Periodic attractors of nonautonomous flat-topped tent systems

Luís Silva

doi:10.3934/dcdsb.2018243

Article Contents

2019, Volume 24, Issue 4: 1867-1874. Doi: 10.3934/dcdsb.2018243

This issue Previous Article Spreading-vanishing dichotomy in information diffusion in online social networks with intervention Next Article Evolutionarily stable dispersal strategies in a two-patch advective environment

Periodic attractors of nonautonomous flat-topped tent systems

Luís Silva^,

ISEL - Instituto Superior de Engenharia de Lisboa, Mathematics Department and CIMA - Research Centre for Mathematics and Applications, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal

* Corresponding author: lfs@adm.isel.pt
* Corresponding author: lfs@adm.isel.pt

We would like to thank the referee for several valuable suggestions

Received: November 30, 2017

Revised: February 28, 2018

Early access: August 2018

Published: January 2019

The author was partially supported by FCT Fundação para a Ciência e a Tecnologia, Portugal, through the project UID/MAT/04674/2013, CIMA and ISEL.

Abstract Full Text(HTML) Figure(2) Related Papers Cited by

Abstract

In this work we will consider a family of nonautonomous dynamical systems $x_{k+1} = f_k(x_k,\lambda)$, $\lambda \in [-1,1]^{\mathbb{N}_0}$, generated by a one-parameter family of flat-topped tent maps $g_{\alpha}(x)$, i.e., $f_k(x,\lambda) = g_{\lambda_k}(x)$ for all $k\in \mathbb{N}_0$. We will reinterpret the concept of attractive periodic orbit in this context, through the existence of some periodic, invariant and attractive nonautonomous sets and establish sufficient conditions over the parameter sequences for the existence of such periodic attractors.

Keywords:

Mathematics Subject Classification: Primary: 37B55, 37E05, 37G35; Secondary: 37E15.

Citation:

Full Text(HTML)

Figure 1. Construction of the sets $A_i$ for $X=RL0$, $X^R=(RLL)^{\infty}$ and $X^L=RLR(RLL)^{\infty}$

Download: Full-size image PowerPoint slide

Figure 2. A bifurcation diagram with a sequence of nonautonomous $p$-periodic $\lambda$-attractors with period doubling periods $p = 3,6,\ldots$, for a sequence of sequences $\lambda$, such that $\lambda_{3n+1} = \lambda_{3n+2} = 1$, $\lambda_0$ varies from $-0.6$ to $-0.5$ and $\lambda_{3n} = \lambda_0 +0.01r_n$, where $r_n$ ia a random integer between $0$ and $9$

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	N. Franco, L. Silva and P. Simões, Symbolic dynamics and renormalization of nonautonomous $k$ periodic dynamical systems, Journal of Difference Equations and Applications, 19 (2013), 27-38. doi: 10.1080/10236198.2011.611804.
[2]	J. Franke and A. Yakubu, Population models with periodic recruitment functions and survival rates, Journal of Difference Equations and Applications, 11 (2005), 1169-1184. doi: 10.1080/10236190500386275.
[3]	L. Glass and W. Zeng, Bifurcations in flat-topped maps and the control of cardiac chaos, International Journal of Bifurcation and Chaos, 4 (1994), 1061-1067. doi: 10.1142/S0218127494000770.
[4]	J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys., 209 (2000), 123-178. doi: 10.1007/s002200050018.
[5]	C. Pötzsche, Bifurcations in nonautonomous dynamical systems: Results and tools in discrete time, in Proceedings of the International Workshop Future Directions in Difference Equations (eds. E. Liz and V. Mañosa), Universidade de Vigo, Vigo, 69 (2011), 163-212.
[6]	L. Silva, J. L. Rocha and M. T. Silva, Bifurcations of 2-periodic nonautonomous stunted tent systems, Int. J. Bifurcation Chaos, 27 (2017), 1730020 [17 pages]. doi: 10.1142/S0218127417300208.
[7]	A. Rădulescu, The connected isentropes conjecture in a space of quartic polynomials, Discrete Contin. Dyn. Syst., 19 (2007), 139-175. doi: 10.3934/dcds.2007.19.139.
[8]	C. Wagner and R. Stoop, Renormalization approach to optimal limiter control in 1-D chaotic systems, Journal of Statistical Physics, 106 (2002), 97-106. doi: 10.1023/A:1013120112236.