American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Entire solutions originating from monotone fronts for nonlocal dispersal equations with bistable nonlinearity
  • DCDS-B Home
  • This Issue
  • Next Article
    Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force
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  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)
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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} $
Table Options

Download as excel

[1]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[2]

Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020399
[3]

The Editors. The 2019 Michael Brin Prize in Dynamical Systems. Journal of Modern Dynamics, 2020, 16: 349-350. doi: 10.3934/jmd.2020013
[4]

Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021004
[5]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129
[6]

Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021012
[7]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[8]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[9]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[10]

Chongyang Liu, Meijia Han, Zhaohua Gong, Kok Lay Teo. Robust parameter estimation for constrained time-delay systems with inexact measurements. Journal of Industrial & Management Optimization, 2021, 17 (1) : 317-337. doi: 10.3934/jimo.2019113
[11]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108
[12]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[13]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334
[14]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278
[15]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[16]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003
[17]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[18]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[19]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[20]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (60)
  • HTML views (260)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences