Publication TypeJournal Article
Year of Publication2011
Secondary TitleCommunications in Applied Analysis
Start Page283
Date Published08/2011
Type of Workscientific: mathematics
We illustrate Mather’s theorem and its applications to the classification of Aubry-Mather sets. We discuss the equivalence of various definitions of Aubry-Mather set available in the literature.
Refereed DesignationRefereed
Full Text


[1] S. Aubry, P. Y. LeDaeron, The discrete Frenkel-Kontorova model and its extension I: exact result for the ground states, Phys. D 8 (1983), 381–422.
[2] V. Bangert, Mather sets for twist maps and geodesics on tori, in Dynam. Report. Ser. Dynam.  Systems Appl. 1, Wiley, Chichester, 1988.
[3] A. Capietto, W. Dambrosio, B. Liu, On the boundedness of solutions to a nonlinear singular oscillator, Z. Angew. Math. Phys.–ZAMP 60 (2009), 1007-1034.
[4] A. Capietto, W. Dambrosio, X. Wang, Quasi-periodic solutions of a damped reversible oscillator at resonance, Differential Integral Equation 22 (2009), 1033–1046.
[5] A. Capietto, B. Liu, Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Anal. 56 (2004), 105–117.
[6] A. Capietto, N. Soave, Quasi-periodic solutions to some planar reversible differential systems at resonance, in preparation.
[7] S. N. Chow, M. L. Pei, Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Anal. 25 (1995), 905–931.
[8] J. Denzler, Mather sets for plane Hamiltonian systems, Z. Angew. Math. Phys.–ZAMP 38 (1987), 791–812.
[9] A. Fathi, A. Siconolfi, Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation, Invent. Math. 155 (2004), 363–388.
[10] A. Fonda, J. Mawhin, Planar differential systems at resonance, Adv. Differential Equations 11 (2006), 197–214.
[11] G. Forni, J. N. Mather, Action minimizing orbits in Hamiltonian systems, in: Transition to chaos in classical and quantum mechanics, Montecatini Terme 1991, Lecture Notes in Mathematics, Springer, Berlin, 1994.
[12] A. Katok, Some remarks on Birkhoff and Mather twist map theorem, Ergodic Theory Dynamical Systems 2 (1982), 183–194.
[13] A. Katok, Periodic and quasi-periodic orbits for twist maps. in: Dynamical Systems and Chaos, Springer Lecture Notes in Physics 179 (1983), 47–65.
[14] M. Kunze, T. Kupper, B. Liu, Boundedness and unboundedness of solutions for reversible oscillators at resonance, Nonlinearity 14 (2001), 1105–1122.
[15] M.Y. Jiang, On a theorem by Mather and Aubry-Mather sets for planar hamiltonian systems, Sci. China Ser. A 42 (1999), 1121–1128.
[16] M.Y. Jiang, Subharmonic solutions of second order subquadratic hamiltonian systems with potential changing sign, J. Math. Anal. Appl. 244 (2000), 291–303.
[17] M.-Y. Jiang, M.L. Pei, Mather sets for plane superlinear Hamiltonian systems, Dynam. Systems Appl. 2 (1993), 189–199.
[18] B. Liu, Boundedness of solutions for semilinear Duffing equations, J. Differential Equations 145 (1998), 119–144.
[19] B. Liu, On Littlewood’s boundedness problem for sublinear Duffing equations, Trans. Amer. Math. Soc. 353 (2001), 1567–1585.
[20] B. Liu, Quasiperiodic solutions of semilinear Li ́enard equations. Discrete Contin. Dyn. Syst. 12 (2005), 137–160.
[21] B. Liu, J.J. Song, Invariant curves of reversible mappings with small twist, Acta Math. Sin. 20 (2004), 15–24.
[22] B. Liu, R. Wang, Quasi-periodic solutions for reversible oscillators at resonance, preprint.
[23] B. Liu, F. Zanolin, Boundedness of solutions of nonlinear differential equations, J. Differential Equations 144 (1998), 66–98
[24] J. N. Mather, Existence of quasi-periodic orbits for twist homeomorfism of the annulus, Topology 21 (1982), 457–467.
[25] J. N. Mather, Denjoy minimal sets for area preserving diffeomorphisms, Comment. Math. Helv. 60 (1985), 508–557.
[26] J. N. Mather, Minimal measures, Comment. Math. Helv. 64 (1989), 459–485.
[27] J. N. Mather, Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), 207–263.
[28] J. N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), 169–207.
[29] J. Moser, Recent developments in the theory of Hamiltonian systems, SIAM Rev. 28 (1986), 459–485.
[30] J. Moser, Monotone twist mappings and the calculus of variation, Ergodic Theory Dynamical Systems 6 (1986), 401–413.
[31] J. Moser, Selected chapters in the calculus of variations. Lecture notes by Oliver Knill, Birkhauser Verlag, Basel, 2003.
[32] R. Ortega, Twist mappings, invariant curves and periodic differential equations, Proceedings Autumn School Nonlinear Anal. and Differential Equations, Lisbon 1998. Ed. L. Sanchez, Birkhauser, Basel Boston (2000), 85–112.
[33] Z. Nitecki, Differentiable dynamics: an introduction to the orbit structure of diffeomorphisms, MIT Press, 1971.
[34] M. L. Pei, Mather sets for superlinear Duffing equations, Sci. China Ser. A 36 (1993), 524–537.
[35] M. L. Pei, Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations. J. Differential Equations 113 (1994), 106–127.
[36] D. Qian, X. Sun, Invariant tori for asymptotically linear impact oscillators. Sci. China Ser. A 49 (2006), 669–687.
[37] J. V. Rogel, Early Aubry-Mather theory. Informal talks delivered at the Summer colloquium of the Computational Science Department at the National University of Singapore (2001).
[38] M. Sevryuk, Reversible systems, Lecture Notes in Mathematics, Springer, Berlin New York, 1986.
[39] N. Soave,Teoria di Aubry-Mather e applicazioni ad equazioni differenziali ordinarie, Master Degree thesis, University of Torino, 2010.