Time scale-induced asynchronous discrete dynamical systems

Stefan Siegmund; Petr Stehlík

doi:10.3934/dcdsb.2020151

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February 2021, 26(2): 1011-1029. doi: 10.3934/dcdsb.2020151

Time scale-induced asynchronous discrete dynamical systems

Stefan Siegmund ^1, and Petr Stehlík ^2,,

1.	Center for Dynamics & Institute for Analysis, Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany
2.	Dept. of Mathematics and NTIS, University of West Bohemia, Univerzitní 8, 30614 Pilsen, Pilsen, Czech Republic

^* Corresponding author: Petr Stehlík

Received July 2019 Revised February 2020 Published February 2021 Early access May 2020

We study two coupled discrete-time equations with different (asynchronous) periodic time scales. The coupling is of the type sample and hold, i.e., the state of each equation is sampled at its update times and held until it is read as an input at the next update time for the other equation. We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.

Keywords: Asynchronous dynamical systems, stability, time scales.

Mathematics Subject Classification: Primary: 37E99, 39A30; Secondary: 34N05, 91B64, 93C55.

Citation: Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151

References:

[1]	D. Aubry and G. Puel, Two-timescale homogenization method for the modeling of material fatigue, IOP Conference Series: Materials Science and Engineering, 10 (2010), Article no. 012113. doi: 10.1088/1757-899X/10/1/012113. Google Scholar
[2]	G. M. Baudet, Asynchronous iterative methods for multiprocessors, Journal of the ACM (JACM), 25 (1978), 226-244. doi: 10.1145/322063.322067. Google Scholar
[3]	D. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, 1989. Includes corrections (1997). Athena Scientific, Belmont, MA, 2014. Google Scholar
[4]	M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0201-1. Google Scholar
[5]	R. Bru, L. Elsner and M. Neumann., Convergence of infinite products of matrices and inner-outer iteration schemes, Electronic Transactions on Numerical Analysis, 2 (1994), 183-193. Google Scholar
[6]	D. Chazan and W. Miranker, Chaotic relaxation, Linear Algebra and its Applications, 2 (1969), 199-222. doi: 10.1016/0024-3795(69)90028-7. Google Scholar
[7]	T. S. Doan, A. Kalauch and S. Siegmund, A constructive approach to linear Lyapunov functions for positive switched systems using Collatz-Wielandt sets, IEEE Transactions on Automatic Control, 58 (2013), 748–751. doi: 10.1109/TAC.2012.2209270. Google Scholar
[8]	S. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac, 37 (1994), 401-413. Google Scholar
[9]	A. Frommer and D. B. Szyld, On asynchronous iterations, Journal of Computational and Applied Mathematics, 123 (2000), 201–216. Numerical analysis 2000, Vol. III. Linear algebra. doi: 10.1016/S0377-0427(00)00409-X. Google Scholar
[10]	A. Hassibi, S. P. Boyd and J. P. How, Control of asynchronous dynamical systems with rate constraints on events, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 1345–1351. doi: 10.1109/CDC.1999.830133. Google Scholar
[11]	H. Heaton and Y. Censor, Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems, Journal of Global Optimization, 74 (2019), 95-119. doi: 10.1007/s10898-019-00747-4. Google Scholar
[12]	K. Heliövaara, R. Väisänen and C. Simon, Evolutionary ecology of periodical insects, Trends in Ecology and Evolution, 9 (1994), 475-480. Google Scholar
[13]	N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778. Google Scholar
[14]	S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math, 18 (1990), 18-56. doi: 10.1007/BF03323153. Google Scholar
[15]	E. Kaszkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhäuser Boston, 2000. doi: 10.1007/978-1-4612-1346-8. Google Scholar
[16]	W. Kelley and A. Peterson, Difference Equations. An Introduction with Applications, Academic Press, London, 2001. Google Scholar
[17]	P. Klemperer, Competition when Consumers have Switching Costs: An Overview with Applications to Industrial Organization, Macroeconomics, and International Trade, The Review of Economic Studies, 62 (1995), 515-539. doi: 10.2307/2298075. Google Scholar
[18]	A. F. Kleptsyn, M. A. Krasnosel'skiĭ, N. A. Kuznetsov and V. S. Kozyakin, Desynchronization of linear systems, Mathematics and Computers in Simulation, 26 (1984), 423-431. doi: 10.1016/0378-4754(84)90106-X. Google Scholar
[19]	V. Kozyakin, A short introduction to asynchronous systems, In Proceedings of the Sixth International Conference on Difference Equations, 2004,153–165. Google Scholar
[20]	R. Lagunoff and A. Matsui, Asynchronous choice in repeated coordination games, Econometrica, 65 (1997), 1467-1477. doi: 10.2307/2171745. Google Scholar
[21]	C. Lorand and P. Bauer, A factorization approach to the analysis of asynchronous interconnected discrete-time systems with arbitrary clock ratios, In Proceedings of the American Control Conference, 2004,349–354. Google Scholar
[22]	C. Lorand and P. Bauer., Interconnected discrete-time systems with incommensurate clock frequencies, In Proceedings of the IEEE Conference on Decision and Control, 2004,935–940. Google Scholar
[23]	J. Libich and P. Stehlík, Endogenous Monetary Commitment, Economic Letters, 112 (2011), 103-106. doi: 10.1016/j.econlet.2011.03.030. Google Scholar
[24]	J. Libich and P. Stehlík, Incorporating rigidity and commitment in the timing structure of macroeconomic games, Economic Modelling, 27 (2010), 767-781. doi: 10.1016/j.econmod.2010.01.020. Google Scholar
[25]	H. Lütkepohl, Handbook of Matrices, John Wiley & Sons, Ltd., Chichester, 1996. Google Scholar
[26]	J. D. Murray, Mathematical Biology II, Springer, 2003. Google Scholar
[27]	K. Ogata, Discrete-time Control Systems, Prentice Hall Englewood Cliffs, NJ, 1995. Google Scholar
[28]	C. Pötzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst., 9 (2003), 1223-1241. doi: 10.3934/dcds.2003.9.1223. Google Scholar
[29]	R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Review, 49 (2007), 545-592. doi: 10.1137/05063516X. Google Scholar
[30]	W. Shou, C. T. Bergstrom, A. K. Chakraborty and F. K. Skinner, Theory, models and biology, ELife, 4 (2015), e07158. doi: 10.7554/eLife.07158. Google Scholar
[31]	A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl., 385 (2012), 534-550. doi: 10.1016/j.jmaa.2011.06.068. Google Scholar
[32]	Y. Su, A. Bhaya, E. Kaszkurewicz and V. S. Kozyakin, Further results on convergence of asynchronous linear iterations, Linear Algebra and its Applications, 281 (1998), 11-24. doi: 10.1016/S0024-3795(98)10030-7. Google Scholar
[33]	J. Tobin, Money and Finance in the Macroeconomic Process, Journal of Money, Credit and Banking, 14 (1982), 171–204. doi: 10.2307/1991638. Google Scholar
[34]	Q. Yu and J. Fish, Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading, Computational Mechanics, 29 (2002), 199-211. doi: 10.1007/s00466-002-0334-y. Google Scholar

show all references

References:

[1]	D. Aubry and G. Puel, Two-timescale homogenization method for the modeling of material fatigue, IOP Conference Series: Materials Science and Engineering, 10 (2010), Article no. 012113. doi: 10.1088/1757-899X/10/1/012113. Google Scholar
[2]	G. M. Baudet, Asynchronous iterative methods for multiprocessors, Journal of the ACM (JACM), 25 (1978), 226-244. doi: 10.1145/322063.322067. Google Scholar
[3]	D. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, 1989. Includes corrections (1997). Athena Scientific, Belmont, MA, 2014. Google Scholar
[4]	M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. doi: 10.1007/978-1-4612-0201-1. Google Scholar
[5]	R. Bru, L. Elsner and M. Neumann., Convergence of infinite products of matrices and inner-outer iteration schemes, Electronic Transactions on Numerical Analysis, 2 (1994), 183-193. Google Scholar
[6]	D. Chazan and W. Miranker, Chaotic relaxation, Linear Algebra and its Applications, 2 (1969), 199-222. doi: 10.1016/0024-3795(69)90028-7. Google Scholar
[7]	T. S. Doan, A. Kalauch and S. Siegmund, A constructive approach to linear Lyapunov functions for positive switched systems using Collatz-Wielandt sets, IEEE Transactions on Automatic Control, 58 (2013), 748–751. doi: 10.1109/TAC.2012.2209270. Google Scholar
[8]	S. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac, 37 (1994), 401-413. Google Scholar
[9]	A. Frommer and D. B. Szyld, On asynchronous iterations, Journal of Computational and Applied Mathematics, 123 (2000), 201–216. Numerical analysis 2000, Vol. III. Linear algebra. doi: 10.1016/S0377-0427(00)00409-X. Google Scholar
[10]	A. Hassibi, S. P. Boyd and J. P. How, Control of asynchronous dynamical systems with rate constraints on events, Proceedings of the 38th IEEE Conference on Decision and Control, 1999, 1345–1351. doi: 10.1109/CDC.1999.830133. Google Scholar
[11]	H. Heaton and Y. Censor, Asynchronous sequential inertial iterations for common fixed points problems with an application to linear systems, Journal of Global Optimization, 74 (2019), 95-119. doi: 10.1007/s10898-019-00747-4. Google Scholar
[12]	K. Heliövaara, R. Väisänen and C. Simon, Evolutionary ecology of periodical insects, Trends in Ecology and Evolution, 9 (1994), 475-480. Google Scholar
[13]	N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778. Google Scholar
[14]	S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus, Results Math, 18 (1990), 18-56. doi: 10.1007/BF03323153. Google Scholar
[15]	E. Kaszkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhäuser Boston, 2000. doi: 10.1007/978-1-4612-1346-8. Google Scholar
[16]	W. Kelley and A. Peterson, Difference Equations. An Introduction with Applications, Academic Press, London, 2001. Google Scholar
[17]	P. Klemperer, Competition when Consumers have Switching Costs: An Overview with Applications to Industrial Organization, Macroeconomics, and International Trade, The Review of Economic Studies, 62 (1995), 515-539. doi: 10.2307/2298075. Google Scholar
[18]	A. F. Kleptsyn, M. A. Krasnosel'skiĭ, N. A. Kuznetsov and V. S. Kozyakin, Desynchronization of linear systems, Mathematics and Computers in Simulation, 26 (1984), 423-431. doi: 10.1016/0378-4754(84)90106-X. Google Scholar
[19]	V. Kozyakin, A short introduction to asynchronous systems, In Proceedings of the Sixth International Conference on Difference Equations, 2004,153–165. Google Scholar
[20]	R. Lagunoff and A. Matsui, Asynchronous choice in repeated coordination games, Econometrica, 65 (1997), 1467-1477. doi: 10.2307/2171745. Google Scholar
[21]	C. Lorand and P. Bauer, A factorization approach to the analysis of asynchronous interconnected discrete-time systems with arbitrary clock ratios, In Proceedings of the American Control Conference, 2004,349–354. Google Scholar
[22]	C. Lorand and P. Bauer., Interconnected discrete-time systems with incommensurate clock frequencies, In Proceedings of the IEEE Conference on Decision and Control, 2004,935–940. Google Scholar
[23]	J. Libich and P. Stehlík, Endogenous Monetary Commitment, Economic Letters, 112 (2011), 103-106. doi: 10.1016/j.econlet.2011.03.030. Google Scholar
[24]	J. Libich and P. Stehlík, Incorporating rigidity and commitment in the timing structure of macroeconomic games, Economic Modelling, 27 (2010), 767-781. doi: 10.1016/j.econmod.2010.01.020. Google Scholar
[25]	H. Lütkepohl, Handbook of Matrices, John Wiley & Sons, Ltd., Chichester, 1996. Google Scholar
[26]	J. D. Murray, Mathematical Biology II, Springer, 2003. Google Scholar
[27]	K. Ogata, Discrete-time Control Systems, Prentice Hall Englewood Cliffs, NJ, 1995. Google Scholar
[28]	C. Pötzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst., 9 (2003), 1223-1241. doi: 10.3934/dcds.2003.9.1223. Google Scholar
[29]	R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King, Stability criteria for switched and hybrid systems, SIAM Review, 49 (2007), 545-592. doi: 10.1137/05063516X. Google Scholar
[30]	W. Shou, C. T. Bergstrom, A. K. Chakraborty and F. K. Skinner, Theory, models and biology, ELife, 4 (2015), e07158. doi: 10.7554/eLife.07158. Google Scholar
[31]	A. Slavík, Dynamic equations on time scales and generalized ordinary differential equations, J. Math. Anal. Appl., 385 (2012), 534-550. doi: 10.1016/j.jmaa.2011.06.068. Google Scholar
[32]	Y. Su, A. Bhaya, E. Kaszkurewicz and V. S. Kozyakin, Further results on convergence of asynchronous linear iterations, Linear Algebra and its Applications, 281 (1998), 11-24. doi: 10.1016/S0024-3795(98)10030-7. Google Scholar
[33]	J. Tobin, Money and Finance in the Macroeconomic Process, Journal of Money, Credit and Banking, 14 (1982), 171–204. doi: 10.2307/1991638. Google Scholar
[34]	Q. Yu and J. Fish, Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading, Computational Mechanics, 29 (2002), 199-211. doi: 10.1007/s00466-002-0334-y. Google Scholar

Figure 1. Time scales $ \mathbb{T}_3 $ and $ \mathbb{T}_5 $ of a $ (3,5) $-asynchronous discrete dynamical system (3)

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Figure 2. Time scales related to dynamically equivalent (2, 3)- and (6, 1)-asynchronous discrete dynamical systems from Example 7.3

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Table 1. 9 possible forms of the one-step evolution operator $ A(t) $, $ t,\sigma(t)\in \mathbb{T} $ associated with the system (8), see Corollary 2. The pictograms illustrate each quadruple $ (i,j,k,\ell) = \big(1_{ \mathbb{T}_{\mu}}(t), 1_{ \mathbb{T}_{\mu}}(\sigma(t)), 1_{ \mathbb{T}_{\nu}}(t), 1_{ \mathbb{T}_{\nu}}(\sigma(t))\big) $, squares correspond to $ \mathbb{T}_\mu $, circles to $ \mathbb{T}_\nu $, the left symbols to time $ t\in \mathbb{T} $ and the right symbols to $ \sigma(t)\in \mathbb{T} $

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