# American Institute of Mathematical Sciences

## Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise

 Department of Mathematics, Hokkaido university, Sapporo, Hokkaido, 060-0808, Japan

Received  April 2017 Revised  July 2017 Published  February 2018

We consider the perturbed dynamical system applied to non expanding piecewise linear maps on $[0, 1]$ which describe simplified dynamics of a single neuron. It is known that the Markov operator generated by this perturbed system has asymptotic periodicity with period $n≥1$. In this paper, we give a sufficient condition for $n>1$, asymptotic periodicity, and for $n = 1$, asymptotic stability. That is, we show that there exists a threshold of noises $θ_{*}$ such that the Markov operator generated by this perturbed system displays asymptotic periodicity (asymptotic stability) if a maximum value of noises is less (greater) than $θ_{*}$. This result indicates that an existence of phenomenon called mode-locking is mathematically clarified for this perturbed system.

Citation: Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018055
##### References:
 [1] T.M.Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York-Heidelberg, 1976. doi: 10.1007/978-1-4612-0999-7. [2] J. Berstel, Recent results on Sturmian words, In Developments in Language Theory, 2 (1996), 13-24. doi: 10.1142/9789814531153. [3] J. Berstel, Sturmian and episturmian words. In Algebraic informatics, Springer Berlin Heidelberg, (2007), 23-47. doi: 10.1007/978-3-540-75414-5_2. [4] V. Berthé, A. De Luca and C. Reutenauer, On an involution of Christoffel words and Sturmian morphisms, European Journal of Combinatorics, 29 (2008), 535-553. doi: 10.1016/j.ejc.2007.03.001. [5] G. I. Bischi, L. Gardini and F. Tramontana, Bifurcation curves in discontinuous maps, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 249-267. doi: 10.3934/dcdsb.2010.13.249. [6] T. C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull, 36 (1993), 15-21. doi: 10.4153/CMB-1993-003-6. [7] E. R. Caianiello, Outline of a theory of thought-processes and thinking machines, Journal of Theoretical Biology, 1 (1961), 204-235. doi: 10.1016/0022-5193(61)90046-7. [8] E. J. Ding and P. C. Hemmer, Exact treatment of mode locking for a piecewise linear map, Journal of Statistical Physics, 46 (1987), 99-110. doi: 10.1007/BF01010333. [9] G.Essl, Circle maps as simple oscillators for complex behavior: I.Basics, In In Proceedings of the International Computer Music Conference (ICMC), 2006. [10] D. Faranda, J. M. Freitas, P. Guiraud and S. Vaienti, Sampling local properties of attractors via extreme value theory, Chaos, Solitons & Fractals, 74 (2015), 55-66. doi: 10.1016/j.chaos.2015.01.016. [11] D.Faranda, J.M.Freitas, P.Guiraud and S.Vaienti, Statistical properties of random dynamical systems with contracting direction Journal of Physics A: Mathematical and Theoretical, 49 (2016), 204001, 17pp. doi: 10.1088/1751-8113/49/20/204001. [12] 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. doi: 10.1016/0167-2789(83)90119-7. [13] G. Gómez, J. M. Mondelo and C. Simó, A collocation method for the numerical Fourier analysis of quasi-periodic functions. I. Numerical tests and examples, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 41-74. doi: 10.3934/dcdsb.2010.14.41. [14] T. Inoue, Invariant measures for position dependent random maps with continuous random parameters, Studia Math., 208 (2012), 11-29. doi: 10.4064/sm208-1-2. [15] Y. Iwata and T. Ogihara, Random perturbations of non-singular transformations on $[0, 1]$, Hokkaido Mathematical Journal, 42 (2013), 269-291. doi: 10.14492/hokmj/1372859588. [16] T.Kaijser, Stochastic perturbations of iterations of a simple, non-expanding, nonperiodic, piecewise linear, interval-map, preprint, arXiv:1606.00741. [17] J. P. Keener, Chaotic behavior in piecewise continuous difference equations, Transactions of the American Mathematical Society, 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3. [18] A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Transactions of the American Mathematical Society, 286 (1984), 751-764. doi: 10.1090/S0002-9947-1984-0760984-4. [19] A.Lasota and M.C.Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Second edition. Applied Mathematical Sciences, 97.Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4. [20] M. McGuinness, Y. Hong, D. Galletly and P. Larsen, Arnold tongues in human cardiorespiratory systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004), 1-6. doi: 10.1063/1.1620990. [21] J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Biological Cybernetics, 10 (1972), 155-164. doi: 10.1007/BF00290514. [22] F. Nakamura, Periodicity of non-expanding piecewise linear maps and effects of random noises, Dynamical Systems, 30 (2015), 450-467. doi: 10.1080/14689367.2015.1073225. [23] M.Oku and K.Aihara, Numerical analysis of transient and periodic dynamics in single and coupled Nagumo-Sato models International Journal of Bifurcation and Chaos, 22 (2012), 1230021, 15 pp. doi: 10.1142/S0218127412300212. [24] A.Panchuk, I.Sushko, B.Schenke and V.Avrutin, Bifurcation structures in a bimodal piecewise linear map: Regular dynamics International Journal of Bifurcation and Chaos, 23 (2013), 1330040, 24pp. doi: 10.1142/S0218127413300401. [25] N. Provatas and M.C. Mackey, Noise-induced asymptotic periodicity in a piecewise linear map, Journal of Statistical Physics, 63 (1991), 585-612. doi: 10.1007/BF01029201.

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##### References:
 [1] T.M.Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York-Heidelberg, 1976. doi: 10.1007/978-1-4612-0999-7. [2] J. Berstel, Recent results on Sturmian words, In Developments in Language Theory, 2 (1996), 13-24. doi: 10.1142/9789814531153. [3] J. Berstel, Sturmian and episturmian words. In Algebraic informatics, Springer Berlin Heidelberg, (2007), 23-47. doi: 10.1007/978-3-540-75414-5_2. [4] V. Berthé, A. De Luca and C. Reutenauer, On an involution of Christoffel words and Sturmian morphisms, European Journal of Combinatorics, 29 (2008), 535-553. doi: 10.1016/j.ejc.2007.03.001. [5] G. I. Bischi, L. Gardini and F. Tramontana, Bifurcation curves in discontinuous maps, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 249-267. doi: 10.3934/dcdsb.2010.13.249. [6] T. C. Brown, Descriptions of the characteristic sequence of an irrational, Canad. Math. Bull, 36 (1993), 15-21. doi: 10.4153/CMB-1993-003-6. [7] E. R. Caianiello, Outline of a theory of thought-processes and thinking machines, Journal of Theoretical Biology, 1 (1961), 204-235. doi: 10.1016/0022-5193(61)90046-7. [8] E. J. Ding and P. C. Hemmer, Exact treatment of mode locking for a piecewise linear map, Journal of Statistical Physics, 46 (1987), 99-110. doi: 10.1007/BF01010333. [9] G.Essl, Circle maps as simple oscillators for complex behavior: I.Basics, In In Proceedings of the International Computer Music Conference (ICMC), 2006. [10] D. Faranda, J. M. Freitas, P. Guiraud and S. Vaienti, Sampling local properties of attractors via extreme value theory, Chaos, Solitons & Fractals, 74 (2015), 55-66. doi: 10.1016/j.chaos.2015.01.016. [11] D.Faranda, J.M.Freitas, P.Guiraud and S.Vaienti, Statistical properties of random dynamical systems with contracting direction Journal of Physics A: Mathematical and Theoretical, 49 (2016), 204001, 17pp. doi: 10.1088/1751-8113/49/20/204001. [12] 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. doi: 10.1016/0167-2789(83)90119-7. [13] G. Gómez, J. M. Mondelo and C. Simó, A collocation method for the numerical Fourier analysis of quasi-periodic functions. I. Numerical tests and examples, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 41-74. doi: 10.3934/dcdsb.2010.14.41. [14] T. Inoue, Invariant measures for position dependent random maps with continuous random parameters, Studia Math., 208 (2012), 11-29. doi: 10.4064/sm208-1-2. [15] Y. Iwata and T. Ogihara, Random perturbations of non-singular transformations on $[0, 1]$, Hokkaido Mathematical Journal, 42 (2013), 269-291. doi: 10.14492/hokmj/1372859588. [16] T.Kaijser, Stochastic perturbations of iterations of a simple, non-expanding, nonperiodic, piecewise linear, interval-map, preprint, arXiv:1606.00741. [17] J. P. Keener, Chaotic behavior in piecewise continuous difference equations, Transactions of the American Mathematical Society, 261 (1980), 589-604. doi: 10.1090/S0002-9947-1980-0580905-3. [18] A. Lasota, T. Y. Li and J. A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Transactions of the American Mathematical Society, 286 (1984), 751-764. doi: 10.1090/S0002-9947-1984-0760984-4. [19] A.Lasota and M.C.Mackey, Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics, Second edition. Applied Mathematical Sciences, 97.Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-4286-4. [20] M. McGuinness, Y. Hong, D. Galletly and P. Larsen, Arnold tongues in human cardiorespiratory systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, 14 (2004), 1-6. doi: 10.1063/1.1620990. [21] J. Nagumo and S. Sato, On a response characteristic of a mathematical neuron model, Biological Cybernetics, 10 (1972), 155-164. doi: 10.1007/BF00290514. [22] F. Nakamura, Periodicity of non-expanding piecewise linear maps and effects of random noises, Dynamical Systems, 30 (2015), 450-467. doi: 10.1080/14689367.2015.1073225. [23] M.Oku and K.Aihara, Numerical analysis of transient and periodic dynamics in single and coupled Nagumo-Sato models International Journal of Bifurcation and Chaos, 22 (2012), 1230021, 15 pp. doi: 10.1142/S0218127412300212. [24] A.Panchuk, I.Sushko, B.Schenke and V.Avrutin, Bifurcation structures in a bimodal piecewise linear map: Regular dynamics International Journal of Bifurcation and Chaos, 23 (2013), 1330040, 24pp. doi: 10.1142/S0218127413300401. [25] N. Provatas and M.C. Mackey, Noise-induced asymptotic periodicity in a piecewise linear map, Journal of Statistical Physics, 63 (1991), 585-612. doi: 10.1007/BF01029201.
The region of the parameter space $(\alpha,\beta)$ in which $S_{\alpha,\beta}$ has a periodic point with period $n = 2,3,4,5$.
Asymptotic periodicity illustrated. Here we show histograms obtain after iterating 5,000,000 initial points uniformly distributed on $[0,1]$ with $\alpha = 1/2, \beta = 4/7$, and $\theta = 1/14$ in Equation (2) for (a) $t = 200$; (b) $t = 201$; (c) $t = 202$; and (d) $t = 203$. A correspondence of the histograms for $t = 200$ and $t = 203$ indicates that the sequence of densities has period 3.
Asymptotic stability illustrated. Here we show histograms obtain after 200 iterating 5,000,000 initial points uniformly distributed on $[0,1]$ with $\alpha = 1/2, \beta = 4/7$, and $\theta = 1/14+0.02$ in Equation (2).
Asymptotic stability illustrated. Here we show histograms obtain after (a)200; (b)1,000; (c)10,000; (d)100,000 iterating 5,000,000 initial points uniformly distributed on $[0,1]$ with $\alpha = 1/2, \beta = 17/30$, and $\theta = 1/15$ in Equation (2).
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