Specification properties for non-autonomous discrete systems
Keywords
Non-autonomous discrete system, induced systems, specification, topological mixingAbstract
In this paper notions of strong specification property and quasi-weak specification property for non-autonomous discrete systems are introduced and studied. It is shown that these properties are dynamical properties and are preserved under finite product. It is proved that a $k$-periodic non-autonomous system on intervals having weak specification is Devaney chaotic. Moreover, it is shown that if the system has strong specification then the result is true in general. Specification properties of induced systems on hyperspaces and probability measures spaces are also studied. Examples/counterexamples are provided wherever necessary to support results obtained.References
S. Bartoll, F. Martı́ nez Giménez, and A. Peris, The specification property for backward shifts, J. Difference Equ. Appl. 18 (2012), 599–605.
W. Bauer and K. Sigmund, Topological dynamics of transformations induced on the space of probability measures, Monatsh. Math. 79 (1975), 81–92.
R. Bowen, Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc. 154 (1971), 377–397.
F.A.B. Coutinho, M.N. Burattini, 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.
M. Dateyama, The almost weak specification property for ergodic group automorphisms of abelian groups, J. Math. Soc. Japan 42 (1990), 341–351.
P.J. Huber and E.M. Ronchetti, Robust Statistics, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, second ed., 2009.
A. Illanes and S.B. Nadler, Jr., Hyperspaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 216, Marcel Dekker, Inc., New York, 1999.
S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam. 4 (1996),205–233.
M. Kulczycki, D. Kwietniak and P. Oprocha, On almost specification and average shadowing properties, Fund. Math. 224 (2014), 241–278.
T. Y. Li and J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992.
M. Mazur and P. Oprocha, Subshifts, rotations and the specification property, Topol. Methods Nonlinear Anal. 46 (2015), 799–812.
R. Memarbashi and H. Rasuli, Notes on the dynamics of nonautonomous discrete dynamical systems, J. Adv. Res. Dyn. Control Syst. 6 (2014), 8–17.
A. Miralles, M. Murillo-Arcila and M. Sanchis, Sensitive dependence for nonautonomous discrete dynamical systems, J. Math. Anal. Appl. 463 (2018), 268–275.
C.-E. Pfister and W.G. Sullivan, Large deviations estimates for dynamical systems without the specification property. Applications to the β-shifts, Nonlinearity, 18 (2005), 237–261.
M. Salman and R. Das, Multi-sensitivity and other stronger forms of sensitivity in nonautonomous discrete systems, Chaos Solitons Fractals 115 (2018), 341–348.
M. Salman and R. Das, Dynamics of weakly mixing nonautonomous systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 29 (2019), 1950123 (11 pp.).
I. Sánchez, M. Sanchis and H. Villanueva, Chaos in hyperspaces of nonautonomous discrete systems, Chaos Solitons Fractals 94 (2017), 68–74.
J.N. Sarkooh and F. Ghane, Specification and thermodynamic properties of nonautonomous dynamical systems, arXiv preprint arXiv:1712.06109, (2017).
S. Shah, R. Das and T. Das, Specification property for topological spaces, J. Dynam. Control Syst. 22 (2016), 615–622.
P. Sharma and M. Raghav, Dynamics of non-autonomous discrete dynamical systems, Topology Proc. 52 (2018), 45–59.
K. Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc. 190 (1974), 285–299.
D. Thompson, The irregular set for maps with the specification property has full topological pressure, Dyn. Syst. 25 (2010), 25–51.
C. Tian and G. Chen, Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals 28 (2006), 1067–1075.
R. Vasisht and R. Das, On stronger forms of sensitivity in non-autonomous systems, Taiwanese J. Math. 22 (2018), 1139–1159.
Y. Zhang, S. Gao and Y. Liu, Analysis of a nonautonomous model for migratory birds with saturation incidence rate, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1659–1672.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
