zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
KAM-stable Hamiltonians. (English) Zbl 0951.37038
An original method of studying small perturbations of highly degenerate Hamiltonians is presented. The approach is based on embedding the perturbated Hamiltonian in a family of Hamiltonians depending on an external multidimensional parameter μ and thus achieve full control of the frequencies. The invariant tori of the original Hamiltonian system is picked out using the fact that the invariant tori of the whole family constitute a Whitney-smooth foliation.
37K55Perturbations, KAM for infinite-dimensional systems
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37C55Periodic and quasiperiodic flows and diffeomorphisms
70H08Nearly integrable Hamiltonian systems, KAM theory
70H09Perturbation theories (mechanics of particles and systems)
[1]J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d’après Kolmogorov, Arnol’d, Moser, Rüssmann, Zehnder, Herman, Pöschel,...). In:Sémin. Bourbaki 1984/85;Astérisque 133–134 (1986), 113–157.
[2]A. N. Kolmogorov, On the persistence of conditionally periodic motions under a small change of the Hamilton function. (Russian)Dokl. Akad. Nauk SSSR 98 (1954), No. 4, 527–530. English translation in: Stochastic Behavior in Classical and Quantum Hamiltonian Systems,Lect. Notes Phys., Ed. G. Casati and J. Ford,Springer-Verlag, Berlin,93 (1979), 51–56.
[3]V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian.Russ. Math. Surveys 18 (1963), No. 5, 9–36. · doi:10.1070/RM1963v018n05ABEH004130
[4]–, Small denominators and problems of stability of motion in classical and celestial mechanics.Russ. Math. Surveys 18 (1963), No. 6, 85–191. · doi:10.1070/RM1963v018n06ABEH001143
[5]J. Pöschel, Integrability of Hamiltonian systems on Cantor sets.Commun. Pure Appl. Math. 35 (1982), No. 5, 653–696. · Zbl 0542.58015 · doi:10.1002/cpa.3160350504
[6]V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and celestial Mechanics. Dynamical Systems, III, Encyclopaedia Math. Sic., 3,Springer-Verlag, Berlin, 1988.
[7]A. D. Bruno, Local methods in nonlinear differential equations.Springer-Verlag, Berlin, 1989.
[8]–, On nondegeneracy conditions in Kolmogorov’s theorem.Sov. Math. Dokl. 45 (1992), No. 1, 221–225.
[9]H. Rüssmann, Non-degeneracy in the perturbation theory of integrable dynamical systems. In: Number Theory and Dynamical Systems,London Math. Soc. Lect. Notes Series. Ed. M. M. Dodson and J. A. G. Vickers,Cambridge University Press, Cambridge,134 (1989), 5–18.
[10]–, Nondegeneracy in the perturbation theory of integrable dynamical systems. In: Stochastics, Algebra and Analysis in Classical and Quantum Dynamics,Math. and its Appl. Ed. S. Albeverio, Ph. Blanchard, and D. Testard,Kluwer Academic, Dordrecht,59 (1990), 211–223.
[11]H. Rüssmann, On Twist-Hamitlonians, talk held on the Colloque international: Mécanique céleste et systèmes hamiltoniens,Marseille, May–June 1990.
[12]J. Xiu, J. You, and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degenearcy.Preprint ETH-Zürich, July 1994.
[13]Ch.-Q. Cheng and Y.-S. Sun, Existence of KAM tori in degenerate Hamiltonian systems.J. Differ. Eqs. 114 (1994), No. 1, 288–335. · Zbl 0813.58050 · doi:10.1006/jdeq.1994.1152
[14]L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems.Ann. Sc. Norm. Super. Pisa Cl. Sci. Ser. IV 15 (1988), No. 1, 115–147.
[15]J. Pöschel, On elliptic lower dimensional tori in Hamiltonian systems.Math. Z. 202 (1989), No. 4, 559–608. · Zbl 0662.58037 · doi:10.1007/BF01221590
[16]G. B. Huitema, Unfoldings of Quasi-Periodic Tori. Proefschrift,Rijksuniversiteit, Groningen, 1988.
[17]H. W. Broer, G. B. Huitema, and F. Takens, Unfoldings of quasiperiodic tori.Mem. Am. Math. Soc. 83 (1990), No. 421, 1–81.
[18]V. F. Lazutkin, The existence of a continuum of closed invariant curves for a convex billiard. (Russian)Usp. Mat. Nauk 27 (1972), No. 3, 201–202.
[19]–, The existence of caustics for a billiard problem in a convex domain.Math. USSR Izv. 7 (1973), No. 1, 185–214. · Zbl 0277.52002 · doi:10.1070/IM1973v007n01ABEH001932
[20]–, Concerning Moser’s theorem on invariant curves. In: Problems in the Dynamical Theory of Seismic Waves Propagation. (Russian)Nauka, Leningrad,14 (1974), 109–120.
[21]–, Convex Billiard and Eigenfunctions of the Laplace Operator. (Russian)Leningrad University Press, Leningrad, 1981.
[22]N. V. Svanidze, Small perturbations of an integrable dynamical system with an integral invariant.Proc. Steklov Math. Inst. 2 (1981), 127–151.
[23]V. F. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions.Ergebnisse Math. und ihrer Grenzgebiete,24,Springer-Verlag,Berlin, 1993.
[24]V. G. Sprindžuk, Metric theory of Diophantine approximations.John Wiley, New York, 1979.
[25]–, Mahler’s Problem in Metric Number Theory.Am. Math. Soc., Providence, RI, 1969.
[26]A. S. Pyartli, Diophantine approximations on submanifolds of Euclidean space.Funct. Anal. Appl. 3 (1969), No. 4, 303–306. · Zbl 0216.04401 · doi:10.1007/BF01076316
[27]V. G. Sprindžuk, Achievements and problems in Diophantine approximation theory.Russ. Math. Surveys 35 (1980), No. 4, 1–80. · Zbl 0463.10020 · doi:10.1070/RM1980v035n04ABEH001861
[28]V. I. Bernik, Diophantine approximations on differentiable manifolds. (Russian)Dokl. Akad. Nauk Beloruss. SSR 33 (1989), No. 8, 681–683.
[29]M. M. Dodson, B. P. Rynne, and J. A. G. Vickers, Metric Diophantine approximation and Hausdorff dimension on manifolds.Math. Proc. Camb. Phil. Soc. 105 (1989), No. 3, 547–558.
[30]–, Dirichlet’s theorem and Diophantine approximation on manifolds.J. Number Theory 36 (1990), No. 1, 85–88. · Zbl 0716.11029 · doi:10.1016/0022-314X(90)90006-D
[31]–, Khintchine-type theorems on manifolds.Acta Arithmetica 57 (1991), No. 2, 115–130.
[32]V. I. Bakhtin, Diophantine approximations on images of mappings. (Russian)Dokl. Akad. Nauk Beloruss. SSR 35 (1991), No. 5, 398–400.
[33]I. O. Parasyuk, The persistence of quasiperiodic motions in reversible multifrequency systems. (Russian)Dokl. Akad. Nauk Ukrain. SSR, Ser. A No. 9 (1982), 19–22.
[34]–, On the persistence of multidimensional invariant tori of Hamiltonian systems.Ukrain. Math. J. 36 (1984), No. 4, 380–385. · Zbl 0561.58017 · doi:10.1007/BF01066558
[35]Zh. Xia, Existence of invariant tori in volume-preserving diffeomorphisms.Ergod. Theory Dyn. Syst. 12 (1992), No. 3, 621–631. · Zbl 0768.58042 · doi:10.1017/S0143385700006969
[36]D. V. Anosov, Averaging in systems of ordinary differential equations with fast-oscillating solutions. (Russian)Izv. Akad. Nauk SSSR 24 (1960), No. 5, 721–742.
[37]V. I. Bakhtin, Averaging in multifrequency systems.Funct. Anal. Appl. 20 (1986), No. 2, 83–88. · Zbl 0611.34035 · doi:10.1007/BF01077261
[38]V. I. Arnold, Geometrical methods in the theory of ordinary differential equations.Springer-Verlag, New York, 1983.
[39]–, Conditions for the applicability, and estimate of the error, of the averaging method for systems which pass through resonances in the course of their evolution.Sov. Math. Dokl. 6 (1965), No. 2, 331–334.
[40]A. I. Neishtadt, On the passage through resonances in the twofrequency problem. (Russian)Dokl. Akad. Nauk SSSR 221 (1975), No. 2, 301–304.
[41]–, On the averaging in multifrequency systems. (Russian)Dokl. Akad. Nauk SSSR 223 (1975), No. 2, 314–317.
[42]–, On the averaging in multifrequency systems. II. (Russian)Dokl. Akad. Nauk SSSR 226 (1976), No. 6, 1295–1298.
[43]H. W. Broer and G. B. Huitema, A proof of the isoenergetic KAM-theorem from the ”ordinary” one.J. Differ. Eqs. 90 (1991), No. 1, 52–60. · Zbl 0721.58020 · doi:10.1016/0022-0396(91)90160-B
[44]J. Moser, Stable and random motions in dynamical systems, with special emphasis on celestial mechanics.Ann. Math. Stud., Princeton University Press, Priceton, NJ,77, 1973.
[45]R. L. Devaney, Reversible diffeomorphisms and flows.Trans. Am. Math. Soc. 218 (1976), 89–113. · doi:10.1090/S0002-9947-1976-0402815-3
[46]V. I. Arnold and M. B. Sevryuk, Oscillations and bifurcations in reversible systems. In: Nonlinear Phenomena in Plasma Physics and Hydrodynamics, Ed. R. Z. Sagdeev,Mir, Moscow, 31–64, 1986.
[47]M. B. Sevryuk, Reversible Systems.Lect. Notes Math. 1211,Springer-Verlag,Berlin, 1986.
[48]–, Lower-dimensional tori in reversible systems.Chaos 1 (1991), No. 2, 160–167. · Zbl 0899.58017 · doi:10.1063/1.165858
[49]J. A. G. Roberts and G. R. W. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems.Phys. Rep. 216 (1992), No. 2-3, 63–177. · doi:10.1016/0370-1573(92)90163-T
[50]G. R. W. Quispel and M. B. Sevryuk, KAM theorems for the product of two involutions of different types.Chaos 3 (1993), No.4, 757–769. · Zbl 1055.37575 · doi:10.1063/1.165935
[51]J. A. G. Roberts and M. Baake, Trace maps as 3D reversible dynamical systems with an invariant.J. Stat. Phys. 74 (1994), No. 3-4, 829–888. · Zbl 0830.58025 · doi:10.1007/BF02188581
[52]J. S. W. Lamb and G. R. W. Quispel, Reversingk-symmetries in dynamical systems.Physica D 73 (1994), No. 4, 277–304. · Zbl 0814.58035 · doi:10.1016/0167-2789(94)90101-5
[53]H. W. Broer and G. B. Huitema, Unfoldings of quasi-periodic tori in reversible systems.J. Dyn. Differ. Eqs. 7 (1995), No. 1, 191–212. · Zbl 0820.58050 · doi:10.1007/BF02218818