Journal articles: 'Dynamical Systems'

1

Hornstein, John, and V. I. Arnold. "Dynamical Systems." American Mathematical Monthly 96, no. 9 (November 1989): 861. http://dx.doi.org/10.2307/2324864.

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2

Chillingworth, D. R. J., D. K. Arrowsmith, and C. M. Place. "Dynamical Systems." Mathematical Gazette 79, no. 484 (March 1995): 233. http://dx.doi.org/10.2307/3620112.

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3

Jacob, G. "Dynamical systems." Mathematics and Computers in Simulation 42, no. 4-6 (November 1996): 639. http://dx.doi.org/10.1016/s0378-4754(97)84413-8.

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4

Rota, Gian-Carlo. "Dynamical systems." Advances in Mathematics 58, no. 3 (December 1985): 322. http://dx.doi.org/10.1016/0001-8708(85)90129-x.

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5

Meiss, James. "Dynamical systems." Scholarpedia 2, no. 2 (2007): 1629. http://dx.doi.org/10.4249/scholarpedia.1629.

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6

Li, Zhiming, Minghan Wang, and Guo Wei. "Induced hyperspace dynamical systems of symbolic dynamical systems." International Journal of General Systems 47, no. 8 (October 3, 2018): 809–20. http://dx.doi.org/10.1080/03081079.2018.1524467.

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7

Nasim, Imran, and Michael E. Henderson. "Dynamically Meaningful Latent Representations of Dynamical Systems." Mathematics 12, no. 3 (February 2, 2024): 476. http://dx.doi.org/10.3390/math12030476.

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Dynamical systems are ubiquitous in the physical world and are often well-described by partial differential equations (PDEs). Despite their formally infinite-dimensional solution space, a number of systems have long time dynamics that live on a low-dimensional manifold. However, current methods to probe the long time dynamics require prerequisite knowledge about the underlying dynamics of the system. In this study, we present a data-driven hybrid modeling approach to help tackle this problem by combining numerically derived representations and latent representations obtained from an autoencoder. We validate our latent representations and show they are dynamically interpretable, capturing the dynamical characteristics of qualitatively distinct solution types. Furthermore, we probe the topological preservation of the latent representation with respect to the raw dynamical data using methods from persistent homology. Finally, we show that our framework is generalizable, having been successfully applied to both integrable and non-integrable systems that capture a rich and diverse array of solution types. Our method does not require any prior dynamical knowledge of the system and can be used to discover the intrinsic dynamical behavior in a purely data-driven way.

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8

Caballero, Rubén, Alexandre N. Carvalho, Pedro Marín-Rubio, and José Valero. "Robustness of dynamically gradient multivalued dynamical systems." Discrete & Continuous Dynamical Systems - B 24, no. 3 (2019): 1049–77. http://dx.doi.org/10.3934/dcdsb.2019006.

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9

Landry, Nicholas W., and Juan G. Restrepo. "Hypergraph assortativity: A dynamical systems perspective." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 5 (May 2022): 053113. http://dx.doi.org/10.1063/5.0086905.

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The largest eigenvalue of the matrix describing a network’s contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.

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10

Akashi, Shigeo. "Embedding of expansive dynamical systems into symbolic dynamical systems." Reports on Mathematical Physics 46, no. 1-2 (August 2000): 11–14. http://dx.doi.org/10.1016/s0034-4877(01)80003-3.

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11

Szpiro, George G. "Measuring dynamical noise in dynamical systems." Physica D: Nonlinear Phenomena 65, no. 3 (May 1993): 289–99. http://dx.doi.org/10.1016/0167-2789(93)90164-v.

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12

Wang, Guangwa, and Yongluo Cao. "Dynamical Spectrum in Random Dynamical Systems." Journal of Dynamics and Differential Equations 26, no. 1 (November 27, 2013): 1–20. http://dx.doi.org/10.1007/s10884-013-9340-3.

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13

van Gelder, Tim. "The dynamical hypothesis in cognitive science." Behavioral and Brain Sciences 21, no. 5 (October 1998): 615–28. http://dx.doi.org/10.1017/s0140525x98001733.

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According to the dominant computational approach in cognitive science, cognitive agents are digital computers; according to the alternative approach, they are dynamical systems. This target article attempts to articulate and support the dynamical hypothesis. The dynamical hypothesis has two major components: the nature hypothesis (cognitive agents are dynamical systems) and the knowledge hypothesis (cognitive agents can be understood dynamically). A wide range of objections to this hypothesis can be rebutted. The conclusion is that cognitive systems may well be dynamical systems, and only sustained empirical research in cognitive science will determine the extent to which that is true.

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14

Lyle, Cory. "Dynamical Systems Theory." International Journal of Communication and Linguistic Studies 10, no. 1 (2013): 47–58. http://dx.doi.org/10.18848/2327-7882/cgp/v10i01/58272.

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15

Erik Fornæss, John. "Sustainable dynamical systems." Discrete & Continuous Dynamical Systems - A 9, no. 6 (2003): 1361–86. http://dx.doi.org/10.3934/dcds.2003.9.1361.

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16

Whitley, D. C., and Peter A. Cook. "Nonlinear Dynamical Systems." Mathematical Gazette 72, no. 459 (March 1988): 69. http://dx.doi.org/10.2307/3618016.

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17

Ali Akbar, K., V. Kannan, and I. Subramania Pillai. "Simple dynamical systems." Applied General Topology 20, no. 2 (October 1, 2019): 307. http://dx.doi.org/10.4995/agt.2019.7910.

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<div class="page" title="Page 1"><div class="layoutArea"><div class="column">In this paper, we study the class of simple systems on R induced by homeomorphisms having finitely many non-ordinary points. We characterize the family of homeomorphisms on R having finitely many non-ordinary points upto (order) conjugacy. For x,y ∈ R, we say x ∼ y on a dynamical system (R,f) if x and y have same dynamical properties, which is an equivalence relation. Said precisely, x ∼ y if there exists an increasing homeomorphism h : R → R such that h ◦ f = f ◦ h and h(x) = y. An element x ∈ R is ordinary in (R,f) if its equivalence class [x] is a neighbourhood of it. </div></div></div>

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18

Willems, Jan C. "Dissipative Dynamical Systems." European Journal of Control 13, no. 2-3 (January 2007): 134–51. http://dx.doi.org/10.3166/ejc.13.134-151.

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19

Gilmore, C. "Linear Dynamical Systems." Irish Mathematical Society Bulletin 0086 (2020): 47–78. http://dx.doi.org/10.33232/bims.0086.47.78.

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20

Martin, Gaven J. "Complex dynamical systems." International Journal of Mathematical Education in Science and Technology 25, no. 6 (November 1994): 879–97. http://dx.doi.org/10.1080/0020739940250613.

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21

Bender, Carl M., Darryl D. Holm, and Daniel W. Hook. "Complexified dynamical systems." Journal of Physics A: Mathematical and Theoretical 40, no. 32 (July 24, 2007): F793—F804. http://dx.doi.org/10.1088/1751-8113/40/32/f02.

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22

Goebel, Rafal, Ricardo G. Sanfelice, and Andrew R. Teel. "Hybrid dynamical systems." IEEE Control Systems 29, no. 2 (April 2009): 28–93. http://dx.doi.org/10.1109/mcs.2008.931718.

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23

Honerkamp, Joseph, and James D. Meiss. "Stochastic Dynamical Systems." Physics Today 47, no. 12 (December 1994): 63–64. http://dx.doi.org/10.1063/1.2808753.

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24

Calogero, F. "Isochronous dynamical systems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1939 (March 28, 2011): 1118–36. http://dx.doi.org/10.1098/rsta.2010.0250.

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This is a terse review of recent results on isochronous dynamical systems, namely systems of (first-order, generally nonlinear) ordinary differential equations (ODEs) featuring an open set of initial data (which might coincide with the entire set of all initial data), from which emerge solutions all of which are completely periodic (i.e. periodic in all their components) with a fixed period (independent of the initial data, provided they are within the isochrony region). A leitmotif of this presentation is that ‘isochronous systems are not rare’. Indeed, it is shown how any (autonomous) dynamical system can be modified or extended so that the new (also autonomous) system thereby obtained is isochronous with an arbitrarily assigned period T , while its dynamics, over time intervals much shorter than the period T , mimics closely that of the original system, or even, over an arbitrarily large fraction of its period T , coincides exactly with that of the original system. It is pointed out that this fact raises the issue of developing criteria providing, for a dynamical system, some kind of measure associated with a finite time scale of the complexity of its behaviour (while the current, standard definitions of integrable versus chaotic dynamical systems are related to the behaviour of a system over infinite time).

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25

Calogero, F. "Isochronous dynamical systems." Applicable Analysis 85, no. 1-3 (January 2006): 5–22. http://dx.doi.org/10.1080/00036810500277926.

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26

Moon, F. C. "Nonlinear Dynamical Systems." Applied Mechanics Reviews 38, no. 10 (October 1, 1985): 1284–86. http://dx.doi.org/10.1115/1.3143693.

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New discoveries have been made recently about the nature of complex motions in nonlinear dynamics. These new concepts are changing many of the ideas about dynamical systems in physics and in particular fluid and solid mechanics. One new phenomenon is the apparently random or chaotic output of deterministic systems with no random inputs. Another is the sensitivity of the long time dynamic history of many systems to initial starting conditions even when the motion is not chaotic. New mathematical ideas to describe this phenomenon are entering the field of nonlinear vibrations and include ideas from topology and analysis such as Poincare´ maps, fractal dimensions, Cantor sets and strange attractors. These new ideas are already making their way into the engineering vibrations laboratory. Further research in this field is needed to extend these new ideas to multi-degree of freedom and continuum vibration problems. Also the loss of predictability in certain nonlinear problems should be studied for its impact on the field of numerical simulation in mechanics of nonlinear materials and structures.

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27

Saavedra, Joel, Ricardo Troncoso, and Jorge Zanelli. "Degenerate dynamical systems." Journal of Mathematical Physics 42, no. 9 (September 2001): 4383–90. http://dx.doi.org/10.1063/1.1389088.

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28

Montenbruck, Jan Maximilian, and Shen Zeng. "Collinear dynamical systems." Systems & Control Letters 105 (July 2017): 34–43. http://dx.doi.org/10.1016/j.sysconle.2017.04.008.

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29

Tresser, Charles, and Patrick A. Worfolk. "Resynchronizing dynamical systems." Physics Letters A 229, no. 5 (May 1997): 293–98. http://dx.doi.org/10.1016/s0375-9601(97)00206-5.

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30

Craciun, Gheorghe, Alicia Dickenstein, Anne Shiu, and Bernd Sturmfels. "Toric dynamical systems." Journal of Symbolic Computation 44, no. 11 (November 2009): 1551–65. http://dx.doi.org/10.1016/j.jsc.2008.08.006.

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31

Bellomo, Nicola, and Ahmed Elaiw. "Nonlinear dynamical systems." Physics of Life Reviews 22-23 (December 2017): 22–23. http://dx.doi.org/10.1016/j.plrev.2017.07.005.

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32

Czitrom, Veronica. "Linear Dynamical Systems." Technometrics 31, no. 1 (February 1989): 125–26. http://dx.doi.org/10.1080/00401706.1989.10488495.

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33

DOBBS, NEIL, and MIKKO STENLUND. "Quasistatic dynamical systems." Ergodic Theory and Dynamical Systems 37, no. 8 (May 12, 2016): 2556–96. http://dx.doi.org/10.1017/etds.2016.9.

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We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behavior as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the ‘obvious’ centering suggested by the initial distribution sometimes fails to yield the expected diffusion.

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34

Hinich, Melvin J. "SAMPLING DYNAMICAL SYSTEMS." Macroeconomic Dynamics 3, no. 4 (December 1999): 602–9. http://dx.doi.org/10.1017/s1365100599013073.

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Linear dynamical systems are widely used in many different fields from engineering to economics. One simple but important class of such systems is called the single-input transfer function model. Suppose that all variables of the system are sampled for a period using a fixed sample rate. The central issue of this paper is the determination of the smallest sampling rate that will yield a sample that will allow the investigator to identify the discrete-time representation of the system. A critical sampling rate exists that will identify the model. This rate, called the Nyquist rate, is twice the highest frequency component of the system. Sampling at a lower rate will result in an identification problem that is serious. The standard assumptions made about the model and the unobserved innovation errors in the model protect the investigators from the identification problem and resulting biases of undersampling. The critical assumption that is needed to identify an undersampled system is that at least one of the exogenous time series be white noise.

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35

Koutsogiannis, Andreas. "Rational dynamical systems." Topology and its Applications 159, no. 7 (April 2012): 1993–2003. http://dx.doi.org/10.1016/j.topol.2011.04.031.

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36

Rota, Gian-Carlo. "Smooth dynamical systems." Advances in Mathematics 56, no. 3 (June 1985): 319. http://dx.doi.org/10.1016/0001-8708(85)90041-6.

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37

Isidori, A. "Nonlinear dynamical systems." Automatica 26, no. 5 (September 1990): 939–40. http://dx.doi.org/10.1016/0005-1098(90)90016-b.

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38

Muraskin, M. "Dynamical lattice systems." Computers & Mathematics with Applications 28, no. 7 (October 1994): 77–95. http://dx.doi.org/10.1016/0898-1221(94)00162-6.

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39

Smith, Peter. "Discrete dynamical systems." Agricultural Systems 42, no. 3 (January 1993): 307–10. http://dx.doi.org/10.1016/0308-521x(93)90060-f.

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40

García-Morales, Vladimir. "Semipredictable dynamical systems." Communications in Nonlinear Science and Numerical Simulation 39 (October 2016): 81–98. http://dx.doi.org/10.1016/j.cnsns.2016.02.022.

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41

Akin, Ethan. "Simplicial dynamical systems." Memoirs of the American Mathematical Society 140, no. 667 (1999): 0. http://dx.doi.org/10.1090/memo/0667.

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42

Longtin, Andre. "Stochastic dynamical systems." Scholarpedia 5, no. 4 (2010): 1619. http://dx.doi.org/10.4249/scholarpedia.1619.

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43

Moehlis, Jeff, and Edgar Knobloch. "Equivariant dynamical systems." Scholarpedia 2, no. 10 (2007): 2510. http://dx.doi.org/10.4249/scholarpedia.2510.

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44

Kolyada, Sergiy, and Ä½ubomÃr Snoha. "Minimal dynamical systems." Scholarpedia 4, no. 11 (2009): 5803. http://dx.doi.org/10.4249/scholarpedia.5803.

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45

Penny, W., Z. Ghahramani, and K. Friston. "Bilinear dynamical systems." Philosophical Transactions of the Royal Society B: Biological Sciences 360, no. 1457 (May 29, 2005): 983–93. http://dx.doi.org/10.1098/rstb.2005.1642.

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In this paper, we propose the use of bilinear dynamical systems (BDS)s for model-based deconvolution of fMRI time-series. The importance of this work lies in being able to deconvolve haemodynamic time-series, in an informed way, to disclose the underlying neuronal activity. Being able to estimate neuronal responses in a particular brain region is fundamental for many models of functional integration and connectivity in the brain. BDSs comprise a stochastic bilinear neurodynamical model specified in discrete time, and a set of linear convolution kernels for the haemodynamics. We derive an expectation-maximization (EM) algorithm for parameter estimation, in which fMRI time-series are deconvolved in an E-step and model parameters are updated in an M-Step. We report preliminary results that focus on the assumed stochastic nature of the neurodynamic model and compare the method to Wiener deconvolution.

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46

Fortuna, Luigi, Arturo Buscarino, and Mattia Frasca. "Imperfect dynamical systems." Chaos, Solitons & Fractals 117 (December 2018): 200. http://dx.doi.org/10.1016/j.chaos.2018.10.016.

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47

Stefański, Krzysztof. "Dynamical systems III." Reports on Mathematical Physics 31, no. 3 (June 1992): 373–75. http://dx.doi.org/10.1016/0034-4877(92)90027-x.

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48

ROSEN, R. "Beyond dynamical systems." Journal of Social and Biological Systems 14, no. 2 (1991): 217–20. http://dx.doi.org/10.1016/0140-1750(91)90337-p.

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49

Barrett, Chris L., Henning S. Mortveit, and Christian M. Reidys. "Sequential dynamical systems." Artificial Life and Robotics 6, no. 4 (December 2002): 167–69. http://dx.doi.org/10.1007/bf02481261.

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50

Hübler, Alfred W. "“Homeopathic” dynamical systems." Complexity 13, no. 3 (January 2008): 8–11. http://dx.doi.org/10.1002/cplx.20220.

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Journal articles on the topic 'Dynamical Systems'

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