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

References:
[1]

Trans. Am. Math. Soc., 307 (1988), 545-568. doi: 10.1090/S0002-9947-1988-0940217-X.  Google Scholar

[2]

J. Reine Engew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79.  Google Scholar

[3]

Nonlinearity, 11 (1998), 949-964. doi: 10.1088/0951-7715/11/4/011.  Google Scholar

[4]

in Dynamics of coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Phys., 671, Springer, Berlin, 2005, 241-263. doi: 10.1007/11360810_10.  Google Scholar

[5]

in Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988, 1-56.  Google Scholar

[6]

Comm. Math. Phys., 199 (1998), 441-470. doi: 10.1007/s002200050508.  Google Scholar

[7]

Arch. Rational Mech. Anal., 107 (1989), 325-345. doi: 10.1007/BF00251553.  Google Scholar

[8]

Phys. Rev. B, 52 (1995), 6451-6457. doi: 10.1103/PhysRevB.52.6451.  Google Scholar

[9]

Advances in Physics, 45 (1996), 505-598. doi: 10.1080/00018739600101557.  Google Scholar

[10]

in Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems, Lecture Notes in Phys., 671, Springer, Berlin, 2005, 209-240. doi: 10.1007/11360810_9.  Google Scholar

[11]

Trans. Am. Math. Soc., 363 (2011), 2571-2598. doi: 10.1090/S0002-9947-2010-05148-7.  Google Scholar

[12]

J. Dynam. Differential Equations, 13 (2001), 757-789. 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]

Trans. Am. Math. Soc., 361 (2009), 2755-2788. doi: 10.1090/S0002-9947-08-04823-X.  Google Scholar

[15]

Physica D, 208 (2005), 172-190. doi: 10.1016/j.physd.2005.06.022.  Google Scholar

[16]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1397-1440. doi: 10.1016/j.anihpc.2010.09.001.  Google Scholar

[17]

Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  Google Scholar

[18]

Second edition, Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-0-8176-8134-0.  Google Scholar

[19]

Ann. I. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344. doi: 10.1016/j.anihpc.2008.11.002.  Google Scholar

[20]

Comm. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683.  Google Scholar

[21]

Mem. Am. Math. Soc., 198 (2009), vi+97 pp. doi: 10.1090/memo/0925.  Google Scholar

[22]

in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0.  Google Scholar

[23]

Nolinearity, 23 (2010), 1873-1886. doi: 10.1088/0951-7715/23/8/005.  Google Scholar

[24]

Comm. Math. Physics, 311 (2012), 513-538. doi: 10.1007/s00220-011-1385-8.  Google Scholar

[25]

Discrete Contin. Dyn. Syst., 6 (2000), 503-518. doi: 10.3934/dcds.2000.6.503.  Google Scholar

[26]

Ergodic Theory Dyn. Sys., 22 (2002), 303-313. doi: 10.1017/S0143385702000147.  Google Scholar

[27]

Nonlinearity, 26 (2013), 2051-2079. 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]

SIAM J. Math. Anal., 15 (1984), 530-534. doi: 10.1137/0515040.  Google Scholar

[30]

Mathematical Surveys and Monographs, Vol. 41, AMS, Providence, 1996. Google Scholar

[31]

Discrete Contin. Dyn. Sys., 28 (2010), 1713-1751. 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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