American Institute of Mathematical Sciences

Advanced
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

[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

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 $X^L=RLR(RLL)^{\infty}$
Figure Options

Download full-size image

Download as 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 $9$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[2]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[3]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[4]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96
[5]

George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43
[6]

Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190
[7]

David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287
[8]

Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473
[9]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020065
[10]

Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705
[11]

H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549
[12]

Alfredo Marzocchi, Sara Zandonella Necca. Attractors for dynamical systems in topological spaces. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 585-597. doi: 10.3934/dcds.2002.8.585
[13]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[14]

Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435
[15]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[16]

Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 43-54. doi: 10.3934/dcds.1998.4.43
[17]

Ahmed Y. Abdallah. Exponential attractors for second order lattice dynamical systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 803-813. doi: 10.3934/cpaa.2009.8.803
[18]

P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213
[19]

Xiaoying Han. Exponential attractors for lattice dynamical systems in weighted spaces. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 445-467. doi: 10.3934/dcds.2011.31.445
[20]

David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (69)
  • HTML views (384)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences