Abstract
Switched server systems are mathematical models of manufacturing, traffic and queueing systems. Recently, it was proved in (Eur J Appl Math 31(4), 682–708, 2020) that there exist switched server systems with 3 buffers (tanks), a server, filling rates \(\rho _1=\rho _2=\rho _3=\frac{1}{3}\) and parameters \(d_1, d_2, d_3>0\) whose \(\omega \)-limit set is a fractal set. In this article, we give an explicit large subset of parameters for which the corresponding switched server systems have no fractal \(\omega \)-limit set. More precisely, the Poincaré map of each system has a finite \(\omega \)-limit set. The approach we use is to study the topological dynamics of a family of piecewise \(\lambda \)-affine contractions that includes the Poincaré maps of the switched server systems as a particular case.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
References
Bugeaud, Y.: Dynamique de certaines applications contractantes, linéaires par morceaux, sur \([0,1)\). C. R. Acad. Sci. Paris Sér. I Math. 317(6), 575–578 (1993)
Bugeaud, Y.: Distribution Modulo one and Diophantine Approximation. Cambridge Tracts in Mathematics, vol. 193. Cambridge University Press, Cambridge (2012)
Bugeaud, Y.: On the expansions of a real number to several integer bases. Rev. Mat. Iberoam. 28(4), 931–946 (2012)
Bugeaud, Y., Conze, J.-P.: Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey. Acta Arith. 88(3), 201–218 (1999)
Calderon, A., Catsigeras, E., Guiraud, P.: A spectral decomposition of the attractor of piecewise-contracting maps of the interval. Ergod. Theory Dyn. Syst. 41(7), 1940–1960 (2021)
Catsigeras, E., Guiraud, P., Meyroneinc, A.: Complexity of injective piecewise contracting interval maps. Ergod. Theory Dyn. Syst. 40(1), 64–88 (2020)
Catsigeras, E., Guiraud, P., Meyroneinc, A., Ugalde, E.: On the asymptotic properties of piecewise contracting maps. Dyn. Syst. 31(2), 107–135 (2016)
Chase, C., Serrano, J., Ramadge, P.J.: Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems. IEEE Trans. Autom. Control 38(1), 70–83 (1993)
Coutinho, R., Fernandez, B., Lima, R., Meyroneinc, A.: Discrete time piecewise affine models of genetic regulatory networks. J. Math. Biol. 52(4), 524–570 (2006)
Ding, E.J., Hemmer, P.C.: Exact treatment of mode locking for a piecewise linear map. J. Stat. Phys. 46(1–2), 99–110 (1987)
Fernandes, F., Pires, B.: A switched server system semiconjugate to a minimal interval exchange. Eur. J. Appl. Math. 31(4), 682–708 (2020)
José Pedro Gaivão and Arnaldo Nogueira: Dynamics of piecewise increasing contractions. Bull. Lond. Math. Soc. 54(2), 482–500 (2022)
Gambaudo, J.-M., Tresser, C.: On the dynamics of quasi-contractions. Bol. Soc. Brasil. Mat. 19(1), 61–114 (1988)
Hata, M.: Dynamics of Caianiello’s equation. J. Math. Kyoto Univer. 22(1), 155–173 (1982/83)
Hertling, P.: Disjunctive \(\omega \)-words and real numbers. J.UCS 2(7), 549–568 (1996)
Janson, S., Öberg, A.: A piecewise contractive dynamical system and Phragmén’s election method. Bull. Soc. Math. France 147(3), 395–441 (2019)
Keener, J.P.: Chaotic behavior in piecewise continuous difference equations. Trans. Am. Math. Soc. 261(2), 589–604 (1980)
Laurent, M., Nogueira, A.: Rotation number of contracted rotations. J. Mod. Dyn. 12, 175–191 (2018)
Nagumo, J., Sato, S.: On a response characteristic of a mathematical neuron model. Kybernetik 10(3), 155–164 (1972)
Nogueira, A., Pires, B.: Dynamics of piecewise contractions of the interval. Ergod. Theory Dyn. Syst. 35(7), 2198–2215 (2015)
Nogueira, A., Pires, B., Rosales, R.A.: Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27(7), 1603–1610 (2014)
Nogueira, A., Pires, B., Rosales, R.A.: Topological dynamics of piecewise \(\lambda \)-affine maps. Ergod. Theory Dyn. Syst. 38(5), 1876–1893 (2018)
Pires, B.: Symbolic dynamics of piecewise contractions. Nonlinearity 32(12), 4871–4889 (2019)
Staiger, L.: How large is the set of disjunctive sequences? In: Combinatorics, Computability and Logic (Constanţa, 2001), Springer Ser. Discrete Math. Theor. Comput. Sci., pp. 215–225. Springer, London, (2001)
Acknowledgements
Part of this work was carried out while the first named author had a postdoctoral position in the University of São Paulo at Ribeirão Preto. He is very thankful for the excellent working conditions there. The authors are very grateful to the reviewers for their careful reading of the manuscript and their valuable comments. The third named author was partially supported by Grant #2019/10269-3 São Paulo Research Foundation (FAPESP).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Antunes, A.d.A., Bugeaud, Y. & Pires, B. Switched Server Systems Whose Parameters are Normal Numbers in Base 4. Qual. Theory Dyn. Syst. 21, 143 (2022). https://doi.org/10.1007/s12346-022-00679-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-022-00679-7