Partial Preservation of Frequencies and Floquet Exponents in KAM Theory

M. B. Sevryuk

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 174-202.

Résumé

Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete proof for the reversible case).

Bibliographie

[1] Arnold V. I., “O klassicheskoi teorii vozmuschenii i probleme ustoichivosti planetnykh sistem”, DAN SSSR, 145:3 (1962), 487–490

[2] Arnold V. I., “Dokazatelstvo teoremy A. N. Kolmogorova o sokhranenii uslovno periodicheskikh dvizhenii pri malom izmenenii funktsii Gamiltona”, UMN, 18:5 (1963), 13–40 | MR

[3] Arnold V. I., “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6 (1963), 91–192 ; 23:6 (1968), 216 | MR | Zbl | MR | Zbl

[4] Arnold V. I., “Obratimye sistemy”, Problemy nelineinykh i turbulentnykh protsessov v fizike, Tr. II Mezhdunar. rabochei gruppy. Ch. 2, Nauk. dumka, Kiev, 1985, 15–21

[5] Arnold V. I., Kozlov V. V., Neishtadt A. I., Matematicheskie aspekty klassicheskoi i nebesnoi mekhaniki, 2-e izd., Editorial URSS, M., 2002

[6] Bibikov Yu. N., Pliss V. A., “O suschestvovanii invariantnykh torov v okrestnosti nulevogo resheniya sistemy obyknovennykh differentsialnykh uravnenii”, Dif. uravneniya, 3:11 (1967), 1864–1881 | MR | Zbl

[7] Bogoyavlenskii O. I., “Neobkhodimye i dostatochnye usloviya dlya dvukh kontseptsii nevyrozhdennosti po Puankare integriruemykh gamiltonovykh sistem”, DAN, 360:3 (1998), 295–298 | MR | Zbl

[8] de la Yave R., Vvedenie v KAM-teoriyu, In-t kompyut. issled., M., Izhevsk, 2003, 176 pp. | MR

[9] Kolmogorov A. N., “O sokhranenii uslovno periodicheskikh dvizhenii pri malom izmenenii funktsii Gamiltona”, DAN SSSR, 98:4 (1954), 527–530 | MR | Zbl

[10] Lazutkin V. F., Vypuklyi billiard i sobstvennye funktsii operatora Laplasa, Izd-vo LGU, L., 1981, 196 pp. | MR | Zbl

[11] Melnikov V. K., “O nekotorykh sluchayakh sokhraneniya uslovno periodicheskikh dvizhenii pri malom izmenenii funktsii Gamiltona”, DAN SSSR, 165:6 (1965), 1245–1248 | MR

[12] Melnikov V. K., “Ob odnom semeistve uslovno periodicheskikh reshenii sistemy Gamiltona”, DAN SSSR, 181:3 (1968), 546–549 | MR

[13] Mozer Yu., “O krivykh, invariantnykh pri otobrazheniyakh koltsa, sokhranyayuschikh ploschad”, Matematika, 6:5 (1962), 51–67

[14] Mozer Yu., “O razlozhenii uslovno periodicheskikh dvizhenii v skhodyaschiesya stepennye ryady”, UMN, 24:2 (1969), 165–211 | MR

[15] Postnikov A. G., Freiman G. A., “Mezhvuzovskii simpozium po teorii chisel, Vladimir, iyun 1968 g.”, UMN, 24:1 (1969), 235–237 | MR

[16] Puankare A., “Novye metody nebesnoi mekhaniki. I: Periodicheskie resheniya. Nesuschestvovanie odnoznachnykh integralov. Asimptoticheskie resheniya”, Izbr. tr. T. 1, Klassiki nauki, Nauka, M., 1971, 7–326

[17] Sevryuk M. B., “Lineinye obratimye sistemy i ikh versalnye deformatsii”, Tr. sem. im. I. G. Petrovskogo, 15 (1991), 33–54 | MR

[18] Sevryuk M. B., “Invariantnye tory gamiltonovykh sistem, nevyrozhdennykh v smysle Ryussmana”, DAN, 346:5 (1996), 590–593 | MR | Zbl

[19] Shmidt V., Diofantovy priblizheniya, Mir, M., 1983, 230 pp. | MR

[20] Yakubovich V. A., Starzhinskii V. M., Lineinye differentsialnye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya, Nauka, M., 1972, 719 pp. | MR

[21] Abdullah K., Albouy A., “On a strange resonance noticed by M. Herman”, Regular and Chaotic Dyn., 6:4 (2001), 421–432 | DOI | MR | Zbl

[22] Arnold V. I., Sevryuk M. B., “Oscillations and bifurcations in reversible systems”, Nonlinear phenomena in plasma physics and hydrodynamics, ed. R. Z. Sagdeev, Mir, M., 1986, 31–64

[23] Bogoyavlenskij O. I., “Conformal symmetries of dynamical systems and Poincaré 1892 concept of iso-energetic non-degeneracy”, C. r. Acad. sci. Paris. Sér. 1, 326:2 (1998), 213–218 | MR | Zbl

[24] Bost J.-B., “Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnold, Moser, Rüssmann, Zehnder, Herman, Pöschel,...)”, Séminaire Bourbaki 1984/85, Exp. 639, Astérisque, 133–134, Soc. math. France, Paris, 1986, 113–157 | MR

[25] Bourgain J., “On Melnikov's persistency problem”, Math. Res. Lett., 4:4 (1997), 445–458 | MR | Zbl

[26] Broer H., Ciocci M.-C., Litvak-Hinenzon A., “Survey on dissipative KAM theory including quasi-periodic bifurcation theory”, Geometric mechanics and symmetry: the Peyresq lectures, LMS Lect. Note Ser., 306, eds. J. Montaldi, T. Ratiu, Cambridge Univ. Press, Cambridge, 2005, 303–355 | MR | Zbl

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

[28] Broer H. W., Huitema G. B., “Unfoldings of quasi-periodic tori in reversible systems”, J. Dyn. and Diff. Equat., 7:1 (1995), 191–212 | DOI | MR | Zbl

[29] 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, Progr. Nonlin. Diff. Equat. and Appl., 19, eds. H. W. Broer, S. A. van Gils, I. Hoveijn, F. Takens, Birkhäuser, Basel, 1996 | MR

[30] Broer H. W., Huitema G. B., Sevryuk M. B., Quasi-periodic motions in families of dynamical systems. Order amidst chaos, Lect. Notes Math., 1645, Springer, Berlin, 1996, xi+195 pp. | MR

[31] Broer H. W., Huitema G. B., Takens F., “Unfoldings of quasi-periodic tori”, Unfoldings and bifurcations of quasi-periodic tori, Mem. AMS, 83, No 421, Amer. Math. Soc., Providence, RI, 1990, 1–81 | MR

[32] Chierchia L., Qian D., “Moser's theorem for lower dimensional tori”, J. Diff. Equat., 206:1 (2004), 55–93 | DOI | MR | Zbl

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

[34] Féjoz J., “Démonstration du ‘théorème d’Arnold' sur la stabilité du système planétaire (d'après Herman)”, Ergod. Theory and Dyn. Syst., 24:5 (2004), 1521–1582 | DOI | MR | Zbl

[35] Hanssmann H., Local and semi-local bifurcations in Hamiltonian dynamical systems: Results and examples, Lect. Notes Math., 1893, Springer, Berlin, 2007, xv+237 pp. | MR | Zbl

[36] Herman M. R., Existence et non existence de tores invariants par des difféomorphismes symplectiques, Sém. Équat. Dériv. Part. 1987/1988, Exp. 14, École Polytech., Palaiseau, 1988, 24 pp. | MR

[37] Herman M. R., “Inégalités “a priori” pour des tores lagrangiens invariants par des difféomorphismes symplectiques”, Publ. Math. IHES, 70 (1989), 47–101 | MR | Zbl

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

[39] Huitema G. B., Unfoldings of quasi-periodic tori: Proefschrift, Rijksuniv. Groningen, Groningen, 1988

[40] Kubichka A. A., Parasyuk I. O., “Bifurcation of a Whitney-smooth family of coisotropic invariant tori of a Hamiltonian system under small deformations of a symplectic structure”, Ukr. Math. J., 53:5 (2001), 701–718 | DOI | MR | Zbl

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

[42] Lazutkin V. F., KAM theory and semiclassical approximations to eigenfunctions, Ergeb. Math. und Grenzgeb. Flg. 3, 24, Springer, Berlin, 1993, ix+387 pp. | MR | Zbl

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

[44] Liu Zh., “Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems”, Nonlin. Anal. Theory, Meth. and Appl., 61:8 (2005), 1319–1342 | DOI | MR | Zbl

[45] Moser J., “Combination tones for Duffing's equation”, Commun. Pure and Appl. Math., 18:1–2 (1965), 167–181 | DOI | MR | Zbl

[46] Moser J., “On the theory of quasiperiodic motions”, SIAM Rev., 8:2 (1966), 145–172 | DOI | MR | Zbl

[47] Popov G., KAM tori and quantum Birkhoff normal forms, Sém. Équat. Dériv. Part. 1999/2000, Exp. 20, École Polytech., Palaiseau, 2000, 13 pp. | MR | Zbl

[48] Popov G., “Invariant tori, effective stability, and quasimodes with exponentially small error terms. I: Birkhoff normal forms”, Ann. H. Poincaré, 1:2 (2000), 223–248 | DOI | MR | Zbl

[49] Popov G., “Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms”, Ann. H. Poincaré, 1:2 (2000), 249–279 | DOI | MR | Zbl

[50] Popov G., “KAM theorem for Gevrey Hamiltonians”, Ergod. Theory and Dyn. Syst., 24:5 (2004), 1753–1786 | DOI | MR | Zbl

[51] Popov G., “KAM theorem and quasimodes for Gevrey Hamiltonians”, Mat. Contemp., 26 (2004), 87–107 | MR | Zbl

[52] Pöschel J., “Integrability of Hamiltonian systems on Cantor sets”, Commun. Pure and Appl. Math., 35:5 (1982), 653–696 | DOI | MR | Zbl

[53] Pöschel J., “A lecture on the classical KAM theorem”, Smooth ergodic theory and its applications, Proc. Symp. Pure Math., 69, eds. A. B. Katok, R. de la Llave, Ya. B. Pesin, H. Weiss, Amer. Math. Soc., Providence, RI, 2001, 707–732 | MR | Zbl

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

[55] Roy N., “The geometry of nondegeneracy conditions in completely integrable systems”, Ann. Fac. Sci. Toulouse. Math. Sér. 6, 15:2 (2006), 383–397 | MR

[56] Rüssmann H., “Non-degeneracy in the perturbation theory of integrable dynamical systems”, Number theory and dynamical systems, LMS Lect. Note Ser., 134, eds. M. M. Dodson, J. A. G. Vickers, Cambridge Univ. Press, Cambridge, 1989, 5–18 | MR

[57] Rüssmann H., “Nondegeneracy in the perturbation theory of integrable dynamical systems”, Stochastics, algebra and analysis in classical and quantum dynamics, Math. and Appl., 59, eds. S. Albeverio, P. Blanchard, D. Testard, Kluwer, Dordrecht, 1990, 211–223 | MR

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

[59] Salamon D. A., “The Kolmogorov–Arnold–Moser theorem”, Math. Phys. Electron. J., 10, Pap. 3 (2004), 37 pp. | MR

[60] Sevryuk M. B., Reversible systems, Lect. Notes Math., 1211, Springer, Berlin, 1986, v+319 pp. | MR | Zbl

[61] Sevryuk M. B., “Lower-dimensional tori in reversible systems”, Chaos, 1:2 (1991), 160–167 | DOI | MR | Zbl

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

[63] Sevryuk M. B., “KAM-stable Hamiltonians”, J. Dyn. and Control Syst., 1:3 (1995), 351–366 | DOI | MR | Zbl

[64] Sevryuk M. B., “The finite-dimensional reversible KAM theory”, Physica D, 112:1–2 (1998), 132–147 | DOI | MR

[65] Sevryuk M. B., “The lack-of-parameters problem in the KAM theory revisited”, Hamiltonian systems with three or more degrees of freedom, NATO Adv. Sci. Inst. C: Math. and Phys. Sci., 533, eds. C. Simó, Kluwer, Dordrecht, 1999, 568–572 | MR | Zbl

[66] Sevryuk M., “The classical KAM theory at the dawn of the twenty-first century”, Moscow Math. J., 3:3 (2003), 1113–1144 | MR | Zbl

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

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

[69] Shih Ch.-W., “Normal forms and versal deformations of linear involutive dynamical systems”, Chin. J. Math., 21:4 (1993), 333–347 | MR | Zbl

[70] Wagener F., “A note on Gevrey regular KAM theory and the inverse approximation lemma”, Dyn. Syst., 18:2 (2003), 159–163 | DOI | MR | Zbl

[71] Xu J., You J., “Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition”, J. math. pures et appl. Sér. 9, 80:10 (2001), 1045–1067 | DOI | MR | Zbl

[72] Xu J., You J., “Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition”, J. Diff. Equat., 235:2 (2007), 609–622 | DOI | MR | Zbl

[73] Yoccoz J.-C., “Travaux de Herman sur les tores invariants”, Séminaire Bourbaki 1991/92, Exp. 754, Astérisque, 206, Soc. math. France, Paris, 1992, 311–344 | MR | Zbl

[74] Zhang D., Xu J., “Gevrey-smoothness of elliptic lower-dimensional invariant tori in Hamiltonian systems under Rüssmann's non-degeneracy condition”, J. Math. Anal. and Appl., 323:1 (2006), 293–312 | DOI | MR | Zbl

[75] Zhang D., Xu J., “On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition”, Discr. and Contin. Dyn. Syst. A, 16:3 (2006), 635–655 | DOI | MR | Zbl