Asymptotic (statistical) periodicity in two-dimensional maps

Fumihiko Nakamura; Michael C. Mackey

doi:10.3934/dcdsb.2021227

Article Contents

2022, Volume 27, Issue 8: 4285-4303. Doi: 10.3934/dcdsb.2021227

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Asymptotic (statistical) periodicity in two-dimensional maps

Fumihiko Nakamura^1, , and
Michael C. Mackey^2,

1.
Kitami Institute of Technology, 165 Koen-cho, Kitami city, Hokkaido, 090-8507, Japan
2.
McGill University, 3655 Promenade Sir William Osler, Montreal, Quebec H3G 1Y6, Canada

Received: November 2020

Revised: July 2021

Early access: September 2021

Published: August 2022

We thank Prof. Pawel Gora (Concordia University) for pointing out a critical flaw in our original proof of our main Theorem 3.1. The work is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the Ministry of Education, Culture, Sports, Science and Technology through Program for Leading Graduate Schools (Hokkaido University "Ambitious Leader's Program")

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Abstract

In this paper we give a new sufficient condition for the existence of asymptotic periodicity of Frobenius–Perron operators corresponding to two–dimensional maps. Asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical system was already known. Our new result enables one to deal with systems having an eigenvalue smaller than one. The key idea for the proof is to use a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system numerically exhibiting asymptotic periodicity with different periods depending on parameter values, and discuss the application of our theorem to the example.

Keywords:

Mathematics Subject Classification: Primary: 37A30, 26A45; Secondary: 37E30.

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Figure 1. Numerical illustration of asymptotic periodicity in (29). We show the support of $ \{P^{500} f_0\} $ for an initial density $ f_0 = 1_{[-5, 5]\times[-5, 5]} $, approximated by $ 1,000\times1,000 $ initial points uniformly distributed on $ [-5, 5]\times[-5, 5] $ and various values of $ \alpha $ with $ \beta = 1.1 $. (a) $ \alpha = 0.0 $, Period $ = 16 $; (b) $ \alpha = 0.1 $, Period $ = 1 $; (c) $ \alpha = 0.14 $, Period $ = 1 $; (d) $ \alpha = 0.25 $, Period $ = 1 $; (e) $ \alpha = 0.34 $, Period $ = 9 $; (f) $ \alpha = 0.4 $, Period $ = 1 $; (g) $ \alpha = 0.54 $, Period $ = 12 $; (h) $ \alpha = 0.57 $, Period $ = 5 $; (i) $ \alpha = 0.64 $, Period $ = 10 $; (j) $ \alpha = 0.8 $, Period $ = 1 $; (k) $ \alpha = 0.99 $, Period $ = 6 $

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Figure 2. As in Figure 1 with $ \beta = 1.1 $. (a) $ \alpha = -0.08 $, Period $ = 8 $; (b) $ \alpha = -0.1 $, Period $ = 1 $; (c) $ \alpha = -0.41 $, Period $ = 1 $; (d) $ \alpha = -0.46 $, Period $ = 7 $; (e) $ \alpha = -0.5 $, Period $ = 1 $; (f) $ \alpha = -0.75 $, Period $ = 3 $; (g) $ \alpha = -0.8 $, Period $ = 3 $; (h) $ \alpha = -1.14 $, Period $ = 1 $;

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Figure 3. As in Figure 1 with $ \beta = 1.02 $. (a) $ \alpha = 0.24 $, Period $ = 13 $; (b) $ \alpha = 0.25 $, Period $ = 1 $; (c) $ \alpha = 0.27 $, Period $ = 35 $; (d) $ \alpha = 0.28 $, Period $ = 1 $; (e) $ \alpha = 0.284 $, Period $ = 22 $; (f) $ \alpha = 0.3 $, Period $ = 1 $; (g) $ \alpha = 0.3015 $, Period $ = 31 $; (h) $ \alpha = 0.31 $, Period $ = 1 $;

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Figure 4. The regions $ D_i $, $ i = 0, 1, \cdots, 5 $, and $ C $ are illustrated when $ \ell = 5 $. The fixed point $ (x_L^*, y_L^*) $ is a saddle and $ (x_R^*, y_R^*) $ is an unstable focus,

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Figure 5. The situation can be separated into three cases depending on positions of $ p, q $ and $ 1 $. (a) the case $ p, 1<1 $, (b) the case $ p < 1 < q $, and (c) the case $ 1<p, q $

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Figure 6. Illustrations of the result of iterating the regions $ \{I_i\}_{i = 0}^{\ell+1} $ by $ \tilde{S} $

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Table 1. For each $ \beta $, the value $ \alpha $ which gives the condition for $ q\geq 1 $ are calculated numerically

$\beta$	$\ell$	$\alpha < $	$\beta$	$\ell$	$\alpha < $	$\beta$	$\ell$	$\alpha < $	$\beta$	$\ell$	$\alpha < $

1.01	14	1.85664	1.06	7	1.57519	1.2	4	1.15624	1.7	3	0.53436
1.02	11	1.78516	1.07	7	1.56379	1.3	3	1.03992	1.8	2	0.32593
1.03	9	1.71214	1.08	6	1.48766	1.4	3	0.78308	1.9	2	0.13439
1.04	8	1.65753	1.09	6	1.46841	1.5	3	0.66496	2.0	2	0.00000
1.05	8	1.64245	1.1	5	1.45765	1.6	3	0.58999

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