
-
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
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 |
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.
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. Google Scholar |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. |


![]() |
![]() |
[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
Other articles
by authors
[Back to Top]