# zbMATH — the first resource for mathematics

Construction of analytic KAM-surfaces and effective stability bounds. (English) Zbl 0657.58032
An effective method for finding of KAM-tori of Hamiltonian systems and canonical mappings closed to integrable ones is suggested. The method yields lower estimations for the break-down threshold of the tori under increasing of perturbative parameter. Computer-assisted applications to Chirikov-Green standard map $$(x,y)\mapsto (x+y+\epsilon \sin x,\quad y+\epsilon \sin x)$$ are given and the existence of “gold-mean” KAM- circles for complex values of $$\epsilon$$ with $$| \epsilon | \leq 0.65$$ is stated.
Reviewer: A.Givental’

##### MSC:
 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion 37C55 Periodic and quasi-periodic flows and diffeomorphisms 70H15 Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics
Full Text:
##### References:
 [1] Arnold, V.I.: Proof of a Theorem by A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv.18, 9 (1963) · Zbl 0129.16606 · doi:10.1070/RM1963v018n05ABEH004130 [2] Arnold, V.I., Avez, A.: Ergodic problems of classical mechanics. New York: Benjamin 1968 · Zbl 0167.22901 [3] Aubry, S., Le Daeron, P.Y.: The discrete Frenkel-Kontorova model and its extensions. I. Physica8, 381 (1983) · Zbl 1237.37059 [4] Benettin, G., Galgani, L., Giorgilli, A.: Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part I. Commun. Math. Phys.113, 87 (1987) · Zbl 0646.70013 · doi:10.1007/BF01221399 [5] Benettin, G., Gallavotti, G.: Stability of motions near resonances in quasi-integrable Hamiltonian systems. J. Stat. Phys.44, 293 (1986) · Zbl 0636.70018 · doi:10.1007/BF01011301 [6] Braess, D., Zehnder, E.: On the numerical treatment of a small divisor problem. Numer. Math.39, 269 (1982) · Zbl 0497.65047 · doi:10.1007/BF01408700 [7] Celletti, A., Chierchia, L.: Rigorous estimates for a computer-assisted KAM theory. J. Math. Phys.28, 2078 (1987) · Zbl 0651.58011 · doi:10.1063/1.527418 [8] Celletti, A., Falcolini, C., Porzio, A.: Rigorous numerical stability estimates for the existence of KAM tori in a forced pendulum. Ann. Inst. Henri Poincaré47, 85 (1987) · Zbl 0636.70017 [9] Chirikov, B.V.: A universal instability of many dimensional oscillator systems. Phys. Rep.52, 263 (1979) · doi:10.1016/0370-1573(79)90023-1 [10] Eckmann, J.-P., Wittwer, P.: Computer methods and Borel summability applied to Feigenbaum’s equation. Lecture Notes in Physics, Vol. 227. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0598.58040 [11] Eliasson, H.: Absolutely convergent series expansions for quasiperiodic motions, preprint. Univ. of Stockholm (1987) [12] Escande, D.F.: Stochasticity in classical Hamiltonian systems: Universal aspects. Phys. Rep.121, 165 (1985) · Zbl 0643.58010 · doi:10.1016/0370-1573(85)90019-5 [13] Escande, D.F.: Private communication [14] Escande, D.F., Doveil, F.: Renormalization method for computing the threshold of the largescale stochastic instability in two degrees of freedom Hamiltonian systems. J. Stat. Phys.26, 257 (1981) · doi:10.1007/BF01013171 [15] Falcolini, C.: Private communication [16] Gallavotti, G.: Perturbation theory for classical Hamiltonian systems. In: Scaling and self-similarity in physics. Fröhlich, J. (ed.) PPh. 7, Boston: Birkhäuser 1984 [17] Greene, J.M.: A method for determining a stochastic transition. J. Math. Phys.20, 1183 (1979) · doi:10.1063/1.524170 [18] Hénon, M., Heiles, C.: The applicability of the third integral of motion: Some numerical experiments. Astron. J.69, 73 (1964) · doi:10.1086/109234 [19] Herman, M.: Sur le courbes invariantes par le difféomorphismes de l’anneau. Astérisque2, 144 (1986) [20] Iooss, G., Helleman, R.H.G., Stora, R. (eds.): Chaotic behaviour of deterministic systems. Amsterdam: North-Holland 1983 · Zbl 0546.00032 [21] Kolmogorov, A.N.: On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl. Akad. Nauk. SSR98, 469 (1954) · Zbl 0056.31502 [22] Lanford III, O.E.: Computer assisted proofs in analysis. Phys. A124, 465 (1984) · Zbl 0599.58036 [23] MacKay, R.S.: Transition to chaos for area-preserving maps. Lecture Notes in Physics, Vol. 247, p. 390. Berlin, Heidelberg, New York: Springer 1985 [24] MacKay, R.S., Percival, I.C.: Converse KAM: Theory and practice. Commun. Math. Phys.98, 469 (1985) · Zbl 0585.58032 · doi:10.1007/BF01209326 [25] Mather, J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology21, 457 (1982) · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4 [26] Mather, J.N.: Non-existence of invariant circles. Ergodic Theory Dyn. Syst.4, 301 (1984) · Zbl 0557.58019 · doi:10.1017/S0143385700002455 [27] Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nach. Akad. Wiss. Göttingen, Math. Phys. K1. II1, 1 (1962) · Zbl 0107.29301 [28] Moser, J.: A rapidly convergent iteration method and non-linear partial differential equations. Ann. Scuola Norm. Sup. Pisa20, 265 (1966) · Zbl 0174.47801 [29] Moser, J.: Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré3, 229 (1986) · Zbl 0609.49029 [30] Moser, J.: Minimal foliations on a torus. Forschungsinstitut für Mathematik, ETH Zürich, September 1987 · Zbl 0689.49036 [31] Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Russ. Math. Surv.32, 1 (1977) · Zbl 0389.70028 · doi:10.1070/RM1977v032n06ABEH003859 [32] Percival, I.C.: Variational principles for invariant tori and cantori, in nonlinear dynamics and the beam-beam interaction. AIP Conference Proceedings 57, p. 302. M. Month, J.C. Herrera (eds.). (1980) [33] Rüssmann, H.: On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. Lecture Notes in Physics, Vol. 38, p. 598. Berlin, Heidelberg, New York: Springer 1975 · Zbl 0319.35017 [34] Rüssmann, H.: On the existence of invariant curves of twist mappings of an annulus. Lecture Notes in Mathematics, Vol. 1007, p. 677. Berlin, Heidelberg, New York: Springer 1983 · Zbl 0531.58041 [35] Salamon, D., Zehnder, E.: KAM theory in configuration space. Preprint 1987 · Zbl 0682.58014 [36] Wayne, C.E.: The KAM theory of systems with short range interactions. I. Commun. Math. Phys.96, 311 (1984) · Zbl 0585.58023 · doi:10.1007/BF01214577 [37] Wayne, C.E.: Bounds on the trajectories of a system of weakly coupled rotators. Commun. Math. Phys.104, 21 (1986) · Zbl 0615.70013 · doi:10.1007/BF01210790 [38] Wisdom, J., Peale, S.J.: The chaotic rotation of Hyperion. Icarus58, 137 (1984) · doi:10.1016/0019-1035(84)90032-0 [39] (no author listed) Vax architecture handbook. Digital Equipment Corporation (1981)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.