Partial preservation of frequencies and floquet exponents of invariant tori in the reversible KAM context~2

M. B. Sevryuk

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 516-541.

Résumé

We consider the persistence of smooth families of invariant tori in the reversible context 2 of KAM theory under various weak nondegeneracy conditions via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question. The nondegeneracy conditions we employ ensure the preservation of any prescribed subsets of the frequencies of the unperturbed tori and of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus).

Bibliographie

[1] G. E. Bredon, Introduction to Compact Transformation Groups, Nauka, Moscow, 1980, (Russian translation) | MR

[2] R. de la Llave, A tutorial on KAM theory, In-t komp. issl., Moscow–Izhevsk, 2003, (Russian translation) | MR

[3] P. E. Conner, E. E. Floyd, Smooth Periodic Mappings, Mir, Moscow, 1969, (Russian translation)

[4] J. E. Marsden, M. McCracken, The Hopf Bifurcation and Its Applications, Mir, Moscow, 1980, (Russian translation) | MR

[5] J. Moser, “On the expansion of quasi-periodic motions in convergent power series”, Progr. Math. Sci., 24:2 (1969), 165–211 (in Russian) | MR

[6] S. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Mir, Moscow, 1980, (Russian translation) | MR

[7] M. B. Sevryuk, “Linear reversible systems and their versal deformations”, Proc. Petrovskii Semin., 15, 1991, 33–54 (in Russian) | MR

[8] M. B. Sevryuk, “Some problems of the KAM-theory: conditionally-periodic motions in typical systems”, Progr. Math. Sci., 50:2 (1995), 111–124 (in Russian) | MR | Zbl

[9] M. B. Sevryuk, “Partial preservation of frequencies and Floquet exponents in KAM theory”, Proc. Math. Inst. Russ. Acad. Sci., 259, 2007, 174–202 (in Russian) | MR | Zbl

[10] Arnold V. I., Kozlov V. V., Neishtadt A. I., Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 2006 | MR

[11] Bredon G. E., Introduction to Compact Transformation Groups, Academic Press, New York, 1972 | MR

[12] Broer H. W., Ciocci M. C., Hanßmann H., Vanderbauwhede A., “Quasi-periodic stability of normally resonant tori”, Phys. D, 238:3 (2009), 309–318 | DOI | MR

[13] Broer H. W., Hoo J., Naudot V., “Normal linear stability of quasi-periodic tori”, J. Differ. Equ, 232:2 (2007), 355–418 | DOI | MR

[14] Broer H. W., Huitema G. B., “Unfoldings of quasi-periodic tori in reversible systems”, J. Dynam. Differ. Equ., 7:1 (1995), 191–212 | DOI | MR

[15] Broer H. W., Huitema G. B., Sevryuk M. B., “Families of quasi-periodic motions in dynamical systems depending on parameters”, Nonlinear Dynamical Systems and Chaos, Birkhäuser, Basel, 1996, 171–211 | DOI | MR

[16] Broer H. W., Huitema G. B., Sevryuk M. B., Quasi-Periodic Motions in Families of Dynamical Systems. Order amidst Chaos, Springer, Berlin, 1996 | MR

[17] Broer H. W., Huitema G. B., Takens F., “Unfoldings of quasi-periodic tori”, Unfoldings and bifurcations of quasi-periodic tori, Mem. Am. Math. Soc, 83, no. 421, 1990, 1–81 | MR

[18] Broer H. W., Sevryuk M. B., “KAM theory: Quasi-periodicity in dynamical systems”, Handbook of Dynamical Systems, v. 3, Elsevier, Amsterdam, 2010, 249–344

[19] Calleja R. C., Celletti A., de la Llave R., “Domains of analyticity and Lindstedt expansions of KAM tori in some dissipative perturbations of Hamiltonian systems”, Nonlinearity, 30:8 (2017), 3151–3202 | DOI | MR

[20] Chow S.-N., Li Y., Yi Y., “Persistence of invariant tori on submanifolds in Hamiltonian systems”, J. Nonlinear Sci., 12:6 (2002), 585–617 | DOI | MR

[21] Conner P. E., Floyd E. E., Differentiable Periodic Maps, Academic Press, New York; Springer, Berlin, 1964 | MR

[22] De la Llave R., “A tutorial on KAM theory”, Proc. Symp. Pure Math., 69 (2001), 175–292 | DOI | MR

[23] Dumas H. S., The KAM Story. A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory, World Scientific, Hackensack, 2014 | MR

[24] González-Enríquez A., Haro À., de la Llave R., Singularity theory for non-twist KAM tori, Mem. Am. Math. Soc., 227, no. 1067, 2014, 115 pp. | MR

[25] Hanßmann H., “Non-degeneracy conditions in KAM theory”, Indag. Math. (N.S.), 22:3–4 (2011), 241–256 | MR

[26] Haro À., Canadell M., Figueras J.-L., Luque A., Mondelo J.-M., The Parameterization Method for Invariant Manifolds. From Rigorous Results to Effective Computations, Springer, Cham, 2016 | MR

[27] Hoveijn I., “Versal deformations and normal forms for reversible and Hamiltonian linear systems”, J. Differ. Equ., 126:2 (1996), 408–442 | DOI | MR

[28] Kong Y., Xu J., “Persistence of lower dimensional hyperbolic tori for reversible system”, Appl. Math. Comput., 236 (2014), 408–421 | MR

[29] Lamb J. S. W., Roberts J. A. G., “Time-reversal symmetry in dynamical systems: a survey”, Phys. D, 112:1–2 (1998), 1–39 | DOI | MR

[30] Li Y., Yi Y., “Persistence of hyperbolic tori in Hamiltonian systems”, J. Differ. Equ., 208:2 (2005), 344–387 | DOI | MR

[31] Liu Zh., “Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems”, Nonlinear Anal., 61:8 (2005), 1319–1342 | DOI | MR

[32] Marsden J. E., McCracken M., The Hopf Bifurcation and Its Applications, Springer, New York, 1976 | MR

[33] Montgomery D., Zippin L., Topological Transformation Groups, R. E. Krieger Publishing, Huntington, 1974 | MR

[34] Morris S. A., Pontryagin Duality and the Structure of Locally Compact Abelian Groups, Cambridge Univ. Press, Cambridge, 1977 | MR

[35] Moser J., “Convergent series expansions for quasi-periodic motions”, Math. Ann, 169:1 (1967), 136–176 | DOI | MR

[36] Quispel G. R. W., Sevryuk M. B., “KAM theorems for the product of two involutions of different types”, Chaos, 3:4 (1993), 757–769 | DOI | MR

[37] Roberts J. A. G., Quispel G. R. W., “Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems”, Phys. Rep., 216:2–3 (1992), 63–177 | DOI | MR

[38] Rüssmann H., “Invariant tori in non-degenerate nearly integrable Hamiltonian systems”, Regul. Chaotic Dyn., 6:2 (2001), 119–204 | DOI | MR

[39] Rüssmann H., “Addendum to ‘Invariant tori in non-degenerate nearly integrable Hamiltonian systems’ ”, Regul. Chaotic Dyn., 10:1 (2005), 21–31 | DOI | MR

[40] Sevryuk M. B., Reversible Systems, Springer, Berlin, 1986 | MR

[41] Sevryuk M. B., “The iteration-approximation decoupling in the reversible KAM theory”, Chaos, 5:3 (1995), 552–565 | DOI | MR

[42] Sevryuk M. B., “Excitation of elliptic normal modes of invariant tori in Hamiltonian systems”, Topics in Singularity Theory, Am. Math. Soc., Providence, 1997, 209–218 | MR

[43] Sevryuk M. B., “Excitation of elliptic normal modes of invariant tori in volume preserving flows”, Global Analysis of Dynamical Systems, Inst. Phys., Bristol, 2001, 339–352 | MR

[44] Sevryuk M. B., “Partial preservation of frequencies in KAM theory”, Nonlinearity, 19:5 (2006), 1099–1140 | DOI | MR

[45] Sevryuk M. B., “Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method”, Discrete Contin. Dyn. Syst., 18:2–3 (2007), 569–595 | DOI | MR

[46] Sevryuk M. B., “KAM tori: persistence and smoothness”, Nonlinearity, 21:10 (2008), T177–T185 | DOI | MR

[47] Sevryuk M. B., “The reversible context 2 in KAM theory: the first steps”, Regul. Chaotic Dyn., 16:1–2 (2011), 24–38 | DOI | MR

[48] Sevryuk M. B., “KAM theory for lower dimensional tori within the reversible context 2”, Mosc. Math. J., 12:2 (2012), 435–455 | MR | Zbl

[49] Sevryuk M. B., “Quasi-periodic perturbations within the reversible context 2 in KAM theory”, Indag. Math. (N.S.), 23:3 (2012), 137–150 | DOI | MR

[50] Sevryuk M. B., “Whitney smooth families of invariant tori within the reversible context 2 of KAM theory”, Regul. Chaotic Dyn., 21:6 (2016), 599–620 | DOI | MR

[51] Sevryuk M. B., “Herman's approach to quasi-periodic perturbations in the reversible KAM context 2”, Mosc. Math. J., 17:4 (2017), 803–823 | MR

[52] Shih C. W., “Normal forms and versal deformations of linear involutive dynamical systems”, Chinese J. Math., 21:4 (1993), 333–347 | MR

[53] Tao T., Poincaré's Legacies, Pages from Year Two of a Mathematical Blog, Part I, Am. Math. Soc., Providence, 2009 | MR

[54] Tits J., ØE uvres, Collected Works, v. IV, Eur. Math. Soc., Zürich, 2013 | MR

[55] Wagener F., “A parametrised version of Moser's modifying terms theorem”, Discrete Contin. Dyn. Syst. Ser. S, 3:4 (2010), 719–768 | DOI | MR

[56] Wang X., Xu J., Zhang D., “Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems”, Discrete Contin. Dyn. Syst. Ser. B, 14:3 (2010), 1237–1249 | DOI | MR

[57] Wang X., Xu J., Zhang D., “A new KAM theorem for the hyperbolic lower dimensional tori in reversible systems”, Acta Appl. Math., 143 (2016), 45–61 | DOI | MR

[58] Wang X., Xu J., Zhang D., “A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems”, Discrete Contin. Dyn. Syst., 37:4 (2017), 2141–2160 | MR

[59] Xu J., Lu X., “General KAM theorems and their applications to invariant tori with prescribed frequencies”, Regul. Chaotic Dyn., 21:1 (2016), 107–125 | DOI | MR | Zbl

[60] Yoccoz J.-C., “Travaux de Herman sur les tores invariants”, Astérisque, 206 (1992), 311–344

[61] Zhang D., Xu J., Wu H., “On invariant tori with prescribed frequency in Hamiltonian systems”, Adv. Nonlinear Stud., 16:4 (2016), 719–735 | DOI | MR