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

## Time scale-induced asynchronous discrete dynamical systems

 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.

Citation: Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete and 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. [2] G. M. Baudet, Asynchronous iterative methods for multiprocessors, Journal of the ACM (JACM), 25 (1978), 226-244.  doi: 10.1145/322063.322067. [3] D. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, 1989. Includes corrections (1997). Athena Scientific, Belmont, MA, 2014. [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. [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. [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. [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. [8] S. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac, 37 (1994), 401-413. [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. [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. [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. [12] K. Heliövaara, R. Väisänen and C. Simon, Evolutionary ecology of periodical insects, Trends in Ecology and Evolution, 9 (1994), 475-480. [13] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778. [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. [15] E. Kaszkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhäuser Boston, 2000. doi: 10.1007/978-1-4612-1346-8. [16] W. Kelley and A. Peterson, Difference Equations. An Introduction with Applications, Academic Press, London, 2001. [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. [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. [19] V. Kozyakin, A short introduction to asynchronous systems, In Proceedings of the Sixth International Conference on Difference Equations, 2004,153–165. [20] R. Lagunoff and A. Matsui, Asynchronous choice in repeated coordination games, Econometrica, 65 (1997), 1467-1477.  doi: 10.2307/2171745. [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. [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. [23] J. Libich and P. Stehlík, Endogenous Monetary Commitment, Economic Letters, 112 (2011), 103-106.  doi: 10.1016/j.econlet.2011.03.030. [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. [25] H. Lütkepohl, Handbook of Matrices, John Wiley & Sons, Ltd., Chichester, 1996. [26] J. D. Murray, Mathematical Biology II, Springer, 2003. [27] K. Ogata, Discrete-time Control Systems, Prentice Hall Englewood Cliffs, NJ, 1995. [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. [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. [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. [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. [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. [33] J. Tobin, Money and Finance in the Macroeconomic Process, Journal of Money, Credit and Banking, 14 (1982), 171–204. doi: 10.2307/1991638. [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.

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. [2] G. M. Baudet, Asynchronous iterative methods for multiprocessors, Journal of the ACM (JACM), 25 (1978), 226-244.  doi: 10.1145/322063.322067. [3] D. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Prentice Hall, 1989. Includes corrections (1997). Athena Scientific, Belmont, MA, 2014. [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. [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. [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. [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. [8] S. Elaydi and S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac, 37 (1994), 401-413. [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. [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. [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. [12] K. Heliövaara, R. Väisänen and C. Simon, Evolutionary ecology of periodical insects, Trends in Ecology and Evolution, 9 (1994), 475-480. [13] N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, 2008. doi: 10.1137/1.9780898717778. [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. [15] E. Kaszkurewicz and A. Bhaya, Matrix Diagonal Stability in Systems and Computation, Birkhäuser Boston, 2000. doi: 10.1007/978-1-4612-1346-8. [16] W. Kelley and A. Peterson, Difference Equations. An Introduction with Applications, Academic Press, London, 2001. [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. [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. [19] V. Kozyakin, A short introduction to asynchronous systems, In Proceedings of the Sixth International Conference on Difference Equations, 2004,153–165. [20] R. Lagunoff and A. Matsui, Asynchronous choice in repeated coordination games, Econometrica, 65 (1997), 1467-1477.  doi: 10.2307/2171745. [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. [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. [23] J. Libich and P. Stehlík, Endogenous Monetary Commitment, Economic Letters, 112 (2011), 103-106.  doi: 10.1016/j.econlet.2011.03.030. [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. [25] H. Lütkepohl, Handbook of Matrices, John Wiley & Sons, Ltd., Chichester, 1996. [26] J. D. Murray, Mathematical Biology II, Springer, 2003. [27] K. Ogata, Discrete-time Control Systems, Prentice Hall Englewood Cliffs, NJ, 1995. [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. [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. [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. [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. [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. [33] J. Tobin, Money and Finance in the Macroeconomic Process, Journal of Money, Credit and Banking, 14 (1982), 171–204. doi: 10.2307/1991638. [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.
Time scales $\mathbb{T}_3$ and $\mathbb{T}_5$ of a $(3,5)$-asynchronous discrete dynamical system (3)
Time scales related to dynamically equivalent (2, 3)- and (6, 1)-asynchronous discrete dynamical systems from Example 7.3
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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