vasisht and Ruchi Das

Full-text: Open access


In this paper, some stronger forms of transitivity in a non-autonomous discrete dynamical system $(X,f_{1,\infty})$ generated by a sequence $(f_n)$ of continuous self maps converging uniformly to $f$, are studied. The concepts of thick sensitivity, ergodic sensitivity and multi-sensitivity for non-autonomous discrete dynamical systems, which are all stronger forms of sensitivity, are defined and studied. It is proved that under certain conditions, if the rate of convergence at which $(f_n)$ converges to $f$ is “sufficiently fast”, then various forms of sensitivity and transitivity for the non-autonomous system $(X,f_{1,\infty})$ and the autonomous system $(X,f)$ coincide. Also counter examples are given to support results.

Article information

Taiwanese J. Math., Volume 22, Number 5 (2018), 1139-1159.

Received: 15 December 2017
Revised: 6 April 2018
Accepted: 23 April 2018
First available in Project Euclid: 26 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx]
Secondary: 37B55: Nonautonomous dynamical systems

non-autonomous dynamical systems sensitivity transitivity


Vasisht, Radhika; Das, Ruchi. On Stronger Forms of Sensitivity in Non-autonomous Systems. Taiwanese J. Math. 22 (2018), no. 5, 1139--1159. doi:10.11650/tjm/180406. https://projecteuclid.org/euclid.twjm/1524708018

Export citation


  • M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.
  • J. Dvořáková, Chaos in nonautonomous discrete dynamical systems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 12, 4649–4652.
  • Fatmawati and H. Tasman, An optimal treatment control of TB-HIV coinfection, Int. J. Math. Math. Sci. 2016 (2016), Art. ID 8261208, 11 pp.
  • S. García-Ferreira and M. Sanchis, The Ellis semigroup of a nonautonomous discrete dynamical system, Quaest. Math. 40 (2017), no. 6, 753–767.
  • N. Kaur, M. Ghosh and S. S. Bhatia, Modeling the spread of HIV in a stage structured population: effect of awareness, Int. J. Biomath. 5 (2012), no. 5, 1250040, 18 pp.
  • S. Kolyada and L'. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996), no. 2-3, 205–233.
  • R. Li, A note on stronger forms of sensitivity for dynamical systems, Chaos Solitons Fractals 45 (2012), no. 6, 753–758.
  • ––––, A note on uniform convergence and transitivity, Chaos Solitons Fractals 45 (2012), no. 6, 759–764.
  • ––––, A note on shadowing with chain transitivity, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 7, 2815–2823.
  • ––––, The large deviations theorem and ergodic sensitivity, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 4, 819–825.
  • ––––, Several sufficient conditions for a map and a semi-flow to be ergodically sensitive, Dyn. Syst. 33 (2018), no. 2, 348–360.
  • R. Li and Y. Shi, Stronger forms of sensitivity for measure-preserving maps and semiflows on probability spaces, Abstr. Appl. Anal. 2014 (2014), Art. ID 769523, 10 pp.
  • R. Li and H. Wang, Erratum to “A note on uniform convergence and transitivity" [Chaos, Solitons and Fractals 45 (2012) 759–764], Chaos Solitons Fractals 59 (2014), 112–118.
  • R. Li, Y. Zhao and H. Wang, Furstenberg families and chaos on uniform limit maps, J. Nonlinear Sci. Appl. 10 (2017), no. 2, 805–816.
  • H. Liu, L. Liao and L. Wang, Thickly syndetical sensitivity of topological dynamical system, Discrete Dyn. Nat. Soc. 2014 (2014), Art. ID 583431, 4 pp.
  • L. Liu and Y. Sun, Weakly mixing sets and transitive sets for non-autonomous discrete systems, Adv. Difference Equ. 2014 (2014), 217–225.
  • T. Lu and G. Chen, Proximal and syndetical properties in nonautonomous discrete systems, J. Appl. Anal. Comput. 7 (2017), no. 1, 92–101.
  • R. Memarbashi and H. Rasuli, Notes on the dynamics of nonautonomous discrete dynamical systems, J. Adv. Res. Dyn. Control Syst. 6 (2014), no. 2, 8–17.
  • T. K. S. Moothathu, Stronger forms of sensitivity for dynamical systems, Nonlinearity 20 (2007), no. 9, 2115–2126.
  • P. Sharma and M. Raghav, On dynamics generated by a uniformly convergent sequence of maps, arXiv:1703.06640.
  • M. Štefánková, Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval, Discrete Contin. Dyn. Syst. 36 (2016), no. 6, 3435–3443.
  • D. Thakkar and R. Das, Topological stability of a sequence of maps on a compact metric space, Bull. Math. Sci. 4 (2014), no. 1, 99–111.
  • ––––, Spectral decomposition theorem in equicontinuous nonautonomous discrete dynamical systems, J. Difference Equ. Appl. 22 (2016), no. 5, 676–686.
  • X. Wang, X. Wu and G. Chen, Sufficient conditions for ergodic sensitivity, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3404–3408.
  • X. Wu, R. Li and Y. Zhang, The multi-$\mathcal{F}$-sensitivity and $(\mathcal{F}_1,\mathcal{F}_2)$-sensitivity for product systems, J. Nonlinear Sci. Appl. 9 (2016), no. 6, 4364–4370.
  • C. Yang and Z. Li, On the sensitivities dependence in non-autonomous dynamical systems, arXiv:1602.00075.