A Study on Some Properties for Weak Stability of Non-Autonomous Discrete Dynamical Systems
DOI:
https://doi.org/10.46754/jmsi.2023.12.005Keywords:
Topological dynamics, Induced systems, Weak stability (stableness), Non-autonomous discrete dynamical systems (NADDS), Shadowing propertyAbstract
The interest for non-autonomous discrete dynamical systems has been increasing in recent years, because they are adequate to tell of real activities. For examples, when the mapping is disturbed in each iteration because of external factors, or model of some phenomena in physics, biology and economy, in specific, the population of human and weather forecasting, plus to solve problems generated in mathematics. In mathematics, stability theory addresses the stability of solutions of trajectories of dynamical systems under small perturbations of initial conditions. Besides that, the topological dynamics is the main method used in this paper, since we studied the non-autonomous discrete dynamical systems on a topological space. Next, we present a conception of weak stability (stableness) of non-autonomous discrete dynamical systems (NADDS). We evaluate a collection of weak stable points, and survey the connection among weak stableness and shadowing property. We also consider the connection among weak stableness of a non-autonomous discrete dynamical system and its induced system. In conclusion, the results of this paper will provide some helps in the modelling of a general dynamical system.
References
Acosta, G., & Sanchis, M. (2019). A note on nonautonomous discrete dynamical systems. In Descriptive Topology and Functional Analysis II (pp. 29-41). Springer.
Balibrea, F., & Oprocha, P. (2012). Weak mixing and chaos in nonautonomous discrete systems. Applied Mathematics Letters, 25, 1135-1141.
Banks, J. (2005). Chaos for induced hyperspace maps. Chaos, Solitons Fractals, 25(3), 681-685. https://doi.org/10.1016/j.chaos.2004.11.089
Bauer, W., & Sigmund, K. (1975). Topological dynamics of transformations induced on the space of probability measures. Monatshefte fr Mathematik, 79, 81-92.
Bernardes Jr., N. C., & Vermersch, R. M. (2016). On the dynamics of induced maps on the space of probability measures. Transactions of the American Mathematical Society, 368(11), 7703-7725.
Bernardes Jr., N. C., Peris, A., & Rodenas, F. (2017). Set-valued chaos in linear dynamics. Integral Equations Operator Theory, 88, 451-463.
Cánovas, J. S. (2011). Li-Yorke chaos in a class of nonautonomous discrete systems. Journal of Difference Equations and Applications, 17, 479-486.
Daghar, A., & Marzougui, H. (2023). Dynamics of monotone maps on regular curves. Topology and Its Applications, 324, 108352.
Dvoráková, J. (2012). Chaos in nonautonomous discrete dynamical systems. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4649-4652. https://doi.org/10.1016/j.cnsns.2012.06.005
Fedeli, A. (2005). On chaotic set-valued discrete dynamical systems. Chaos, Solitons Fractals, 23, 1381-1384.
Fernández, L., & Good, C. (2016). Shadowing for induced maps of hyperspaces. Fundamenta Mathematicae, 235, 277-286.
Garg, M., & Das, R. (2016). Relations of the almost average shadowing property with ergodicity and proximality. Chaos, Solitons Fractals, 91, 430-433.
Guirao, J. L. G., Kwietniak, D., Lampart, M., Oprocha, P., & Peris, A. (2009). Chaos on hyperspaces. Nonlinear Analysis Theory, Methods & Applications, 71, 1-8.
Honary, B., & Bahabadi, A. Z. (2008). Orbital shadowing property. Bulletin of the Korean Mathematical Society, 45, 645-650.
Huang, Q. L., Shi, Y. M., & Zhang, L. J. (2015). Sensitivity of non-autonomous discrete dynamical systems. Applied Mathematical Letters, 39, 31-34.
Kawaguchi, N. (2016). Entropy points of continuous maps with the sensitivity and the shadowing property. Topology and its Applications, 210, 8-15.
Kolyada, S., & Snoha, L. (1996). Topological entropy of nonautonomous dynamical systems. Random & Computational Dynamics, 4(2/3), 205-233.
Kwietniak, D., & Oprocha, P. (2007). Topological entropy and chaos for maps induced on hyperspaces. Chaos, Solitons Fractals, 33, 76-86.
Lan, Y. Y., & Peris, A. (2018). Weak stability of non-autonomous discrete dynamical systems. Topology and its Applications, 250, 53-60.
Li, J., Yan, K. S., & Ye, X. D. (2015). Recurrence properties and disjointness on the induced spaces. Discrete and Continuous Dynamical Systems, 45, 1059-1073.
Liu, H., Shi, E. H., & Liao, G. F. (2009). Sensitivity of set-valued discrete systems. Nonlinear Analysis Theory, Methods & Applications, 71, 6122-6125.
Liu, H., Lei, F. C., & Wang, L. D. (2013). Li-Yorke sensitivity of set-valued discrete systems. Journal of Applied Mathematics, 2013, 1-6. https://doi.org/10.1155/2013/260856
Meiss, J. (2007). Dynamical systems. Scholarpedia, 2(2), 1629. Retrieved from http://www.scholarpedia.org/article/Dynamical_systems.
Miralles, A., Murillo-Arcila, M., & Sanchis, M. (2018). Sensitive dependence for nonautonomous discrete dynamical systems. Journal of Mathematical Analysis and Applications, 463, 268-275.
Murillo-Arcila, M., & Peris, A. (2013). Mixing properties for nonautonomous linear dynamics and invariant sets. Applied Mathematical Letters, 26, 215-218.
Niu, Y. X. (2011). The average-shadowing property and strong ergodicity. Journal of Mathematical Analysis and Applications, 376, 528-534.
Peris, A. (2005). Set-valued discrete chaos. Chaos, Solitons Fractals, 26, 19-23.
Pilyugin, S. Yu. (1997). Shadowing in structurally stable flows. Journal of Differential Equations, 140, 238-265.
Pilyugin, S. Yu., & Plamenevskaya, O. B. (1999). Shadowing is generic. Topology and Its Applications, 97, 253-266.
Rasouli, H. (2016). On the shadowing property of nonautonomous discrete systems. International Journal of Nonlinear Analysis and Applications, 7(1), 271-277.
Sakai, K. (2003). Various shadowing properties for positively expansive maps. Topology and Its Applications, 131, 15-31.
Schermerhorn, A. C., & Cummings, M. E. (2008). Transactional family dynamics: A new framework for conceptualizing family influence processes. Advances in Child Development and Behaviour, 36, 187-250.
Scheinerman, & Edward, R. (1996). Invitation to dynamical systems, New Jersey: Prentice-Hall Inc.,
Shi, Y. M., & Chen, G. R. (2007). Chaos of time-varying discrete dynamical systems. Journal of Difference Equations and Applications, 15(5), 429-449.
Wang, Y. G., & Wei, G. (2007). Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. Topology and its Applications, 155, 56-68.
Wang, Y. G., Wei, G., & Campbell, W. H. (2009). Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. Topology and Its Applications, 156, 803-811.
Willard, S. (1970). General topology. Addison-Wesley Publishing Company, Inc.
Wu, X. X., Oprocha, P., & Chen, G. R. (2016). On various definitions of shadowing with average error in tracing. Nonlinearity, 29(7), 1942-1972.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 Journal of Mathematical Sciences and Informatics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
