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The Aubry-Mather theorem for driven generalized elastic chains. (English) Zbl 1293.34017
This paper studies uniformly (DC) or periodically (AC) driven generalized infinite elastic chains with gradient dynamics. The authors first show that the union of supports of all space-time invariant measures, denoted by \(\mathcal{A}\), projects injectively to a dynamical system on a two dimensional cylinder. They also prove the existence of space-time ergodic measures supported on a set of rotationally ordered configurations with an arbitrary rotation number. This shows that the Aubry-Mather structure of ground states persists if an arbitrary AC or DC force is applied. The set \(\mathcal{A}\) attracts almost surely in probability configurations with bounded spacing. In the DC case, \(\mathcal{A}\) consists entirely of equilibria and uniformly sliding solutions. The key tool is a new weak Lyapunov function on the space of translationally invariant probability measures on the state space, which counts intersections.

34A33 Ordinary lattice differential equations
34C12 Monotone systems involving ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D45 Attractors of solutions to ordinary differential equations
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37E40 Dynamical aspects of twist maps
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[1] S. Angenent, The dynamics of rotating waves in scalar reaction diffusion equations,, Trans. Am. Math. Soc., 307, 545, (1988) · Zbl 0696.35086
[2] S. Angenent, The zeroset of a solution of a parabolic equation,, J. Reine Engew. Math., 390, 79, (1988) · Zbl 0644.35050
[3] C. Baesens, Gradient dynamics of tilted Frenkel-Kontorova models,, Nonlinearity, 11, 949, (1998) · Zbl 0906.58037
[4] C. Baesens, Spatially extended systems with monotone dynamics (continuous time),, in Dynamics of coupled Map Lattices and of Related Spatially Extended Systems, 241, (2005) · Zbl 1111.34031
[5] V. Bangert, Mather sets for twist geodesics on tori,, in Dynamics Reported, 1, (1988) · Zbl 0664.53021
[6] J.-P. Eckmann, Coarsening by Ginzburg-Landau dynamics,, Comm. Math. Phys., 199, 441, (1998) · Zbl 1057.35508
[7] B. Fiedler, A Poincaré-Bendixson theorem for scalar reaction diffusion equations,, Arch. Rational Mech. Anal., 107, 325, (1989) · Zbl 0704.35070
[8] J. J. Mazo, Stability of metastable structures in dissipative ac dynamics of the Frenkel-Kontorova model,, Phys. Rev. B, 52, 6451, (1995)
[9] L. M. Floría, Dissipative dynamics of the Frenkel-Kontorova model,, Advances in Physics, 45, 505, (1996)
[10] L. M. Floría, The Frenkel-Kontorova model,, in Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, 209, (2005) · Zbl 1195.82052
[11] T. Gallay, Diffusive stability of oscillations in reaction-diffusion systems,, Trans. Am. Math. Soc., 363, 2571, (2011) · Zbl 1227.35056
[12] T. Gallay, Energy flow in formally gradient partial differential equations on unbounded domains,, J. Dynam. Differential Equations, 13, 757, (2001) · Zbl 1003.35085
[13] T. Gallay, Distribution of energy and convergence to equilibria in extended dissipative systems,, preprint · Zbl 1338.35045
[14] R. W. Ghrist, Scalar parabolic PDEs and braids,, Trans. Am. Math. Soc., 361, 2755, (2009) · Zbl 1172.35039
[15] B. Hu, Rotation number of the overdamped Frenkel-Kontorova model with ac-driving,, Physica D, 208, 172, (2005) · Zbl 1086.34044
[16] R. Joly, Generic Morse-Smale property for the parabolic equation on the circle,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27, 1397, (2010) · Zbl 1213.35046
[17] A. Katok, <em>Introduction to the Modern Theory of Dynamical Systems</em>,, Encyclopedia of Mathematics and its Applications, (1995) · Zbl 0878.58020
[18] S. G. Krantz, <em>A Primer of Real Analytic Functions</em>,, Second edition, (2002) · Zbl 1015.26030
[19] R. de la Llave, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations,, Ann. I. H. Poincaré Anal. Non Linéaire, 26, 1309, (2009) · Zbl 1171.35372
[20] J. Mather, Minimal measures,, Comm. Math. Helv., 64, 375, (1989) · Zbl 0689.58025
[21] A. Mielke, Multi-pulse evolution and space-time chaos in dissipative systems,, Mem. Am. Math. Soc., 198, (2009) · Zbl 1163.37003
[22] A. Miranville, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, 103, (2008) · Zbl 1221.37158
[23] W.-X. Qin, Dynamics of the Frenkel-Kontorova model with irrational mean spacing,, Nolinearity, 23, 1873, (2010) · Zbl 1201.37104
[24] W.-X. Qin, Existence and modulation of uniform sliding states in driven and overdamped particle chains,, Comm. Math. Physics, 311, 513, (2012) · Zbl 1246.82065
[25] S. Slijepčević, Extended gradient systems: Dimension one,, Discrete Contin. Dyn. Syst., 6, 503, (2000) · Zbl 1009.37004
[26] S. Slijepčević, The shear-rotation interval of twist maps,, Ergodic Theory Dyn. Sys., 22, 303, (2002) · Zbl 1043.37034
[27] S. Slijepčević, The energy flow of discrete extended gradient systems,, Nonlinearity, 26, 2051, (2013) · Zbl 1309.37075
[28] S. Slijepčević, An ergodic Poincaré-Bendixson theorem for scalar reaction diffusion equations,, in preparation.
[29] J. Smillie, Competitive and cooperative tridiagonal systems of differential equations,, SIAM J. Math. Anal., 15, 530, (1984) · Zbl 0546.34007
[30] H. L. Smith, <em>Monotone dynamical systems</em>,, Mathematical Surveys and Monographs, (1996)
[31] D. Turaev, Analytical proof of space-time chaos in Ginzburg-Landau equation,, Discrete Contin. Dyn. Sys., 28, 1713, (2010) · Zbl 1213.35376
[32] S. Zelik, Formally gradient reaction-diffusion systems in \(\mathbbR^n\) have zero spatio-temporal entropy,, Discrete Contin. Dyn. Sys., 2003, 960 · Zbl 1058.35089
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