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The Aubry-Mather theorem for driven generalized elastic chains
1. | Department of Mathematics, Bijenička 30, Zagreb |
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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