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 and Continuous Dynamical Systems, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
References:
[1]

S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Am. Math. Soc., 307 (1988), 545-568. doi: 10.1090/S0002-9947-1988-0940217-X.

[2]

S. Angenent, The zeroset of a solution of a parabolic equation, J. Reine Engew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79.

[3]

C. Baesens and R. S. Mackay, Gradient dynamics of tilted Frenkel-Kontorova models, Nonlinearity, 11 (1998), 949-964. doi: 10.1088/0951-7715/11/4/011.

[4]

C. Baesens, Spatially extended systems with monotone dynamics (continuous time), 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.

[5]

V. Bangert, Mather sets for twist geodesics on tori, in Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988, 1-56.

[6]

J.-P. Eckmann and J. Rougemont, Coarsening by Ginzburg-Landau dynamics, Comm. Math. Phys., 199 (1998), 441-470. doi: 10.1007/s002200050508.

[7]

B. Fiedler and J. Mallet-Paret, A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal., 107 (1989), 325-345. doi: 10.1007/BF00251553.

[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), 6451-6457. doi: 10.1103/PhysRevB.52.6451.

[9]

L. M. Floría and J. J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model, Advances in Physics, 45 (1996), 505-598. doi: 10.1080/00018739600101557.

[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, Lecture Notes in Phys., 671, Springer, Berlin, 2005, 209-240. doi: 10.1007/11360810_9.

[11]

T. Gallay and A. Scheel, Diffusive stability of oscillations in reaction-diffusion systems, Trans. Am. Math. Soc., 363 (2011), 2571-2598. doi: 10.1090/S0002-9947-2010-05148-7.

[12]

T. Gallay and S. Slijepčević, Energy flow in formally gradient partial differential equations on unbounded domains, J. Dynam. Differential Equations, 13 (2001), 757-789. doi: 10.1023/A:1016624010828.

[13]

T. Gallay and S. Slijepčević, Distribution of energy and convergence to equilibria in extended dissipative systems, preprint, arXiv:1212.1573.

[14]

R. W. Ghrist and R. C. Vandervost, Scalar parabolic PDEs and braids, Trans. Am. Math. Soc., 361 (2009), 2755-2788. doi: 10.1090/S0002-9947-08-04823-X.

[15]

B. Hu, W.-X. Qin and Z. Zheng, Rotation number of the overdamped Frenkel-Kontorova model with ac-driving, Physica D, 208 (2005), 172-190. doi: 10.1016/j.physd.2005.06.022.

[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), 1397-1440. doi: 10.1016/j.anihpc.2010.09.001.

[17]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

[18]

S. G. Krantz and R. Parks, A Primer of Real Analytic Functions, 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.

[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), 1309-1344. doi: 10.1016/j.anihpc.2008.11.002.

[20]

J. Mather, Minimal measures, Comm. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683.

[21]

A. Mielke and S. Zelik, Multi-pulse evolution and space-time chaos in dissipative systems, Mem. Am. Math. Soc., 198 (2009), vi+97 pp. doi: 10.1090/memo/0925.

[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, Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[23]

W.-X. Qin, Dynamics of the Frenkel-Kontorova model with irrational mean spacing, Nolinearity, 23 (2010), 1873-1886. doi: 10.1088/0951-7715/23/8/005.

[24]

W.-X. Qin, Existence and modulation of uniform sliding states in driven and overdamped particle chains, Comm. Math. Physics, 311 (2012), 513-538. doi: 10.1007/s00220-011-1385-8.

[25]

S. Slijepčević, Extended gradient systems: Dimension one, Discrete Contin. Dyn. Syst., 6 (2000), 503-518. doi: 10.3934/dcds.2000.6.503.

[26]

S. Slijepčević, The shear-rotation interval of twist maps, Ergodic Theory Dyn. Sys., 22 (2002), 303-313. doi: 10.1017/S0143385702000147.

[27]

S. Slijepčević, The energy flow of discrete extended gradient systems, Nonlinearity, 26 (2013), 2051-2079. doi: 10.1088/0951-7715/26/7/2051.

[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 (1984), 530-534. doi: 10.1137/0515040.

[30]

H. L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, Vol. 41, AMS, Providence, 1996.

[31]

D. Turaev and S. Zelik, Analytical proof of space-time chaos in Ginzburg-Landau equation, Discrete Contin. Dyn. Sys., 28 (2010), 1713-1751. doi: 10.3934/dcds.2010.28.1713.

[32]

S. Zelik, Formally gradient reaction-diffusion systems in $\mathbb{R}^{N}$ have zero spatio-temporal entropy, Discrete Contin. Dyn. Sys., 2003, suppl., 960-966.

show all references

References:
[1]

S. Angenent and B. Fiedler, The dynamics of rotating waves in scalar reaction diffusion equations, Trans. Am. Math. Soc., 307 (1988), 545-568. doi: 10.1090/S0002-9947-1988-0940217-X.

[2]

S. Angenent, The zeroset of a solution of a parabolic equation, J. Reine Engew. Math., 390 (1988), 79-96. doi: 10.1515/crll.1988.390.79.

[3]

C. Baesens and R. S. Mackay, Gradient dynamics of tilted Frenkel-Kontorova models, Nonlinearity, 11 (1998), 949-964. doi: 10.1088/0951-7715/11/4/011.

[4]

C. Baesens, Spatially extended systems with monotone dynamics (continuous time), 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.

[5]

V. Bangert, Mather sets for twist geodesics on tori, in Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988, 1-56.

[6]

J.-P. Eckmann and J. Rougemont, Coarsening by Ginzburg-Landau dynamics, Comm. Math. Phys., 199 (1998), 441-470. doi: 10.1007/s002200050508.

[7]

B. Fiedler and J. Mallet-Paret, A Poincaré-Bendixson theorem for scalar reaction diffusion equations, Arch. Rational Mech. Anal., 107 (1989), 325-345. doi: 10.1007/BF00251553.

[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), 6451-6457. doi: 10.1103/PhysRevB.52.6451.

[9]

L. M. Floría and J. J. Mazo, Dissipative dynamics of the Frenkel-Kontorova model, Advances in Physics, 45 (1996), 505-598. doi: 10.1080/00018739600101557.

[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, Lecture Notes in Phys., 671, Springer, Berlin, 2005, 209-240. doi: 10.1007/11360810_9.

[11]

T. Gallay and A. Scheel, Diffusive stability of oscillations in reaction-diffusion systems, Trans. Am. Math. Soc., 363 (2011), 2571-2598. doi: 10.1090/S0002-9947-2010-05148-7.

[12]

T. Gallay and S. Slijepčević, Energy flow in formally gradient partial differential equations on unbounded domains, J. Dynam. Differential Equations, 13 (2001), 757-789. doi: 10.1023/A:1016624010828.

[13]

T. Gallay and S. Slijepčević, Distribution of energy and convergence to equilibria in extended dissipative systems, preprint, arXiv:1212.1573.

[14]

R. W. Ghrist and R. C. Vandervost, Scalar parabolic PDEs and braids, Trans. Am. Math. Soc., 361 (2009), 2755-2788. doi: 10.1090/S0002-9947-08-04823-X.

[15]

B. Hu, W.-X. Qin and Z. Zheng, Rotation number of the overdamped Frenkel-Kontorova model with ac-driving, Physica D, 208 (2005), 172-190. doi: 10.1016/j.physd.2005.06.022.

[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), 1397-1440. doi: 10.1016/j.anihpc.2010.09.001.

[17]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.

[18]

S. G. Krantz and R. Parks, A Primer of Real Analytic Functions, 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.

[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), 1309-1344. doi: 10.1016/j.anihpc.2008.11.002.

[20]

J. Mather, Minimal measures, Comm. Math. Helv., 64 (1989), 375-394. doi: 10.1007/BF02564683.

[21]

A. Mielke and S. Zelik, Multi-pulse evolution and space-time chaos in dissipative systems, Mem. Am. Math. Soc., 198 (2009), vi+97 pp. doi: 10.1090/memo/0925.

[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, Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0.

[23]

W.-X. Qin, Dynamics of the Frenkel-Kontorova model with irrational mean spacing, Nolinearity, 23 (2010), 1873-1886. doi: 10.1088/0951-7715/23/8/005.

[24]

W.-X. Qin, Existence and modulation of uniform sliding states in driven and overdamped particle chains, Comm. Math. Physics, 311 (2012), 513-538. doi: 10.1007/s00220-011-1385-8.

[25]

S. Slijepčević, Extended gradient systems: Dimension one, Discrete Contin. Dyn. Syst., 6 (2000), 503-518. doi: 10.3934/dcds.2000.6.503.

[26]

S. Slijepčević, The shear-rotation interval of twist maps, Ergodic Theory Dyn. Sys., 22 (2002), 303-313. doi: 10.1017/S0143385702000147.

[27]

S. Slijepčević, The energy flow of discrete extended gradient systems, Nonlinearity, 26 (2013), 2051-2079. doi: 10.1088/0951-7715/26/7/2051.

[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 (1984), 530-534. doi: 10.1137/0515040.

[30]

H. L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, Vol. 41, AMS, Providence, 1996.

[31]

D. Turaev and S. Zelik, Analytical proof of space-time chaos in Ginzburg-Landau equation, Discrete Contin. Dyn. Sys., 28 (2010), 1713-1751. doi: 10.3934/dcds.2010.28.1713.

[32]

S. Zelik, Formally gradient reaction-diffusion systems in $\mathbb{R}^{N}$ have zero spatio-temporal entropy, Discrete Contin. Dyn. Sys., 2003, suppl., 960-966.

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