Milnor attractors and period incrementing on pattern iterations of flat top tent maps
DOI:
https://doi.org/10.12775/TMNA.2024.023Keywords
Non-autonomous dynamical systems, interval maps, attractors, symbolic dynamics, bifurcation diagramsAbstract
In this paper, we consider the family of non-autonomous dynamical systems obtained by iterating tent maps with a flat top of constant value $u$ introduced at instants $i$ such that $s_i=0$, where $s$ is a binary sequence that we call the iteration pattern. We introduce symbolic dynamics and study the kneading invariants for these dynamical systems. More precisely, we define the kneading invariants $K(u,s)$ as symbolic sequences and study sufficient conditions for a symbolic sequence to be a kneading invariant $K(u,s)$ for some $u$, for each iteration pattern $s$. Finally, we describe the parameters $u$ for which there are Milnor attractors for iteration patterns $s$ such that $s_{np}=0$ for all $n$, as limits of parameter sequences corresponding to local attractors, organized according to a period-incrementing structure.References
V. Avrutin, B. Futter, L. Gardini and M. Schanz, Unstable orbits and Milnor attractors in the discontinuous flat top tent map, ESAIM Proc. 36 (2012), 126–158.
V. Avrutin, L. Gardini, I. Sushko and F. Tramontana, Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures, World Scientific Series on Nonlinear Science, Series A, Vol. 95, 2019.
P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston, 2009.
N.J. Corron, S.D. Pethel, and B.A. Hopper, Controlling chaos with simple limiters, Phys. Rev. Lett. 84 (2000), no. 17, 3835–3838.
M.-F. Danca, P. Bourke and M. Romera, Graphical exploration of the connectivity sets of alternated julia sets, Nonlinear Dynamics 73( 2013), no. 1–2, 1155–1163.
M.-F. Danca, M. Romera and G. Pastor Alternated julia sets and connectivity properties, Internat. J. Bifur. Chaos 19 (2009), no. 6, 2123–2129.
L. Glass and W. Zeng, Bifurcations in flat-topped maps and the control of cardiac chaos, Internat. J. Bifur. Chaos 4 (1994), 1061–1067.
X.-Z. He and F.H. Westerhoff, Commodity markets, price limiters and speculative price dynamics, J. Econ. Dyn. Control 29 (2005), 1577–1596.
F. M. Hilker and F.H. Westeroff, Paradox of simple limiter control, Phys. Rev. E 73 (2006), 052901.
J. Milnor, On the concept of attractor, Commun. Math. Phys. 99 (1985), 177–195.
J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems (College Park, MD, 1986–1987), Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563.
K. Myneni, T.A. Barr, N.J. Corron and S.D. Pethel New method for the control of fast chaotic oscillations, Phys. Rev. Lett. 83 (1999), 2175–2178.
C. Pötzsche, Bifurcations in non-autonomous dynamical systems: results and tools in discrete time, Proceedings of the International Workshop Future Directions in Difference Equations (E. Liz and V. Mañosa, eds.), Universidade de Vigo, Vigo, 2011, pp. 163–212.
L. Silva, Periodic attractors of non-autonomous flat-topped tent systems, Discrete Contin. Dyn. Sys. B 24 (2018), 1867–1874.
L. Silva, Attractors in pattern iterations of flat top tent maps, Mathematics 11 (2023), no. 12, 2677.
R. Stoop and C. Wagner, Scaling properties of simple limiter control, Phys. Rev. Lett. 90 (2003), 154101–154104.
R. Stoop and C. Wagner, Optimized chaos control with simple limiters, Phys. Rev. E 63 (2000), 17201–17203.
R. Stoop and C. Wagner, Renormalization approach to optimal limiter control in 1-D chaotic systems, J. Stat. Phys. 106 (2002), 97–106.
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Luís Silva, Teresa M. Silva
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
