# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021227
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## Asymptotic (statistical) periodicity in two-dimensional maps

 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

Fund Project: 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")

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.

Citation: Fumihiko Nakamura, Michael C. Mackey. Asymptotic (statistical) periodicity in two-dimensional maps. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021227
##### References:
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Hardy, On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters, Quart. J. Math. Oxford, 37 (1905/1906), 53-79.   Google Scholar [13] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar [14] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z, 180 (1982), 119-140.  doi: 10.1007/BF01215004.  Google Scholar [15] S. Ito, S. Tanaka and H. Nakada, On unimodal linear transformations and chaos. I, Tokyo J. Math., 2 (1979) 221–239. doi: 10.3792/pjaa.55.231.  Google Scholar [16] S. Ito, S. Tanaka and H. Nakada, On unimodal linear transformations and chaos. II, Tokyo J. Math., 2 (1979), 241-259.  doi: 10.3836/tjm/1270216321.  Google Scholar [17] J. P. Keener, Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. 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Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.  doi: 10.1007/BF00290514.  Google Scholar [29] F. Nakamura, Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2457-2473.  doi: 10.3934/dcdsb.2018055.  Google Scholar [30] H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems, Phys. D, 57 (1992), 39-57.  doi: 10.1016/0167-2789(92)90087-4.  Google Scholar [31] N. Provatas and M. C. Mackey, Asymptotic periodicity and banded chaos, Phys. D, 53 (1991), 295-318.  doi: 10.1016/0167-2789(91)90067-J.  Google Scholar [32] N. Provatas and M. C. Mackey, Noise-induced asymptotic periodicity in a piecewise linear map, J. Statist. Phys., 63 (1991), 585-612.  doi: 10.1007/BF01029201.  Google Scholar [33] H. Shigematsu, H. Mori, T. Yoshida and H. Okamoto, Analytic study of power spectra of the tent maps near band-splitting transitions, J. Statist. Phys., 30 (1983), 649-679.  doi: 10.1007/BF01009682.  Google Scholar [34] I. Sushko and L. Gardini, Center bifurcation for two-dimensional border collision normal form, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 1029-1050.  doi: 10.1142/S0218127408020823.  Google Scholar [35] G. Światek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128.  doi: 10.1007/BF01218263.  Google Scholar [36] H. Toyokawa, $\sigma$-finite invariant densities for eventually conservative Markov operators, Discrete Contin. Dyn. Syst., 40 (2020), 2641-2669.  doi: 10.3934/dcds.2020144.  Google Scholar [37] G. Vitali, Sulle funzione integrali, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 40 (1904/1905), 1021-1034.   Google Scholar [38] T. Yoshida, H. Mori and H. Shigematsu, Analytic study of chaos of the tent map: Band structures, power spectra, and critical behaviors, J. Statist. Phys., 31 (1983), 279-308.  doi: 10.1007/BF01011583.  Google Scholar

show all references

##### References:
 [1] C. R. Adams and J. A. Clarkson, Properties of functions $f(x, y)$ of bounded variation, Transactions of the American Mathematical Society, 36 (1934), 711-730.  doi: 10.2307/1989819.  Google Scholar [2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar [3] B. Ashton and I. Doust, Functions of bounded variation on compact subsets of the plane, Studia Math., 169 (2005), 163-188.  doi: 10.4064/sm169-2-5.  Google Scholar [4] L. Boltzmann, Lectures on Gas Theory, Calif. 1964.  Google Scholar [5] P. L. Boyland, Bifurcations of circle maps: Arnol'd tongues, bistability and rotation intervals, Communications in Mathematical Physics, 106 (1986), 353-381.  doi: 10.1007/BF01207252.  Google Scholar [6] V. V. Chistyakov and Y. V. Tretyachenko, Maps of several variables of finite total variation. I. Mixed differences and the total variation, J. Math. Anal. Appl., 370 (2010), 672-686.  doi: 10.1016/j.jmaa.2010.04.055.  Google Scholar [7] Z. Elhadj and J. C. Sprott, A new simple 2-D piecewise linear map, J. Syst. Sci. Complex., 23 (2010), 379-389.  doi: 10.1007/s11424-010-7184-z.  Google Scholar [8] J. W. Gibbs, Elementary Principles in Statistical Mechanics, Dover, New York, 1960.  Google Scholar [9] J. Giménez, N. Merentes and M. Vivas, Functions of bounded variation on compact subsets of $\Bbb C$, Comment. Math., 54 (2014), 3-19.  doi: 10.14708/cm.v54i1.757.  Google Scholar [10] L. Glass, M. R. Guevara, A. Shrier and R. Perez, Bifurcation and chaos in a periodically stimulated cardiac oscillator, Physica D: Nonlinear Phenomena, 7 (1983), 89-101.   Google Scholar [11] P. Góra and A. Boyarsky, Absolutely continuous invariant measures for piecewise expanding $C^2$ transformations in $R^N$, Israel J. Math., 67 (1989), 272-286.  doi: 10.1007/BF02764946.  Google Scholar [12] G. H. Hardy, On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters, Quart. J. Math. Oxford, 37 (1905/1906), 53-79.   Google Scholar [13] M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar [14] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z, 180 (1982), 119-140.  doi: 10.1007/BF01215004.  Google Scholar [15] S. Ito, S. Tanaka and H. Nakada, On unimodal linear transformations and chaos. I, Tokyo J. Math., 2 (1979) 221–239. doi: 10.3792/pjaa.55.231.  Google Scholar [16] S. Ito, S. Tanaka and H. Nakada, On unimodal linear transformations and chaos. II, Tokyo J. Math., 2 (1979), 241-259.  doi: 10.3836/tjm/1270216321.  Google Scholar [17] J. P. Keener, Chaotic behavior in piecewise continuous difference equations, Trans. Amer. Math. Soc., 261 (1980), 589-604.  doi: 10.1090/S0002-9947-1980-0580905-3.  Google Scholar [18] J. Komorník, Asymptotic periodicity of Markov and related operators, Dynamics Reported, 2 (1993), 31-68.  doi: 10.1007/978-3-642-61232-9_2.  Google Scholar [19] J. Komorník, Asymptotic periodicity of the iterates of weakly constrictive Markoy operators, Tohoku Math. J., 38 (1986), 15-27.  doi: 10.2748/tmj/1178228533.  Google Scholar [20] J. Komorník and A. Lasota, Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math., 35 (1987), 321-327.   Google Scholar [21] A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc., 286 (1984), 751-764.  doi: 10.1090/S0002-9947-1984-0760984-4.  Google Scholar [22] A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Applied Mathematical Sciences, 97. Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4.  Google Scholar [23] A. Lasota and M. C. Mackey, Noise and statistical periodicity, Phys. D, 28 (1987), 143-154.  doi: 10.1016/0167-2789(87)90125-4.  Google Scholar [24] A. Lasota and J. A. Yorke, Statistical periodicity of deterministic systems, Časopis pro pěstování matematiky, 111 (1986), 1–13. doi: 10.21136/CPM.1986.118256.  Google Scholar [25] J. Losson and M. C. Mackey, Coupled map lattices as models of deterministic and stochastic differential delay equations, Phys. Rev. E (3), 52 (1995), 115-128.  doi: 10.1103/PhysRevE.52.115.  Google Scholar [26] R. Lozi, Un attracteur étrange (?) du type attracteur de Hénon, Le Journal de Physique Colloques, 39 (1978), C5–9. doi: 10.1051/jphyscol:1978505.  Google Scholar [27] M. C. Mackey, Time's Arrow: The Origins of Thermodynamic Behaviour, Springer-Verlag, Berlin, New York, Heidelberg, 1992. doi: 10.1007/978-1-4613-9524-9.  Google Scholar [28] J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Kybernetik, 10 (1972), 155-164.  doi: 10.1007/BF00290514.  Google Scholar [29] F. Nakamura, Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2457-2473.  doi: 10.3934/dcdsb.2018055.  Google Scholar [30] H. E. Nusse and J. A. Yorke, Border-collision bifurcations including "period two to period three" for piecewise smooth systems, Phys. D, 57 (1992), 39-57.  doi: 10.1016/0167-2789(92)90087-4.  Google Scholar [31] N. Provatas and M. C. Mackey, Asymptotic periodicity and banded chaos, Phys. D, 53 (1991), 295-318.  doi: 10.1016/0167-2789(91)90067-J.  Google Scholar [32] N. Provatas and M. C. Mackey, Noise-induced asymptotic periodicity in a piecewise linear map, J. Statist. Phys., 63 (1991), 585-612.  doi: 10.1007/BF01029201.  Google Scholar [33] H. Shigematsu, H. Mori, T. Yoshida and H. Okamoto, Analytic study of power spectra of the tent maps near band-splitting transitions, J. Statist. Phys., 30 (1983), 649-679.  doi: 10.1007/BF01009682.  Google Scholar [34] I. Sushko and L. Gardini, Center bifurcation for two-dimensional border collision normal form, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 1029-1050.  doi: 10.1142/S0218127408020823.  Google Scholar [35] G. Światek, Rational rotation numbers for maps of the circle, Comm. Math. Phys., 119 (1988), 109-128.  doi: 10.1007/BF01218263.  Google Scholar [36] H. Toyokawa, $\sigma$-finite invariant densities for eventually conservative Markov operators, Discrete Contin. Dyn. Syst., 40 (2020), 2641-2669.  doi: 10.3934/dcds.2020144.  Google Scholar [37] G. Vitali, Sulle funzione integrali, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 40 (1904/1905), 1021-1034.   Google Scholar [38] T. Yoshida, H. Mori and H. Shigematsu, Analytic study of chaos of the tent map: Band structures, power spectra, and critical behaviors, J. Statist. Phys., 31 (1983), 279-308.  doi: 10.1007/BF01011583.  Google Scholar
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 $= 6$
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 $= 1$;">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 $= 1$;
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 $= 1$;">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 $= 1$;
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,
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 $1<p, q$
Illustrations of the result of iterating the regions $\{I_i\}_{i = 0}^{\ell+1}$ by $\tilde{S}$
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
 $\beta$ $\ell$ $\alpha <$ $\beta$ $\ell$ $\alpha <$ $\beta$ $\ell$ $\alpha <$ $\beta$ $\ell$ $\alpha <$ 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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