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Journal articles on the topic 'KAM and Nekhoroshev theory'

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Published: 4 June 2021

Last updated: 31 January 2023

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1

Benettin, Giancarlo, Francesco Fassò, and Massimiliano Guzzo. "Nekhoroshev-Stability ofL4andL5in the Spatial Restricted Problem." International Astronomical Union Colloquium 172 (1999): 445–46. http://dx.doi.org/10.1017/s0252921100073097.

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The Lagrangian equilateral pointsL4andL5of the restricted circular three-body problem are elliptic for all values of the reduced massμbelow Routh’s critical massμR≈ .0385. In the spatial case, because of the possibility of Arnold diffusion, KAM theory does not provide Lyapunov-stability. Nevertheless, one can consider the so-called ‘Nekhoroshev-stability’: denoting byda convenient distance from the equilibrium point, one asks whetherfor any small єe > 0, with positiveaandb. Until recently this problem, as more generally the problem of Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems, was studied only under some arithmetic conditions on the frequencies, and thus onμ(see e.g .Giorgilli, 1989). Our aim was instead considering all values ofμup toμR. As a matter of fact, Nekhoroshev-stability of elliptic equilibria, without any arithmetic assumption on the frequencies, was proved recently under the hypothesis that the fourth order Birkhoff normal form of the Hamiltonian exists and satisfies a ‘quasi-convexity’ assumption (Fassòet al, 1998; Guzzoet al, 1998; Niedermann, 1998).

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2

Guzzo, Massimiliano. "Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems." International Astronomical Union Colloquium 172 (1999): 443–44. http://dx.doi.org/10.1017/s0252921100073085.

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Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

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3

Wiggins, S., and A. M. Mancho. "Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori." Nonlinear Processes in Geophysics 21, no. 1 (February 4, 2014): 165–85. http://dx.doi.org/10.5194/npg-21-165-2014.

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Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

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4

Li, Yong, and Yingfei Yi. "Nekhoroshev and KAM Stabilities in Generalized Hamiltonian Systems." Journal of Dynamics and Differential Equations 18, no. 3 (July 15, 2006): 577–614. http://dx.doi.org/10.1007/s10884-006-9025-2.

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5

Bounemoura, Abed, and Stéphane Fischler. "A Diophantine duality applied to the KAM and Nekhoroshev theorems." Mathematische Zeitschrift 275, no. 3-4 (May 22, 2013): 1135–67. http://dx.doi.org/10.1007/s00209-013-1174-5.

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6

Bounemoura, Abed, and Laurent Niederman. "Generic Nekhoroshev theory without small divisors." Annales de l’institut Fourier 62, no. 1 (2012): 277–324. http://dx.doi.org/10.5802/aif.2706.

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7

Moan, Per Christian. "On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth." Nonlinearity 17, no. 1 (September 29, 2003): 67–83. http://dx.doi.org/10.1088/0951-7715/17/1/005.

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8

MacKay, R. S., and I. C. Percival. "Converse KAM: Theory and practice." Communications in Mathematical Physics 98, no. 4 (December 1985): 469–512. http://dx.doi.org/10.1007/bf01209326.

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9

Salamon, Dietmar, and Eduard Zehnder. "KAM theory in configuration space." Commentarii Mathematici Helvetici 64, no. 1 (December 1989): 84–132. http://dx.doi.org/10.1007/bf02564665.

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10

Delshams, Amadeu, and Pere Gutiérrez. "Effective Stability and KAM Theory." Journal of Differential Equations 128, no. 2 (July 1996): 415–90. http://dx.doi.org/10.1006/jdeq.1996.0102.

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11

Henrici, Andreas. "Nekhoroshev Stability for the Dirichlet Toda Lattice." Symmetry 10, no. 10 (October 16, 2018): 506. http://dx.doi.org/10.3390/sym10100506.

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In this work, we prove a Nekhoroshev-type stability theorem for the Toda lattice with Dirichlet boundary conditions, i.e., with fixed ends. The Toda lattice is a member of the family of Fermi-Pasta-Ulam (FPU) chains, and in view of the unexpected recurrence phenomena numerically observed in these chains, it has been a long-standing research aim to apply the theory of perturbed integrable systems to these chains, in particular to the Toda lattice which has been shown to be a completely integrable system. The Dirichlet Toda lattice can be treated mathematically by using symmetries of the periodic Toda lattice. Precisely, by treating the phase space of the former system as an invariant subset of the latter one, namely as the fixed point set of an important symmetry of the periodic lattice, the results already obtained for the periodic lattice can be used to obtain analogous results for the Dirichlet lattice. In this way, we transfer our stability results for the periodic lattice to the Dirichlet lattice. The Nekhoroshev theorem is a perturbation theory result which does not have the probabilistic character of related theorems, and the lattice with fixed ends is more important for applications than the periodic one.

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12

Giorgilli, Antonio, Ugo Locatelli, and Marco Sansottera. "Kolmogorov and Nekhoroshev theory for the problem of three bodies." Celestial Mechanics and Dynamical Astronomy 104, no. 1-2 (March 3, 2009): 159–73. http://dx.doi.org/10.1007/s10569-009-9192-7.

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13

Chen, Yuhan, Takashi Matsubara, and Takaharu Yaguchi. "KAM Theory Meets Statistical Learning Theory: Hamiltonian Neural Networks with Non-zero Training Loss." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 6 (June 28, 2022): 6322–32. http://dx.doi.org/10.1609/aaai.v36i6.20582.

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Many physical phenomena are described by Hamiltonian mechanics using an energy function (Hamiltonian). Recently, the Hamiltonian neural network, which approximates the Hamiltonian by a neural network, and its extensions have attracted much attention. This is a very powerful method, but theoretical studies are limited. In this study, by combining the statistical learning theory and KAM theory, we provide a theoretical analysis of the behavior of Hamiltonian neural networks when the learning error is not completely zero. A Hamiltonian neural network with non-zero errors can be considered as a perturbation from the true dynamics, and the perturbation theory of the Hamilton equation is widely known as KAM theory. To apply KAM theory, we provide a generalization error bound for Hamiltonian neural networks by deriving an estimate of the covering number of the gradient of the multi-layer perceptron, which is the key ingredient of the model. This error bound gives a sup-norm bound on the Hamiltonian that is required in the application of KAM theory.

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14

Berti, Massimiliano. "KAM Theory for Partial Differential Equations." Analysis in Theory and Applications 35, no. 3 (June 2019): 235–67. http://dx.doi.org/10.4208/ata.oa-0013.

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15

Llave, R. de la, A. González, À. Jorba, and J. Villanueva. "KAM theory without action-angle variables." Nonlinearity 18, no. 2 (January 22, 2005): 855–95. http://dx.doi.org/10.1088/0951-7715/18/2/020.

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16

Giorgilli, Antonio. "Classical constructive methods in KAM theory." Planetary and Space Science 46, no. 11-12 (November 1998): 1441–51. http://dx.doi.org/10.1016/s0032-0633(98)00045-2.

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17

Sevryuk, M. B. "The finite-dimensional reversible KAM theory." Physica D: Nonlinear Phenomena 112, no. 1-2 (January 1998): 132–47. http://dx.doi.org/10.1016/s0167-2789(97)00207-8.

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18

Bricmont, Jean, Krzysztof Gawędzki, and Antti Kupiainen. "KAM Theorem and Quantum Field Theory." Communications in Mathematical Physics 201, no. 3 (April 1, 1999): 699–727. http://dx.doi.org/10.1007/s002200050573.

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19

Knauf, A. "Closed orbits and converse KAM theory." Nonlinearity 3, no. 3 (August 1, 1990): 961–73. http://dx.doi.org/10.1088/0951-7715/3/3/019.

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20

Evans, Lawrence Craig. "New identities for Weak KAM theory." Chinese Annals of Mathematics, Series B 38, no. 2 (March 2017): 379–92. http://dx.doi.org/10.1007/s11401-017-1074-9.

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21

Hanßmann, Heinz. "Non-degeneracy conditions in kam theory." Indagationes Mathematicae 22, no. 3-4 (December 2011): 241–56. http://dx.doi.org/10.1016/j.indag.2011.09.005.

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22

Cardaliaguet, Pierre, and Marco Masoero. "Weak KAM theory for potential MFG." Journal of Differential Equations 268, no. 7 (March 2020): 3255–98. http://dx.doi.org/10.1016/j.jde.2019.09.060.

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23

Glenn, Jerry, and Lothar Walsdorf. "Über Berge kam ich." World Literature Today 62, no. 4 (1988): 650. http://dx.doi.org/10.2307/40144597.

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24

Glenn, Jerry, and Ernst Meister. "Es kam die Nachricht." World Literature Today 66, no. 1 (1992): 131. http://dx.doi.org/10.2307/40147961.

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25

Davini, Andrea, and Maxime Zavidovique. "Weak KAM theory for nonregular commuting Hamiltonians." Discrete & Continuous Dynamical Systems - B 18, no. 1 (2013): 57–94. http://dx.doi.org/10.3934/dcdsb.2013.18.57.

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26

Bogdanov, Rifkat Ibragimovich, and Mihail Rifkatovich Bogdanov. "Numerical Data in Weakly-Dissipative KAM-Theory." Advanced Materials Research 875-877 (February 2014): 880–84. http://dx.doi.org/10.4028/www.scientific.net/amr.875-877.880.

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In the article are presented as figures numerical data calculation of the basic thermodynamic variables such as dependence of the thermodynamic potentials from the temperature and pressure, and the geometric characteristics of the dynamics like the center of mass of trial particle. The dynamics is described by a simple Euler discretization of family of vector fields arising in the Bogdanov-Takens bifurcation.

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27

Haro, Àlex. "Converse KAM theory for monotone positive symplectomorphisms." Nonlinearity 12, no. 5 (August 13, 1999): 1299–322. http://dx.doi.org/10.1088/0951-7715/12/5/306.

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28

Sevryuk, Mikhail B. "Partial preservation of frequencies in KAM theory." Nonlinearity 19, no. 5 (April 10, 2006): 1099–140. http://dx.doi.org/10.1088/0951-7715/19/5/005.

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29

MacKay, R. S., J. D. Meiss, and J. Stark. "Converse KAM theory for symplectic twist maps." Nonlinearity 2, no. 4 (November 1, 1989): 555–70. http://dx.doi.org/10.1088/0951-7715/2/4/004.

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30

Gidea, Marian, James D. Meiss, Ilie Ugarcovici, and Howard Weiss. "Applications of KAM theory to population dynamics." Journal of Biological Dynamics 5, no. 1 (January 2011): 44–63. http://dx.doi.org/10.1080/17513758.2010.488301.

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31

Gomes, Diogo A., and Adam Oberman. "Viscosity solutions methods for converse KAM theory." ESAIM: Mathematical Modelling and Numerical Analysis 42, no. 6 (September 25, 2008): 1047–64. http://dx.doi.org/10.1051/m2an:2008035.

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32

Levi, Mark. "KAM theory for particles in periodic potentials." Ergodic Theory and Dynamical Systems 10, no. 4 (December 1990): 777–85. http://dx.doi.org/10.1017/s0143385700005897.

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AbstractIt is shown that the system of the form x + V′ (x) = p (t) with periodic V and p and with (p) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V ∈ C(5) and p ∈ L1; furthermore, for any solution x(t) there exists a velocity bound c for all time: |x(t)| < c for all t ∈ R. For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.

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33

Bambusi, D., M. Berti, and E. Magistrelli. "Degenerate KAM theory for partial differential equations." Journal of Differential Equations 250, no. 8 (April 2011): 3379–97. http://dx.doi.org/10.1016/j.jde.2010.11.002.

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34

Khesin, Boris, Sergei Kuksin, and Daniel Peralta-Salas. "KAM theory and the 3D Euler equation." Advances in Mathematics 267 (December 2014): 498–522. http://dx.doi.org/10.1016/j.aim.2014.09.009.

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35

Evans, Lawrence C. "Further PDE methods for weak KAM theory." Calculus of Variations and Partial Differential Equations 35, no. 4 (November 18, 2008): 435–62. http://dx.doi.org/10.1007/s00526-008-0214-1.

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36

Yu, Yifeng. "L∞ variational problems and weak KAM theory." Communications on Pure and Applied Mathematics 60, no. 8 (2007): 1111–47. http://dx.doi.org/10.1002/cpa.20173.

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37

Trujillo, Frank. "Uniqueness properties of the KAM curve." Discrete & Continuous Dynamical Systems 41, no. 11 (2021): 5165. http://dx.doi.org/10.3934/dcds.2021072.

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<p style='text-indent:20px;'>Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.</p>

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38

Sevryuk, M. B. "Kam-stable Hamiltonians." Journal of Dynamical and Control Systems 1, no. 3 (July 1995): 351–66. http://dx.doi.org/10.1007/bf02269374.

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39

Chierchia, Luigi, and Gabriella Pinzari. "Properly-degenerate KAM theory (following V. I. Arnold)." Discrete & Continuous Dynamical Systems - S 3, no. 4 (2010): 545–78. http://dx.doi.org/10.3934/dcdss.2010.3.545.

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40

Berti, Massimiliano, Luca Biasco, and Michela Procesi. "KAM theory for the hamiltonian derivative wave equation." Annales scientifiques de l'École normale supérieure 46, no. 2 (2013): 301–73. http://dx.doi.org/10.24033/asens.2190.

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41

Broer, Henk W. "KAM theory: The legacy of Kolmogorov's 1954 paper." Bulletin of the American Mathematical Society 41, no. 04 (February 9, 2004): 507–22. http://dx.doi.org/10.1090/s0273-0979-04-01009-2.

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42

JianSheng, GENG, XU JunXiang, and YOU JianGong. "KAM theory in finite and infinite dimensional spaces." SCIENTIA SINICA Mathematica 47, no. 1 (December 12, 2016): 77–96. http://dx.doi.org/10.1360/n012016-00154.

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43

Walsh, James. "A Window into the World of KAM Theory." Mathematics Magazine 93, no. 4 (August 7, 2020): 244–60. http://dx.doi.org/10.1080/0025570x.2020.1792238.

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44

MADERNA, EZEQUIEL. "On weak KAM theory for N-body problems." Ergodic Theory and Dynamical Systems 32, no. 3 (April 27, 2011): 1019–41. http://dx.doi.org/10.1017/s0143385711000046.

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AbstractWe consider N-body problems with potential 1/r2κ, where κ∈(0,1), including the Newtonian case (κ=1/2). Given R>0 and T>0, we find a uniform upper bound for the minimal action of paths binding, in time T, any two configurations which are contained in some ball of radius R. Using cluster partitions, we obtain from these estimates the Hölder regularity of the critical action potential (i.e. of the minimal action of paths binding two configurations in free time). As an application, we establish the weak KAM theorem for these N-body problems, i.e. we prove the existence of fixed points of the Lax–Oleinik semigroup, and we show that they are global viscosity solutions of the corresponding Hamilton–Jacobi equation. We also prove that there are invariant solutions for the action of isometries on the configuration space.

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45

Evans, Lawrence C. "Towards a Quantum Analog of Weak KAM Theory." Communications in Mathematical Physics 244, no. 2 (January 1, 2004): 311–34. http://dx.doi.org/10.1007/s00220-003-0975-5.

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46

Khanin, Kostya, João Lopes Dias, and Jens Marklof. "Multidimensional Continued Fractions, Dynamical Renormalization and KAM Theory." Communications in Mathematical Physics 270, no. 1 (October 10, 2006): 197–231. http://dx.doi.org/10.1007/s00220-006-0125-y.

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47

Evans, L. C. "Some new PDE methods for weak KAM theory." Calculus of Variations and Partial Differential Equations 17, no. 2 (June 1, 2003): 159–77. http://dx.doi.org/10.1007/s00526-002-0164-y.

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48

Bernard, Patrick, and Maxime Zavidovique. "Regularization of Subsolutions in Discrete Weak KAM Theory." Canadian Journal of Mathematics 65, no. 4 (August 1, 2013): 740–56. http://dx.doi.org/10.4153/cjm-2012-059-3.

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AbstractWe expose different methods of regularizations of subsolutions in the context of discrete weak KAM theory that allow us to prove the existence and the density of C1,1 subsolutions. Moreover, these subsolutions can be made strict and smooth outside of the Aubry set.

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49

Celletti, Alessandra, and Luigi Chierchia. "Rigorous estimates for a computer‐assisted KAM theory." Journal of Mathematical Physics 28, no. 9 (September 1987): 2078–86. http://dx.doi.org/10.1063/1.527418.

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50

Desogus, Marco, and Elisa Casu. "A SURVEY ON MACROECONOMIC DATA IN THE EUROZONE AND A CONTROL DASHBOARD MODEL BASED ON THE KAM AND NEKHOROSHEV THEOREMS AND THE HÉNON ATTRACTOR." Journal of Academy of Business and Economics 21, no. 3 (October 1, 2021): 67–85. http://dx.doi.org/10.18374/jabe-21-3.6.

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