Journal articles: 'Two-Time scales systems'

1

Agarwal, G. S., and J. Banerji. "Fractional revivals in systems with two time scales." Physical Review A 57, no. 5 (May 1, 1998): 3880–84. http://dx.doi.org/10.1103/physreva.57.3880.

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Borkar, Vivek S. "Stochastic approximation with two time scales." Systems & Control Letters 29, no. 5 (February 1997): 291–94. http://dx.doi.org/10.1016/s0167-6911(97)90015-3.

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Choi, Sung Kyu, and Namjip Koo. "ASYMPTOTIC EQUIVALENCE BETWEEN TWO LINEAR DYNAMIC SYSTEMS ON TIME SCALES." Bulletin of the Korean Mathematical Society 51, no. 4 (July 31, 2014): 1075–85. http://dx.doi.org/10.4134/bkms.2014.51.4.1075.

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Gomez-Exposito, Antonio, Catalina Gomez-Quiles, and Izudin Dzafic. "State Estimation in Two Time Scales for Smart Distribution Systems." IEEE Transactions on Smart Grid 6, no. 1 (January 2015): 421–30. http://dx.doi.org/10.1109/tsg.2014.2335611.

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Xu, Youjun, and Zhiting Xu. "Oscillation criteria for two-dimensional dynamic systems on time scales." Journal of Computational and Applied Mathematics 225, no. 1 (March 2009): 9–19. http://dx.doi.org/10.1016/j.cam.2008.06.010.

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Hassan, Taher. "Oscillation criterion for two-dimensional dynamic systems on time scales." Tamkang Journal of Mathematics 44, no. 3 (October 18, 2012): 227–32. http://dx.doi.org/10.5556/j.tkjm.44.2013.1189.

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The purpose of this paper is to prove oscillation criterion for dynamic system \begin{equation*} u^{\Delta }=pv,\qquad v^{\Delta }=-qu^{\sigma }, \end{equation*}% where $p>0$ and $q$ are rd-continuous functions on a time scale such that $% \sup \mathbb{T=\infty }$ without explicit sign assumptions on $q$ and also without restrictive conditions on the time scale $\mathbb{T}.$

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Hilscher, Roman Šimon, and Petr Zemánek. "Limit circle invariance for two differential systems on time scales." Mathematische Nachrichten 288, no. 5-6 (October 7, 2014): 696–709. http://dx.doi.org/10.1002/mana.201400005.

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Öztürk, Özkan, and Elvan Akın. "Nonoscillation Criteria for Two-Dimensional Time-Scale Systems." Nonautonomous Dynamical Systems 3, no. 1 (January 30, 2016): 1–13. http://dx.doi.org/10.1515/msds-2016-0001.

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AbstractWe study the existence and nonexistence of nonoscillatory solutions of a two-dimensional systemof first-order dynamic equations on time scales. Our approach is based on the Knaster and Schauder fixed point theorems and some certain integral conditions. Examples are given to illustrate some of our main results.

9

Baoguo, Jia. "A new oscillation criterion for two-dimensional dynamic systems on time scales." Tamkang Journal of Mathematics 42, no. 2 (April 14, 2011): 237–44. http://dx.doi.org/10.5556/j.tkjm.42.2011.656.

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Consider the linear dynamic system on time scales\begin{equation}u^\Delta=pv, \quad\quad v^\Delta=-qu^\sigma\end{equation}where $p>0$ and $q$ are rd-continuous functions on a time scale $\mathbb T$ such that $\sup\mathbb T=\infty$. When $p(t)$ is allowed to take on negative values, we establish an oscillation criterion for system (0.1). Our result improves a main result of Fu and Lin [S. C. Fu and M. L. Lin, Oscillation and nonoscillation criteria for linear dynamic systems on time scales, Computers and Mathematics with Applications, 59(2010), 2552-2565].

10

Fu, Zhi-Jun, and Xiao-Yang Dong. "H∞ optimal control of vehicle active suspension systems in two time scales." Automatika 62, no. 2 (April 3, 2021): 284–92. http://dx.doi.org/10.1080/00051144.2021.1935610.

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van der Burg, Erik, John Cass, and David Alais. "Temporal recalibration involves adaptation at two time scales." Multisensory Research 26, no. 1-2 (2013): 60. http://dx.doi.org/10.1163/22134808-000s0038.

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Li, Qiao-Luan, Wing-Sum Cheung, and Xu-Yang Fu. "On Inequalities of Lyapunov for Two-Dimensional Nonlinear Dynamic Systems on Time Scales." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/830595.

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Kolmogorov, Oleg V., Dmitriy V. Prokhorov, Sergey S. Donchenko, and Ekaterina V. Chemesova. "A system of one- and two-way comparisons of time scales using stimulated Raman scattering." Izmeritel`naya Tekhnika, no. 6 (2020): 27–32. http://dx.doi.org/10.32446/0368-1025it.2020-6-27-32.

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Problems requiring high-precision comparisons and synchronization of time scales of remote time and frequency standards are considered. The prospects of using fiber-optic communication lines to solve these problems are considered. Data on maximum operating distances and accuracy characteristics of time scale comparison systems using fiber-optic communication lines are given. It is shown that the range of existing systems is not enough to solve a number of problems. A method for amplifying optical signals in an optical fiber using the effect of stimulated Raman scattering is considered. The application of the Raman amplifier in telecommunication systems is considered. A method is proposed for using the effect of stimulated Raman scattering (SRS) to amplify the optical signals of time scale comparison systems. A scheme is proposed for a system of one- and two-way comparisons of time scales of distant objects, using signal amplification based on the SRS effect realized using two-way counter-pumping. The principle of the system’s operation and the procedure for determining the divergence in the time scales of remote objects are described. The results of estimation of the error and range of systems of one- and two-way comparisons of time scales using Raman amplifier are presented. It is shown that the implementation of Raman-amplification in the equipment of such systems will allow more than three times to increase their range without reducing the accuracy of the systems.

14

Kaymakçalan, Billûr. "Stability analysis in terms of two measures for dynamic systems on time scales." Journal of Applied Mathematics and Stochastic Analysis 6, no. 4 (January 1, 1993): 325–44. http://dx.doi.org/10.1155/s1048953393000280.

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Using the theory of Lyapunov's second method developed earlier for time scales, we extend our stability results to two measures which give rise to unification of several stability concepts in a single set up.

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Yan, Wu, and Fu Jing-Li. "Noether’s theorems of variable mass systems on time scales." Applied Mathematics and Nonlinear Sciences 3, no. 1 (May 29, 2018): 229–40. http://dx.doi.org/10.21042/amns.2018.1.00017.

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AbstractThis paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

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Zhang, Kexue, and Xinzhi Liu. "Stability in Terms of Two Measures for Nonlinear Impulsive Systems on Time Scales." Journal of Applied Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/313029.

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We investigate some stability problems in terms of two measures for nonlinear dynamic systems on time scales with fixed moments of impulsive effects. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Our results include the existing results as special cases when the time scale reduces to the set of real numbers. Particularly, our results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results.

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Tang, Ying, and Guilherme Mazanti. "Stability analysis of coupled linear ODE-hyperbolic PDE systems with two time scales." Automatica 85 (November 2017): 386–96. http://dx.doi.org/10.1016/j.automatica.2017.07.052.

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JANSSON, JOHAN, CLAES JOHNSON, and ANDERS LOGG. "COMPUTATIONAL MODELING OF DYNAMICAL SYSTEMS." Mathematical Models and Methods in Applied Sciences 15, no. 03 (March 2005): 471–81. http://dx.doi.org/10.1142/s0218202505000431.

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In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10–9 computed over the time interval [0,100], and a lattice consisting of large and small point masses.

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Teel, A. R., L. Moreau, and D. Nesic. "A unified framework for input-to-state stability in systems with two time scales." IEEE Transactions on Automatic Control 48, no. 9 (September 2003): 1526–44. http://dx.doi.org/10.1109/tac.2003.816966.

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Yin, G., and Hanqin Zhang. "Countable-state-space Markov chains with two time scales and applications to queueing systems." Advances in Applied Probability 34, no. 3 (September 2002): 662–88. http://dx.doi.org/10.1239/aap/1033662170.

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Motivated by various applications in queueing systems, this work is devoted to continuous-time Markov chains with countable state spaces that involve both fast-time scale and slow-time scale with the aim of approximating the time-varying queueing systems by their quasistationary counterparts. Under smoothness conditions on the generators, asymptotic expansions of probability vectors and transition probability matrices are constructed. Uniform error bounds are obtained, and then sequences of occupation measures and their functionals are examined. Mean square error estimates of a sequence of occupation measures are obtained; a scaled sequence of functionals of occupation measures is shown to converge to a Gaussian process with zero mean. The representation of the variance of the limit process is also explicitly given. The results obtained are then applied to treat Mt/Mt/1 queues and Markov-modulated fluid buffer models.

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Yin, G., and Hanqin Zhang. "Countable-state-space Markov chains with two time scales and applications to queueing systems." Advances in Applied Probability 34, no. 03 (September 2002): 662–88. http://dx.doi.org/10.1017/s0001867800011800.

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Motivated by various applications in queueing systems, this work is devoted to continuous-time Markov chains with countable state spaces that involve both fast-time scale and slow-time scale with the aim of approximating the time-varying queueing systems by their quasistationary counterparts. Under smoothness conditions on the generators, asymptotic expansions of probability vectors and transition probability matrices are constructed. Uniform error bounds are obtained, and then sequences of occupation measures and their functionals are examined. Mean square error estimates of a sequence of occupation measures are obtained; a scaled sequence of functionals of occupation measures is shown to converge to a Gaussian process with zero mean. The representation of the variance of the limit process is also explicitly given. The results obtained are then applied to treat M t /M t /1 queues and Markov-modulated fluid buffer models.

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MARVÁ, M., J. C. POGGIALE, and R. BRAVO DE LA PARRA. "REDUCTION OF SLOW–FAST PERIODIC SYSTEMS WITH APPLICATIONS TO POPULATION DYNAMICS MODELS." Mathematical Models and Methods in Applied Sciences 22, no. 10 (August 13, 2012): 1250025. http://dx.doi.org/10.1142/s021820251250025x.

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This work deals with the approximate reduction of a nonautonomous two time scales ordinary differential equations system with periodic fast dynamics. We illustrate this technique with the analysis of two models belonging to different fields in ecology. On the one hand, we deal with a two patches periodic predator–prey model with a refuge for prey. Considering migrations between patches to be faster than local interaction allows us to study a three-dimensional system by means of a two-dimensional one. On the other hand, a two time scales periodic eco-epidemic model is addressed by considering two competing species, one of them being affected by a periodic SIR epidemic process which is faster than inter-species interactions. The difference between time scales allows us to study the asymptotic behavior of the four-dimensional system by means of a planar, reduced one. Furthermore, we propose a methodology straightforwardly applicable to a very large class of two time scales periodic systems.

23

He, Qi, and G. Yin. "Moderate deviations for time-varying dynamic systems driven by non-homogeneous Markov chains with Two-time Scales." Stochastics 86, no. 3 (October 25, 2013): 527–50. http://dx.doi.org/10.1080/17442508.2013.841695.

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Tian, Xue, and Yi Zhang. "Time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems." Royal Society Open Science 6, no. 11 (November 2019): 191248. http://dx.doi.org/10.1098/rsos.191248.

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The time-scales theory provides a powerful theoretical tool for studying differential and difference equations simultaneously. With regard to Herglotz type variational principle, this generalized variational principle can deal with non-conservative or dissipative problems. Combining the two tools, this paper aims to study time-scales Herglotz type Noether theorem for delta derivatives of Birkhoffian systems. We introduce the time-scales Herglotz type variational problem of Birkhoffian systems firstly and give the form of time-scales Pfaff–Herglotz action for delta derivatives. Then, time-scales Herglotz type Birkhoff’s equations for delta derivatives are derived by calculating the variation of the action. Furthermore, time-scales Herglotz type Noether symmetry for delta derivatives of Birkhoffian systems are defined. According to this definition, time-scales Herglotz type Noether identity and Noether theorem for delta derivatives of Birkhoffian systems are proposed and proved, which can become the ones for delta derivatives of Hamiltonian systems or Lagrangian systems in some special cases. Therefore, it is shown that the results of Birkhoffian formalism are more universal than Hamiltonian or Lagrangian formalism. Finally, the time-scales damped oscillator and a non-Hamiltonian Birkhoffian system are given to exemplify the superiority of the results.

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Lautenschlager, Björn, Sven Pfeiffer, Christian Schmidt, and Gerwald Lichtenberg. "Real-time iterative learning control-two applications with time scales between years and nanoseconds." International Journal of Adaptive Control and Signal Processing 33, no. 2 (November 12, 2018): 424–44. http://dx.doi.org/10.1002/acs.2946.

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Praly, L. "Topological orbital equivalence with asymptotic phase for a two time-scales discrete-time system." Mathematics of Control, Signals, and Systems 3, no. 3 (September 1990): 225–53. http://dx.doi.org/10.1007/bf02551370.

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Wang, Peiguang, and Xia Liu. "Practical stability of impulsive hybrid differential systems in terms of two measures on time scales." Nonlinear Analysis: Theory, Methods & Applications 65, no. 11 (December 2006): 2035–42. http://dx.doi.org/10.1016/j.na.2005.08.034.

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Fu, Zhi-Jun, Wen-Fang Xie, and Wei-Dong Luo. "Robust on-line nonlinear systems identification using multilayer dynamic neural networks with two-time scales." Neurocomputing 113 (August 2013): 16–26. http://dx.doi.org/10.1016/j.neucom.2012.11.041.

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Ma, Qing-Hua, and Josip Pečarić. "The bounds on the solutions of certain two-dimensional delay dynamic systems on time scales." Computers & Mathematics with Applications 61, no. 8 (April 2011): 2158–63. http://dx.doi.org/10.1016/j.camwa.2010.09.001.

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Anderson, Douglas, and William Hall. "Oscillation criteria for two-dimensional systems of first-order linear dynamic equations on time scales." Involve, a Journal of Mathematics 2, no. 1 (March 18, 2009): 1–16. http://dx.doi.org/10.2140/involve.2009.2.1.

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31

Qiang, Cheng-Xiu, Jian-Ping Sun, and Ya-Hong Zhao. "Exponential stability analysis for nonlinear time-varying perturbed systems on time scales." AIMS Mathematics 8, no. 5 (2023): 11131–50. http://dx.doi.org/10.3934/math.2023564.

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<abstract><p>This paper is concerned with the stability of nonlinear time-varying perturbed system on time scales under the assumption that the corresponding linear time-varying nominal system is uniformly exponentially stable. Some less conservative sufficient conditions for uniform exponential stability and uniform practical exponential stability are proposed by imposing different assumptions on the perturbation term. Compared with the traditional exponential stability results of perturbed systems, the time derivatives of related Lyapunov functions in this paper are not required to be negative definite for all time. The main tools employed are two Gronwall's inequalities on time scales. Some examples are also given to illustrate the effectiveness of the theoretical results.</p></abstract>

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Small, Mark A., J. F. Raney, and Terry J. Knapp. "Comparison of Two Reaction-Time Tasks and Their Relation to Intelligence." Perceptual and Motor Skills 65, no. 3 (December 1987): 867–70. http://dx.doi.org/10.2466/pms.1987.65.3.867.

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Two reaction time tasks were compared as measures of information-processing speed. A multiple R between the WAIS—R Full, Performance, and Verbal scales and several reaction time parameters was calculated for 28 college students. Results indicate that the reaction-time task used in exploring the relationships between speed of information processing and IQ can be less complex than those used to date.

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SANZ, LUIS, and RAFAEL BRAVO DE LA PARRA. "TIME SCALES IN A NON-AUTONOMOUS LINEAR DISCRETE MODEL." Mathematical Models and Methods in Applied Sciences 11, no. 07 (October 2001): 1203–35. http://dx.doi.org/10.1142/s0218202501001306.

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In this work we extend approximate aggregation methods in time discrete linear models to the case of time varying environments. Approximate aggregation consists of describing some features of the dynamics of a general system involving many coupled variables in terms of the dynamics of a reduced system with a few "global" variables. We present a time varying discrete model in which we distinguish two processes with different time scales. By defining the global variables as appropriate linear combinations of the state variables, we transform the system into a reduced one. The variables corresponding to the original and reduced systems can be related, therefore allowing one the study of the former in terms of the latter. The property of weak ergodicity, which has to do with the capacity of a system to become asymptotically independent of initial conditions, is explored for the original and reduced systems. The general method is also applied to aggregate a time-dependent multiregional model which appears in the field of population dynamics in two different cases: Fast migration with respect to demography and fast demography with respect to migration.

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LOBRY, CLAUDE, TEWFIK SARI, and SÉFIANE TOUHAMI. "FAST AND SLOW FEEDBACK IN SYSTEMS THEORY." Journal of Biological Systems 07, no. 03 (September 1999): 307–31. http://dx.doi.org/10.1142/s0218339099000206.

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Biological systems are complex systems with feedback. Very often the scales for the dynamics of the system and the dynamics of the feedback are very different. The mathematical tool used to deal with this different time scales is Tikhonov's theorem which permits to reduce the complexity of the system through suitable approximations. This paper presents a theory of two time scales feedback systems phrased in the language of Nonstandard Analysis (NSA), introduced in the sixties by A. Robinson. Our opinion is that this presentation is more understandable for a non-mathematicaly trained reader than the classical one. Th paper is entirely self contained. A short but comprehensive tutorial on NSA is provided and precise definitions for concepts from systems theory are given.

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Simard, SJ. "Fire Severity, Changing Scales, and How Things Hang Together." International Journal of Wildland Fire 1, no. 1 (1991): 23. http://dx.doi.org/10.1071/wf9910023.

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The paper describes attributes of space, time, and process in terms of their relations to wildland fire. It then presents a generic framework, based on eight interrelated scale classes for space, time, and process. The effects of changing scales are discussed in a wildland fire context. A five-layered (society, management, systems, fire, and weather), three-dimensional structure for wildland fire is presented. The paper also discusses inefficiencies and inadequacies inherent in systems with inconsistent scales. It then focuses on the effects of scale differences between fire behavior and fire danger and on an acceptable scale range suggested by the natural evolution of these two systems. The paper then defines fire severity and proposes two types of severity models — situation and extended. Finally, it discusses fundamental differences between situational and extended severity and appropriate space, time, and process attributes for both types of severity models.

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Gu, Mengqi, and Guo-Ping Jiang. "Observability of Discrete-Time Two-Time-Scale Multi-Agent Systems with Heterogeneous Features under Leader-Based Architecture." Mathematics 11, no. 8 (April 18, 2023): 1907. http://dx.doi.org/10.3390/math11081907.

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This paper investigates the observability of discrete-time two-time-scale multi-agent systems with heterogeneous features under leader–follower architecture. First, a singular perturbation difference model for the discussed system is established based on consensus agreement. Second, to eliminate the numerical ill-posed problem that may arise from the singularly perturbed small parameter that distinguishes different time scales in the observability analysis, the order of the system model is reduced using the boundary layer theory of the singular perturbation system to obtain a slow-time-scale subsystem and a fast-time-scale subsystem. Then, based on the matrix theory, some algebraic and graphical features that guarantee the observability of the system are obtained. Finally, the validity of the theoretical results is verified by a numerical example.

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CASUSO, E. "INTEGRAL TREATMENT FOR TIME EVOLUTION: THE GENERAL INTERACTIVITY." International Journal of Modern Physics A 14, no. 20 (August 10, 1999): 3239–52. http://dx.doi.org/10.1142/s0217751x99001524.

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Assuming that the unpredictability associated with many dynamical systems is an artefact of the differential treatment of their time evolution, we propose here an integral treatment as an alternative. We make the assumption that time is two-dimensional, and that the time distribution in the past of observables characterizing the dynamical system, is some characteristic "projection" of its time distribution in the future. We show here how this method can be used to predict the time evolution of several dynamically complex systems over long time intervals. The present work can be considered as the natural next step to the assumption of nonderivability for subatomic dynamical systems to explain the connection between Quantum Mechanics and General Relativity. Here we propose that matter and space–time are not only nonderivable but also show structural discontinuity. Starting with this premise we use continuity and derivability, but only as a first order approximation to reality. Extrapolation to very large or very small scales, or to predictions over long time scales for many natural systems on intermediate scales (human scales), may lead to chaotic behavior, or to nondeterministic or probabilistic theories.

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TURE SAVADKOOHI, A., and C. H. LAMARQUE. "DYNAMICS OF COUPLED DAHL TYPE AND NONSMOOTH SYSTEMS AT DIFFERENT SCALES OF TIME." International Journal of Bifurcation and Chaos 23, no. 07 (July 2013): 1350114. http://dx.doi.org/10.1142/s0218127413501149.

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Vibratory behavior of two coupled oscillators is studied. The main system — Dahl type — is coupled to a very light system with a nonsmooth potential that can be endowed for passively controlling the main system. Invariant manifold of the system at the fast time scale is revealed and the system behavior at slow time scale around the infinity of the fast time scale is detected. This can give us the chance to forecast all possible attractors of the system during energy exchange between the two oscillators.

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Hu, Xing, and Yongkun Li. "Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales." Fractal and Fractional 6, no. 5 (May 15, 2022): 268. http://dx.doi.org/10.3390/fractalfract6050268.

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First, we show the equivalence of two definitions of the left Riemann–Liouville fractional integral on time scales. Then, we establish and characterize fractional Sobolev space with the help of the notion of left Riemann–Liouville fractional derivative on time scales. At the same time, we define weak left fractional derivatives and demonstrate that they coincide with the left Riemann–Liouville ones on time scales. Next, we prove the equivalence of two kinds of norms in the introduced space and derive its completeness, reflexivity, separability, and some embedding. Finally, as an application, by constructing an appropriate variational setting, using the mountain pass theorem and the genus properties, the existence of weak solutions for a class of Kirchhoff-type fractional p-Laplacian systems on time scales with boundary conditions is studied, and three results of the existence of weak solutions for this problem is obtained.

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Yang, Wu, Yan-Wu Wang, Yanjun Shen, and Linqiang Pan. "Cluster synchronization of coupled delayed competitive neural networks with two time scales." Nonlinear Dynamics 90, no. 4 (October 5, 2017): 2767–82. http://dx.doi.org/10.1007/s11071-017-3836-z.

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de La Parra, Rafael Bravo, Eva Sánchez, and Pierre Auger. "Time Scales in Density Dependent Discrete Models." Journal of Biological Systems 05, no. 01 (March 1997): 111–29. http://dx.doi.org/10.1142/s0218339097000096.

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The aim of this work is to extend approximate aggregation methods for multi-time scale systems of nonlinear ordinary differential equations to time discrete models. Approximate aggregation consist on describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete time models with two different time scales, the fast one considered linear and the slow one generally nonlinear. We transform the system to make the global variables appear, and use a version of center manifold theory to build up the aggregated system. Simple forms of the aggregated system are enough for the local study of the asymptotic behaviour of the general system provided that it has certain stability under perturbations. The general method is applied to aggregate a multiregional density dependent Leslie model into a density dependent Leslie model in which the demographic rates are expressed in terms of the equilibrium proportions of individuals in the different patches.

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BAR-YAM, YANEER. "SUM RULE FOR MULTISCALE REPRESENTATIONS OF KINEMATICALLY DESCRIBED SYSTEMS." Advances in Complex Systems 05, no. 04 (December 2002): 409–31. http://dx.doi.org/10.1142/s0219525902000638.

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We derive a sum rule that constrains the scale based decomposition of the trajectories of finite systems of particles. The sum rule reflects a tradeoff between the finer and larger scale collective degrees of freedom. For short duration trajectories, where acceleration is irrelevant, the sum rule can be related to the moment of inertia and the kinetic energy (times a characteristic time squared). Thus, two nonequilibrium systems that have the same kinetic energy and moment of inertia can, when compared to each other, have different scales of behavior, but if one of them has larger scales of behavior than the other, it must compensate by also having smaller scales of behavior. In the context of coherence or correlation, the larger scale of behavior corresponds to the collective motion, while the smaller scales of behavior correspond to the relative motion of correlated particles. For longer duration trajectories, the sum rule includes the full effective moment of inertia of the system in space-time with respect to an external frame of reference, providing the possibility of relating the class of systems that can exist in the same space-time domain.

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OKUDA, H., and I. TSUDA. "A COUPLED CHAOTIC SYSTEM WITH DIFFERENT TIME SCALES: POSSIBLE IMPLICATIONS OF OBSERVATIONS BY DYNAMICAL SYSTEMS." International Journal of Bifurcation and Chaos 04, no. 04 (August 1994): 1011–22. http://dx.doi.org/10.1142/s0218127494000721.

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We present three curious phenomena in coupled chaotic systems with different time scales: a copy, an itinerant motion, and a “firework.” The phenomena obtained may possibly have implications on the interaction between two macroscopic systems, namely the “observing” and the “observed” systems.

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Peng, Chuanjun, Jianwei Xia, Jing Wang, and Hao Shen. "Distributed consensus for nonlinear multi-agent systems with two-time-scales: A hybrid reinforcement learning consensus algorithm." Information Sciences 641 (September 2023): 119091. http://dx.doi.org/10.1016/j.ins.2023.119091.

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Zhang, Xinli, and Shanliang Zhu. "Oscillation for a Nonlinear Dynamic System with a Forced Term on Time Scales." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/747838.

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We consider a class of two-dimensional nonlinear dynamic system with a forced term on a time scale𝕋and obtain sufficient conditions for all solutions of the system to be oscillatory. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that have been established by Saker, 2005, since our results are not restricted to the case whereb(t)≠0for allt∈𝕋andg(u)=u. Some examples are given to illustrate the results.

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Ren, Xiang, and Fei Hao. "Model-based dual-stage event-triggered control of linear system with two time scales." International Journal of Systems Science 51, no. 3 (February 11, 2020): 424–39. http://dx.doi.org/10.1080/00207721.2020.1716099.

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Postavaru, Octavian, and Antonela Toma. "Symmetries for Nonconservative Field Theories on Time Scale." Symmetry 13, no. 4 (March 26, 2021): 552. http://dx.doi.org/10.3390/sym13040552.

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Symmetries and their associated conserved quantities are of great importance in the study of dynamic systems. In this paper, we describe nonconservative field theories on time scales—a model that brings together, in a single theory, discrete and continuous cases. After defining Hamilton’s principle for nonconservative field theories on time scales, we obtain the associated Lagrange equations. Next, based on the Hamilton’s action invariance for nonconservative field theories on time scales under the action of some infinitesimal transformations, we establish symmetric and quasi-symmetric Noether transformations, as well as generalized quasi-symmetric Noether transformations. Once the Noether symmetry selection criteria are defined, the conserved quantities for the nonconservative field theories on time scales are identified. We conclude with two examples to illustrate the applicability of the theory.

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Xue, Wenqian, Jialu Fan, Victor G. Lopez, Jinna Li, Yi Jiang, Tianyou Chai, and Frank L. Lewis. "New Methods for Optimal Operational Control of Industrial Processes Using Reinforcement Learning on Two Time Scales." IEEE Transactions on Industrial Informatics 16, no. 5 (May 2020): 3085–99. http://dx.doi.org/10.1109/tii.2019.2912018.

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Li, Xiang-Jie, and Bing-Qi Zhu. "Interaction between the Westerlies and Asian Monsoons in the Middle Latitudes of China: Review and Prospect." Atmosphere 15, no. 3 (February 25, 2024): 274. http://dx.doi.org/10.3390/atmos15030274.

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The westerly circulation and the monsoon circulation are the two major atmospheric circulation systems affecting the middle latitudes of the Northern Hemisphere (NH), which have significant impacts on climate and environmental changes in the middle latitudes. However, until now, people’s understanding of the long-term paleoenvironmental changes in the westerly- and monsoon-controlled areas in China’s middle latitudes is not uniform, and the phase relationship between the two at different time scales is also controversial, especially the exception to the “dry gets drier, wet gets wetter” paradigm in global warming between the two. Based on the existing literature data published, integrated paleoenvironmental records, and comprehensive simulation results in recent years, this study systematically reviews the climate and environmental changes in the two major circulation regions in the mid-latitudes of China since the Middle Pleistocene, with a focus on exploring the phase relationship between the two systems at different time scales and its influencing mechanism. Through the reanalysis and comparative analysis of the existing data, we conclude that the interaction and relationship between the two circulation systems are relatively strong and close during the warm periods, but relatively weak during the cold periods. From the perspective of orbital, suborbital, and millennium time scales, the phase relationship between the westerly and Asian summer monsoon (ASM) circulations shows roughly in-phase, out-of-phase, and anti-phase transitions, respectively. There are significant differences between the impacts of the westerly and ASM circulations on the middle-latitude regions of northwest China, the Qinghai–Tibet Plateau, and eastern China. However, under the combined influence of varied environmental factors such as BHLSR (boreal high-latitude solar radiation), SST (sea surface temperature), AMOC (north Atlantic meridional overturning circulation), NHI (Northern Hemisphere ice volume), NAO (North Atlantic Oscillation), ITCZ (intertropical convergence zone), WPSH (western Pacific subtropical high), TIOA (tropical Indian Ocean anomaly), ENSO (El Niño/Southern Oscillation), CGT/SRP (global teleconnection/Silk Road pattern), etc., there is a complex and close coupling relationship between the two, and it is necessary to comprehensively consider their “multi-factor’s joint-action” mechanism and impact, while, in general, the dynamic mechanisms driving the changes of the westerly and ASM circulations are not the same at different time scales, such as orbital, suborbital, centennial to millennium, and decadal to interannual, which also leads to the formation of different types of phase relationships between the two at different time scales. Future studies need to focus on the impact of this “multi-factor linkage mechanism” and “multi-phase relationship” in distinguishing the interaction between the westerly and ASM circulation systems in terms of orbital, suborbital, millennium, and sub-millennium time scales.

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Zomer, Judith Y., Bart Vermeulen, and Antonius J. F. Hoitink. "Coexistence of two dune scales in a lowland river." Earth Surface Dynamics 11, no. 6 (December 13, 2023): 1283–98. http://dx.doi.org/10.5194/esurf-11-1283-2023.

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Abstract. A secondary scale of bedforms, superimposed on larger, primary dunes, has been observed in fluvial systems worldwide. This notwithstanding, very little is known about the morphological behavior and characteristics of this secondary scale. This study aims to better characterize and understand how two dune scales coexist in fluvial systems and how both scales adapt over time and space, considering their interdependence. The study is based on analysis of a large biweekly multibeam echo sounding dataset from the river Waal, a lowland sand-bedded river. Results reveal that the secondary dune scale is ubiquitous across space and time and not limited to specific flow or transport conditions. Whereas primary dunes lengthen during low flows, secondary dune height, lee slope angle, and length correlate with discharge. Secondary dune size and migration strongly depend on the primary dune lee slope angle and height. Secondary dunes can migrate over the lee slope of low-angled primary dunes, and their height is inversely correlated to the upstream primary dune height and lee slope angle. In the Waal river, a lateral variation in bed grain size, attributed to shipping, largely affects dune morphology. Primary dunes are lower and less often present in the southern lane, where grain sizes are smaller. Here, secondary bedforms are more developed. At peak discharge, secondary bedforms even become the dominant scale, whereas primary dunes entirely disappear but are re-established during lower flows.

Journal articles on the topic 'Two-Time scales systems'

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