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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2019, 24(4): 1867-1874.
doi: 10.3934/dcdsb.2018243
Periodic attractors of nonautonomous flat-topped tent systems
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 |
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.
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
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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. |
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J. Milnor and C. Tresser,
On entropy and monotonicity for real cubic maps, Comm. Math. Phys., 209 (2000), 123-178.
doi: 10.1007/s002200050018. |
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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. |
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. |
| [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. |
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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