# American Institute of Mathematical Sciences

July  2014, 34(7): 2983-3011. doi: 10.3934/dcds.2014.34.2983

## The Aubry-Mather theorem for driven generalized elastic chains

 1 Department of Mathematics, Bijenička 30, Zagreb

Received  May 2013 Revised  October 2013 Published  December 2013

We consider uniformly (DC) or periodically (AC) driven generalized infinite elastic chains (a generalized Frenkel-Kontorova model) with gradient dynamics. We 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 2-dimensional cylinder. We also prove existence of space-time ergodic measures supported on a set of rotationaly ordered configurations with an arbitrary (rational or irrational) 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.
Citation: Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
##### References:
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Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). Google Scholar [18] S. G. Krantz and R. Parks, A Primer of Real Analytic Functions,, Second edition, (2002). doi: 10.1007/978-0-8176-8134-0. Google Scholar [19] R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations,, Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309. doi: 10.1016/j.anihpc.2008.11.002. Google Scholar [20] J. Mather, Minimal measures,, Comm. Math. Helv., 64 (1989), 375. doi: 10.1007/BF02564683. Google Scholar [21] A. Mielke and S. Zelik, Multi-pulse evolution and space-time chaos in dissipative systems,, Mem. Am. Math. Soc., 198 (2009). doi: 10.1090/memo/0925. Google Scholar [22] A. Miranville and S. 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Slijepčević, The energy flow of discrete extended gradient systems,, Nonlinearity, 26 (2013), 2051. doi: 10.1088/0951-7715/26/7/2051. Google Scholar [28] S. Slijepčević, An ergodic Poincaré-Bendixson theorem for scalar reaction diffusion equations,, in preparation., (). Google Scholar [29] J. Smillie, Competitive and cooperative tridiagonal systems of differential equations,, SIAM J. Math. Anal., 15 (1984), 530. doi: 10.1137/0515040. Google Scholar [30] H. L. Smith, Monotone dynamical systems,, Mathematical Surveys and Monographs, (1996). Google Scholar [31] D. Turaev and S. Zelik, Analytical proof of space-time chaos in Ginzburg-Landau equation,, Discrete Contin. Dyn. Sys., 28 (2010), 1713. doi: 10.3934/dcds.2010.28.1713. Google Scholar [32] S. Zelik, Formally gradient reaction-diffusion systems in $\mathbbR^n$ have zero spatio-temporal entropy,, Discrete Contin. Dyn. Sys., 2003 (): 960. Google Scholar

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##### References:
 [1] S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations,, Trans. Am. Math. Soc., 307 (1988), 545. doi: 10.1090/S0002-9947-1988-0940217-X. Google Scholar [2] S. Angenent, The zeroset of a solution of a parabolic equation,, J. Reine Engew. Math., 390 (1988), 79. doi: 10.1515/crll.1988.390.79. Google Scholar [3] C. Baesens and R. S. Mackay, Gradient dynamics of tilted Frenkel-Kontorova models,, Nonlinearity, 11 (1998), 949. doi: 10.1088/0951-7715/11/4/011. Google Scholar [4] C. Baesens, Spatially extended systems with monotone dynamics (continuous time),, in Dynamics of coupled Map Lattices and of Related Spatially Extended Systems, (2005), 241. doi: 10.1007/11360810_10. Google Scholar [5] V. Bangert, Mather sets for twist geodesics on tori,, in Dynamics Reported, (1988), 1. Google Scholar [6] J.-P. Eckmann and J. Rougemont, Coarsening by Ginzburg-Landau dynamics,, Comm. Math. Phys., 199 (1998), 441. doi: 10.1007/s002200050508. Google Scholar [7] B. Fiedler and J. Mallet-Paret, A Poincaré-Bendixson theorem for scalar reaction diffusion equations,, Arch. Rational Mech. Anal., 107 (1989), 325. doi: 10.1007/BF00251553. Google Scholar [8] J. J. Mazo, F. Falo and L. M. Floría, Stability of metastable structures in dissipative ac dynamics of the Frenkel-Kontorova model,, Phys. Rev. B, 52 (1995), 6451. doi: 10.1103/PhysRevB.52.6451. Google Scholar [9] L. M. Floría and J. J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model,, Advances in Physics, 45 (1996), 505. doi: 10.1080/00018739600101557. Google Scholar [10] L. M. Floría, C. Baesens and J. Gómez-Gardeñez, The Frenkel-Kontorova model,, in Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, (2005), 209. doi: 10.1007/11360810_9. Google Scholar [11] T. Gallay and A. Scheel, Diffusive stability of oscillations in reaction-diffusion systems,, Trans. Am. Math. Soc., 363 (2011), 2571. doi: 10.1090/S0002-9947-2010-05148-7. Google Scholar [12] T. Gallay and S. Slijepčević, Energy flow in formally gradient partial differential equations on unbounded domains,, J. Dynam. Differential Equations, 13 (2001), 757. doi: 10.1023/A:1016624010828. Google Scholar [13] T. Gallay and S. Slijepčević, Distribution of energy and convergence to equilibria in extended dissipative systems,, preprint, (). Google Scholar [14] R. W. Ghrist and R. C. Vandervost, Scalar parabolic PDEs and braids,, Trans. Am. Math. Soc., 361 (2009), 2755. doi: 10.1090/S0002-9947-08-04823-X. Google Scholar [15] B. Hu, W.-X. Qin and Z. Zheng, Rotation number of the overdamped Frenkel-Kontorova model with ac-driving,, Physica D, 208 (2005), 172. doi: 10.1016/j.physd.2005.06.022. Google Scholar [16] R. Joly and G. Raugel, Generic Morse-Smale property for the parabolic equation on the circle,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1397. doi: 10.1016/j.anihpc.2010.09.001. Google Scholar [17] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Encyclopedia of Mathematics and its Applications, (1995). Google Scholar [18] S. G. Krantz and R. Parks, A Primer of Real Analytic Functions,, Second edition, (2002). doi: 10.1007/978-0-8176-8134-0. Google Scholar [19] R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations,, Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309. doi: 10.1016/j.anihpc.2008.11.002. Google Scholar [20] J. Mather, Minimal measures,, Comm. Math. Helv., 64 (1989), 375. doi: 10.1007/BF02564683. Google Scholar [21] A. Mielke and S. Zelik, Multi-pulse evolution and space-time chaos in dissipative systems,, Mem. Am. Math. Soc., 198 (2009). doi: 10.1090/memo/0925. Google Scholar [22] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains,, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, (2008), 103. doi: 10.1016/S1874-5717(08)00003-0. Google Scholar [23] W.-X. Qin, Dynamics of the Frenkel-Kontorova model with irrational mean spacing,, Nolinearity, 23 (2010), 1873. doi: 10.1088/0951-7715/23/8/005. Google Scholar [24] W.-X. Qin, Existence and modulation of uniform sliding states in driven and overdamped particle chains,, Comm. Math. Physics, 311 (2012), 513. doi: 10.1007/s00220-011-1385-8. Google Scholar [25] S. Slijepčević, Extended gradient systems: Dimension one,, Discrete Contin. Dyn. Syst., 6 (2000), 503. doi: 10.3934/dcds.2000.6.503. Google Scholar [26] S. Slijepčević, The shear-rotation interval of twist maps,, Ergodic Theory Dyn. Sys., 22 (2002), 303. doi: 10.1017/S0143385702000147. Google Scholar [27] S. Slijepčević, The energy flow of discrete extended gradient systems,, Nonlinearity, 26 (2013), 2051. doi: 10.1088/0951-7715/26/7/2051. Google Scholar [28] S. Slijepčević, An ergodic Poincaré-Bendixson theorem for scalar reaction diffusion equations,, in preparation., (). Google Scholar [29] J. Smillie, Competitive and cooperative tridiagonal systems of differential equations,, SIAM J. Math. Anal., 15 (1984), 530. doi: 10.1137/0515040. Google Scholar [30] H. L. Smith, Monotone dynamical systems,, Mathematical Surveys and Monographs, (1996). Google Scholar [31] D. Turaev and S. Zelik, Analytical proof of space-time chaos in Ginzburg-Landau equation,, Discrete Contin. Dyn. Sys., 28 (2010), 1713. doi: 10.3934/dcds.2010.28.1713. Google Scholar [32] S. Zelik, Formally gradient reaction-diffusion systems in $\mathbbR^n$ have zero spatio-temporal entropy,, Discrete Contin. Dyn. Sys., 2003 (): 960. Google Scholar
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