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Journal articles on the topic 'Slow-fast systems'

Author: Grafiati

Published: 4 June 2021

Last updated: 19 February 2022

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1

Omelchenko, I., M. Rosenblum, and A. Pikovsky. "Synchronization of slow-fast systems." European Physical Journal Special Topics 191, no. 1 (December 2010): 3–14. http://dx.doi.org/10.1140/epjst/e2010-01338-4.

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2

da Silva, Paulo R., and Jaime R. de Moraes. "Piecewise-Smooth Slow–Fast Systems." Journal of Dynamical and Control Systems 27, no. 1 (March 4, 2020): 67–85. http://dx.doi.org/10.1007/s10883-020-09480-8.

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3

Rossetto, Bruno, Thierry Lenzini, Sofiane Ramdani, and Gilles Suchey. "Slow-Fast Autonomous Dynamical Systems." International Journal of Bifurcation and Chaos 08, no. 11 (November 1998): 2135–45. http://dx.doi.org/10.1142/s0218127498001765.

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In this paper, we consider a class of slow-fast autonomous dynamical systems, i.e. systems having a small parameter ∊ multiplying a component of velocity. At first, the singular perturbation method (∊ = 0+) is recalled. Then we consider the case ∊ ≠ 0. Starting from a working hypothesis and particularly in the case of a singular approximation, our purpose is to show that there exists slow manifolds which can be defined as the slow manifolds of a so-called tangent linear system. The method allowed us to plot the slow manifold and to go further into the qualitative study and the geometric characterization of attractors. As an example, we give the explicit slow manifold equation of the van der Pol limit cycle. The value of the parameter corresponding to bifurcations is computed. Other third order systems are also treated. The method is extended to dynamical systems with no small parameter, and, therefore, which have no singular approximations, but have at least one real and negative eigenvalue in a large domain. It is numerically shown from the Lorenz model and from a laser model that there exists slow manifolds which can be defined as the slow manifods of a so-called tangent linear system, as in the previous cases. The implicit equation of these slow manifolds has been calculated too.

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4

Ginoux, Jean-Marc. "Slow Invariant Manifolds of Slow–Fast Dynamical Systems." International Journal of Bifurcation and Chaos 31, no. 07 (June 15, 2021): 2150112. http://dx.doi.org/10.1142/s0218127421501121.

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Slow–fast dynamical systems, i.e. singularly or nonsingularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow–fast dynamical system.

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5

Brännström, N., and V. Gelfreich. "Drift of slow variables in slow-fast Hamiltonian systems." Physica D: Nonlinear Phenomena 237, no. 22 (November 2008): 2913–21. http://dx.doi.org/10.1016/j.physd.2008.05.001.

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6

NYE, V. A. "An Analysis of Fast-Slow Systems." IMA Journal of Mathematical Control and Information 2, no. 4 (1985): 295–317. http://dx.doi.org/10.1093/imamci/2.4.295.

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7

Kasthuri, Praveen, Induja Pavithran, Abin Krishnan, Samadhan A. Pawar, R. I. Sujith, Rohan Gejji, William Anderson, Norbert Marwan, and Jürgen Kurths. "Recurrence analysis of slow–fast systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 30, no. 6 (June 2020): 063152. http://dx.doi.org/10.1063/1.5144630.

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8

Bouchet, Freddy, Tobias Grafke, Tomás Tangarife, and Eric Vanden-Eijnden. "Large Deviations in Fast–Slow Systems." Journal of Statistical Physics 162, no. 4 (January 21, 2016): 793–812. http://dx.doi.org/10.1007/s10955-016-1449-4.

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9

Rinaldi, Sergio, and Alessandra Gragnani. "Destabilizing factors in slow–fast systems." Ecological Modelling 180, no. 4 (December 2004): 445–60. http://dx.doi.org/10.1016/j.ecolmodel.2003.05.001.

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10

Llibre, Jaume, Paulo R. da Silva, and Marco A. Teixeira. "Sliding Vector Fields via Slow--Fast Systems." Bulletin of the Belgian Mathematical Society - Simon Stevin 15, no. 5 (November 2008): 851–69. http://dx.doi.org/10.36045/bbms/1228486412.

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11

Bertram, Richard, and Jonathan E. Rubin. "Multi-timescale systems and fast-slow analysis." Mathematical Biosciences 287 (May 2017): 105–21. http://dx.doi.org/10.1016/j.mbs.2016.07.003.

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12

Shah, Kushal, Dmitry Turaev, Vassili Gelfreich, and Vered Rom-Kedar. "Equilibration of energy in slow–fast systems." Proceedings of the National Academy of Sciences 114, no. 49 (November 28, 2017): E10514—E10523. http://dx.doi.org/10.1073/pnas.1706341114.

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Ergodicity is a fundamental requirement for a dynamical system to reach a state of statistical equilibrium. However, in systems with several characteristic timescales, the ergodicity of the fast subsystem impedes the equilibration of the whole system because of the presence of an adiabatic invariant. In this paper, we show that violation of ergodicity in the fast dynamics can drive the whole system to equilibrium. To show this principle, we investigate the dynamics of springy billiards, which are mechanical systems composed of a small particle bouncing elastically in a bounded domain, where one of the boundary walls has finite mass and is attached to a linear spring. Numerical simulations show that the springy billiard systems approach equilibrium at an exponential rate. However, in the limit of vanishing particle-to-wall mass ratio, the equilibration rates remain strictly positive only when the fast particle dynamics reveal two or more ergodic components for a range of wall positions. For this case, we show that the slow dynamics of the moving wall can be modeled by a random process. Numerical simulations of the corresponding springy billiards and their random models show equilibration with similar positive rates.

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13

Engel, Maximilian, and Christian Kuehn. "Discretized fast-slow systems near transcritical singularities." Nonlinearity 32, no. 7 (May 30, 2019): 2365–91. http://dx.doi.org/10.1088/1361-6544/ab15c1.

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14

Brännström, Niklas, Emiliano de Simone, and Vassili Gelfreich. "Geometric shadowing in slow–fast Hamiltonian systems." Nonlinearity 23, no. 5 (April 13, 2010): 1169–84. http://dx.doi.org/10.1088/0951-7715/23/5/008.

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15

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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16

Teufel, Stefan. "Semiclassical approximations for adiabatic slow-fast systems." EPL (Europhysics Letters) 98, no. 5 (June 1, 2012): 50003. http://dx.doi.org/10.1209/0295-5075/98/50003.

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17

Arcidiacono, Luca, Maximilian Engel, and Christian Kuehn. "Discretized fast–slow systems near pitchfork singularities." Journal of Difference Equations and Applications 25, no. 7 (July 3, 2019): 1024–51. http://dx.doi.org/10.1080/10236198.2019.1647185.

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18

Meza-Sarmiento, Ingrid S., Regilene Oliveira, and Paulo R. da Silva. "Quadratic slow-fast systems on the plane." Nonlinear Analysis: Real World Applications 60 (August 2021): 103286. http://dx.doi.org/10.1016/j.nonrwa.2020.103286.

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19

RAMDANI, SOFIANE, BRUNO ROSSETTO, LEON O. CHUA, and RENÉ LOZI. "SLOW MANIFOLDS OF SOME CHAOTIC SYSTEMS WITH APPLICATIONS TO LASER SYSTEMS." International Journal of Bifurcation and Chaos 10, no. 12 (December 2000): 2729–44. http://dx.doi.org/10.1142/s0218127400001808.

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In this work we deal with slow–fast autonomous dynamical systems. We initially define them as being modeled by systems of differential equations having a small parameter multiplying one of their velocity components. In order to analyze their solutions, some being chaotic, we have proposed a mathematical analytic method based on an iterative approach [Rossetto et al., 1998]. Under some conditions, this method allows us to give an analytic equation of the slow manifold. This equation is obtained by considering that the slow manifold is locally defined by a plane orthogonal to the tangent system's left fast eigenvector. In this paper, we give another method to compute the slow manifold equation by using the tangent system's slow eigenvectors. This method allows us to give a geometrical characterization of the attractor and a global qualitative description of its dynamics. The method used to compute the equation of the slow manifold has been extended to systems having a real and negative eigenvalue in a large domain of the phase space, as it is the case with the Lorenz system. Indeed, we give the Lorenz slow manifold equation and this allows us to make a qualitative study comparing this model and Chua's model. Finally, we apply our results to derive the slow manifold equations of a nonlinear optical slow–fast system, namely, the optical parametric oscillator model.

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20

Schecter, Stephen, and Christos Sourdis. "Heteroclinic Orbits in Slow–Fast Hamiltonian Systems with Slow Manifold Bifurcations." Journal of Dynamics and Differential Equations 22, no. 4 (June 3, 2010): 629–55. http://dx.doi.org/10.1007/s10884-010-9171-4.

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21

Gedeon, Tomáš, Hiroshi Kokubu, Konstantin Mischaikow, and Hiroe Oka. "The Conley index for fast–slow systems II: Multidimensional slow variable." Journal of Differential Equations 225, no. 1 (June 2006): 242–307. http://dx.doi.org/10.1016/j.jde.2005.11.006.

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22

de Simoi, Jacopo, Carlangelo Liverani, Christophe Poquet, and Denis Volk. "Fast–Slow Partially Hyperbolic Systems Versus Freidlin–Wentzell Random Systems." Journal of Statistical Physics 166, no. 3-4 (September 28, 2016): 650–79. http://dx.doi.org/10.1007/s10955-016-1628-3.

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23

Zhang, Zhengdi, Zhangyao Chen, and Qinsheng Bi. "Modified slow-fast analysis method for slow-fast dynamical systems with two scales in frequency domain." Theoretical and Applied Mechanics Letters 9, no. 6 (November 2019): 358–62. http://dx.doi.org/10.1016/j.taml.2019.05.010.

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24

Morse, Matthew R., and Konstantinos Spiliopoulos. "Moderate deviations for systems of slow-fast diffusions." Asymptotic Analysis 105, no. 3-4 (November 3, 2017): 97–135. http://dx.doi.org/10.3233/asy-171434.

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25

Melis, Ward, and Giovanni Samaey. "Variance-reduced HMM for Stochastic Slow-fast Systems." Procedia Computer Science 80 (2016): 1255–66. http://dx.doi.org/10.1016/j.procs.2016.05.497.

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26

Kremer, E. "Slow motions in systems with fast modulated excitation." Journal of Sound and Vibration 383 (November 2016): 295–308. http://dx.doi.org/10.1016/j.jsv.2016.07.006.

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27

Wang, Ming-xin. "Fast–slow diffusion systems with nonlinear boundary conditions." Nonlinear Analysis: Theory, Methods & Applications 46, no. 6 (November 2001): 893–908. http://dx.doi.org/10.1016/s0362-546x(00)00156-5.

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28

Bykov, V., Y. Cherkinsky, V. Gol’dshtein, N. Krapivnik, and U. Maas. "Fast–slow vector fields of reaction–diffusion systems." IMA Journal of Applied Mathematics 85, no. 1 (February 2020): 67–86. http://dx.doi.org/10.1093/imamat/hxz035.

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Abstract A geometrically invariant concept of fast–slow vector fields perturbed by transport terms describing molecular diffusion is proposed in this paper. It is an extension of our concept of singularly perturbed vector fields for ODEs to reaction–diffusion systems with chemical reactions having wide range of characteristic time scales, while transport processes remain comparatively slow. Under this assumption we developed a decomposition into a fast and slow subsystems. It is assumed that the transport terms for the fast subsystem can be neglected to the leading order. For the slow subsystem we modify a concept of singularly perturbed profiles proposed in our previous works. The results are used to justify and to modify an algorithm of reaction–diffusion manifolds (REDIMs). The modified REDIM method is applied to the Michaelis–Menten model to illustrate the suggested approach.

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29

De Maesschalck, P., and M. Desroches. "Numerical Continuation Techniques for Planar Slow-Fast Systems." SIAM Journal on Applied Dynamical Systems 12, no. 3 (January 2013): 1159–80. http://dx.doi.org/10.1137/120877386.

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30

Bobieński, Marcin, Pavao Mardešić, and Dmitry Novikov. "Pseudo-abelian integrals on slow-fast Darboux systems." Annales de l’institut Fourier 63, no. 2 (2013): 417–30. http://dx.doi.org/10.5802/aif.2765.

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31

Jardón-Kojakhmetov, Hildeberto. "Formal normal form of Ak slow–fast systems." Comptes Rendus Mathematique 353, no. 9 (September 2015): 795–800. http://dx.doi.org/10.1016/j.crma.2015.06.009.

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32

Prohens, R., A. E. Teruel, and C. Vich. "Slow–fast n-dimensional piecewise linear differential systems." Journal of Differential Equations 260, no. 2 (January 2016): 1865–92. http://dx.doi.org/10.1016/j.jde.2015.09.046.

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33

De Simoi, Jacopo, and Carlangelo Liverani. "Limit theorems for fast–slow partially hyperbolic systems." Inventiones mathematicae 213, no. 3 (May 12, 2018): 811–1016. http://dx.doi.org/10.1007/s00222-018-0798-9.

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34

Kuznetsov, Yuri A., Simona Muratori, and Sergio Rinaldi. "Homoclinic bifurcations in slow-fast second order systems." Nonlinear Analysis: Theory, Methods & Applications 25, no. 7 (October 1995): 747–62. http://dx.doi.org/10.1016/0362-546x(94)e0005-2.

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35

Chevyrev, Ilya, Peter K. Friz, Alexey Korepanov, and Ian Melbourne. "Superdiffusive limits for deterministic fast–slow dynamical systems." Probability Theory and Related Fields 178, no. 3-4 (July 16, 2020): 735–70. http://dx.doi.org/10.1007/s00440-020-00988-5.

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Abstract We consider deterministic fast–slow dynamical systems on $$\mathbb {R}^m\times Y$$ R m × Y of the form $$\begin{aligned} {\left\{ \begin{array}{ll} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a\big (x_k^{(n)}\big ) + n^{-1/\alpha } b\big (x_k^{(n)}\big ) v(y_k), \\ y_{k+1} = f(y_k), \end{array}\right. } \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) ) + n - 1 / α b ( x k ( n ) ) v ( y k ) , y k + 1 = f ( y k ) , where $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) . Under certain assumptions we prove convergence of the m-dimensional process $$X_n(t)= x_{\lfloor nt \rfloor }^{(n)}$$ X n ( t ) = x ⌊ n t ⌋ ( n ) to the solution of the stochastic differential equation $$\begin{aligned} \mathrm {d} X = a(X)\mathrm {d} t + b(X) \diamond \mathrm {d} L_\alpha , \end{aligned}$$ d X = a ( X ) d t + b ( X ) ⋄ d L α , where $$L_\alpha $$ L α is an $$\alpha $$ α -stable Lévy process and $$\diamond $$ ⋄ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps f of Pomeau–Manneville type.

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36

Gutman, Semion, and Robert H. Martin. "Porous medium systems with slow and fast reactions." Journal of Differential Equations 74, no. 1 (July 1988): 86–119. http://dx.doi.org/10.1016/0022-0396(88)90020-4.

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37

Khoroshun, A. S. "Control of Takagi–Sugeno Fuzzy Fast/Slow Systems." International Applied Mechanics 54, no. 4 (July 2018): 443–53. http://dx.doi.org/10.1007/s10778-018-0897-8.

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38

Huzak, Renato, Vlatko Crnković, and Domagoj Vlah. "Fractal dimensions and two-dimensional slow-fast systems." Journal of Mathematical Analysis and Applications 501, no. 2 (September 2021): 125212. http://dx.doi.org/10.1016/j.jmaa.2021.125212.

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39

RINALDI, SERGIO. "SYNCHRONY IN SLOW–FAST METACOMMUNITIES." International Journal of Bifurcation and Chaos 19, no. 07 (July 2009): 2447–53. http://dx.doi.org/10.1142/s0218127409024220.

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The synchronization of metacommunities due to dispersal among patches is analyzed in the case of slow–fast populations. The analysis is performed by studying a standard model with the fast population dispersing when special meteorological conditions are present. This assumption fits very well with the peculiar nature of slow–fast systems and implies that metacommunities synchronize if the slow population accelerates during the outbreak of the fast population. This result shows great potential in the study of marine and fresh-water plankton communities as well as in the study of synchronization of insect-pest outbreaks in forests.

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40

Старко, Василь. "Categorization, Fast and Slow." East European Journal of Psycholinguistics 4, no. 1 (June 27, 2017): 205–12. http://dx.doi.org/10.29038/eejpl.2017.4.1.sta.

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The title of this study is inspired by Daniel Kahneman’s best-selling book Thinking, Fast and Slow. In it, the Nobel Prize winner explains in great detail the working of two systems of human reasoning: System 1, which is fast, automatic, associative, subconscious, involuntary and (nearly) effortless, and System 2, which is slow, intentional, logical, conscious, effortful and requires executive control, attention, and concentration. This distinction applies to human categorization as well. Each of the two labels refers, in fact, to a set of systems, which is why the designations Type 1 and Type 2 processes are preferable. The default-interventionist architecture presupposes the constant automatic activation of categories by Type 1 processes and interventions of Type 2 processes if necessary. Type 1 categorization relies on the ‘shallow’ linguistic representation of the world, while Type 2 uses ‘deep’ extralinguistic knowledge. A series of linguistic examples are analyzed to illustrate the differences between Type 1 and Type 2 categorization. A conclusion is drawn about the need to take this distinction into account in psycholinguistic and linguistic research on categorization. References Barrett, F., Tugade, M. M., & Engle, R. (2004). Individual differences in working memorycapacity in dual-process theories of the mind. Psychological Bulletin, 130(4), 553–573. Chaiken, S., & Trope, Y. (Eds.). (1999). Dual-process theories in social psychology. NewYork, NY: Guilford Press. Devine, P. G. (1989). Stereotypes and prejudice: Their automatic and controlledcomponents. Journal of Personality and Social Psychology, 56, 5–18. Evans, J. St. B. T., & Stanovich, K. (2013) Dual-process theories of higher cognition:Advancing the debate. Perspectives on Psychological Science, 8(3), 223–241. Geeraerts, D. (1993). Vagueness’s puzzles, polysemy’s vagaries. Cognitive Linguistics,4(3), 223–272. Heider, Eleanor Rosch (1973). On the internal structure of perceptual and semanticcategories. In: Cognitive Development and the Acquisition of Language, (pp. 111–144).T. E. Moore, (ed.). New York: Academic Press Kahneman, D. (2003). A perspective on judgement and choice. American Psychologist, 58,697–720. Kahneman, D. (2015). Thinking, Fast and Slow. New York: Farrar, Straus and Giroux. Kahneman, D., & Frederick, S. (2002). Representativeness revisited: Attribute substitutionin intuitive judgement. In: Heuristics and Biases: The Psychology of Intuitive Judgment,(pp. 49–81). T. Gilovich, D. Griffin, & D. Kahneman, (eds.). Cambridge, MA: CambridgeUniversity Press. Lakoff, G. (1973). Hedges: A study in meaning criteria and the logic of fuzzy concepts.Journal of Philosophical Logic, 2, 458–508. Lakoff, G. (1987). Women, Fire, and Dangerous Things. Chicago, London: University ofChicago Press. Reber, A. S. (1993). Implicit Learning and Tacit Knowledge. Oxford, England: OxfordUniversity Press. Stanovich, K. E. (1999). Who is Rational? Studies of Individual Differences in Reasoning.Mahwah, NJ: Erlbaum. Stanovich, K. E., & West, R F. (2000). Individual difference in reasoning: implications forthe rationality debate? Behavioural and Brain Sciences, 23, 645–726. Старко В. Категоризаційні кваліфікатори// Проблеми зіставної семантики. 2013,№ 11. С. 132–138.Starko, V. (2013). Katehoryzatsiini kvalifikatory. Problemy Zistavnoyi Semantyky, 11,132–138. Sun, R., Slusarz, P., & Terry, C. (2005). The interaction of the explicit and the implicit inskill learning: A dual-process approach. Psychological Review, 112, 159–192. Teasdale, J. D. (1999). Multi-level theories of cognition–emotion relations. In: Handbookof Cognition and Emotion, (pp. 665–681). T. Dalgleish & M. J. Power, (eds.). Chichester,England: Wiley. Wason, P. C., & Evans, J. St. B. T. (1975). Dual processes in reasoning? Cognition, 3,141–154. Whorf, B. L. (1956). The relation of habitual thought and behavior to language. In:Language, Thought, and Reality. Selected Writings of Benjamin Lee Whorf, (pp. 134–159). Cambridge, Massachusetts: The M.I.T. Press. (originally published in 1941) Wierzbicka, A. (1996). Semantic Primes and Universals. Oxford: Oxford UniversityPress.

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Buzzi, Claudio A., Paulo R. da Silva, and Marco A. Teixeira. "Slow–fast systems on algebraic varieties bordering piecewise-smooth dynamical systems." Bulletin des Sciences Mathématiques 136, no. 4 (June 2012): 444–62. http://dx.doi.org/10.1016/j.bulsci.2011.06.001.

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42

Romanenko, Victor D., and Yu L. Milyavskiy. "Slow and Fast Motion Control Coordination in Multirate Systems." Journal of Automation and Information Sciences 44, no. 5 (2012): 28–37. http://dx.doi.org/10.1615/jautomatinfscien.v44.i5.30.

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43

OTAKE, Yoshie. "RIKEN Compact Neutron Systems with Fast and Slow Neutrons." Plasma and Fusion Research 13 (March 28, 2018): 2401017. http://dx.doi.org/10.1585/pfr.13.2401017.

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44

Linkerhand, Mathias, and Claudius Gros. "Self-organized stochastic tipping in slow-fast dynamical systems." Mathematics and Mechanics of Complex Systems 1, no. 2 (April 16, 2013): 129–47. http://dx.doi.org/10.2140/memocs.2013.1.129.

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45

Kobrin, Alexander, and Vladimir Sobolev. "Integral manifolds of fast-slow systems in nonholonomic mechanics." Procedia Engineering 201 (2017): 556–60. http://dx.doi.org/10.1016/j.proeng.2017.09.610.

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46

GUPTA, KAJARI, and G. AMBIKA. "Dynamics of slow and fast systems on complex networks." Indian Academy of Sciences Conference Series 1, no. 1 (December 18, 2017): 9–15. http://dx.doi.org/10.29195/iascs.01.01.0003.

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47

Kuehn, Christian. "Uncertainty transformation via Hopf bifurcation in fast–slow systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2200 (April 2017): 20160346. http://dx.doi.org/10.1098/rspa.2016.0346.

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Propagation of uncertainty in dynamical systems is a significant challenge. Here we focus on random multiscale ordinary differential equation models. In particular, we study Hopf bifurcation in the fast subsystem for random initial conditions. We show that a random initial condition distribution can be transformed during the passage near a delayed/dynamic Hopf bifurcation: (i) to certain classes of symmetric copies, (ii) to an almost deterministic output, (iii) to a mixture distribution with differing moments and (iv) to a very restricted class of general distributions. We prove under which conditions the cases (i)–(iv) occur in certain classes vector fields.

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48

Michel, G., and G. P. Chini. "Multiple scales analysis of slow–fast quasi-linear systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2223 (March 2019): 20180630. http://dx.doi.org/10.1098/rspa.2018.0630.

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This article illustrates the application of multiple scales analysis to two archetypal quasi-linear systems; i.e. to systems involving fast dynamical modes, called fluctuations, that are not directly influenced by fluctuation–fluctuation nonlinearities but nevertheless are strongly coupled to a slow variable whose evolution may be fully nonlinear. In the first case, fast waves drive a slow, spatially inhomogeneous evolution of their celerity field. Multiple scales analysis confirms that, although the energyE, the angular frequencyωand the modal structure of the waves evolve, the wave actionE/ωis conserved in the absence of forcing and dissipation. In the second system, the fast modes undergo an instability that is saturated through a feedback on the slow variable. A new multi-scale analysis is developed to treat this case. The key technical point, confirmed by the analysis, is that the fluctuation energy and mode structure evolve slowly to ensure that the slow field remains in a state of near marginal stability. These two model systems appear to be generic, being representative of many if not all quasi-linear systems. In each case, numerical simulations of both the full and reduced dynamical systems are performed to highlight the accuracy and efficiency of the multiple scales approach. Python codes are provided as electronic supplementary material.

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de Mello, F. P., J. W. Feltes, T. F. Laskowski, and L. J. Oppel. "Simulating fast and slow dynamic effects in power systems." IEEE Computer Applications in Power 5, no. 3 (July 1992): 33–38. http://dx.doi.org/10.1109/67.143272.

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Gershgorin, Boris, and Andrew Majda. "A nonlinear test model for filtering slow-fast systems." Communications in Mathematical Sciences 6, no. 3 (2008): 611–49. http://dx.doi.org/10.4310/cms.2008.v6.n3.a5.

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