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April  2019, 24(4): 1867-1874. doi: 10.3934/dcdsb.2018243

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

We would like to thank the referee for several valuable suggestions

Received  December 2017 Revised  March 2018 Published  August 2018

Fund Project: 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.

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: Nonautonomous dynamical systems, attractors, symbolic dynamics.
Mathematics Subject Classification: Primary: 37B55, 37E05, 37G35; Secondary: 37E15.
Citation: Luís Silva. Periodic attractors of nonautonomous flat-topped tent systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1867-1874. doi: 10.3934/dcdsb.2018243
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.  Google Scholar

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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.  Google Scholar

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show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Construction of the sets $A_i$ for $X=RL0$, $X^R=(RLL)^{\infty}$ and $X^L=RLR(RLL)^{\infty}$
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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 $9$
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