Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
V.I. Arnol'd, 1963, Proof of a theorem of A.N.Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian, Russ. Math. Surveys 18, no.5, 9–36.
V.I. Arnol'd, 1964, Instability of dynamical systems with several degrees of freedom, Sov. Math. Dokl. 5, 581–585
V.I.Arnol'd and A.Avez, 1968, Ergodic problems of classical mechanics (Benjamin)
S. Aubry, 1983, The twist map, the extended Frenkel-Kontorova model and the devil's staircase, Physica 7D, 240–258
S. Aubry and P.Y. LeDaeron, 1983, The discrete Frenkel-Kontorova model and the extensions: I exact results for the ground-states, Physica 8D, 381–422
D. Bensimon and L.P. Kadanoff, 1984, Extended chaos and disappearance of KAM trajectories, Physica 13D, 82–89
M.V.Berry, 1978, Regular and irregular motion, in Topics in Nonlinear dynamics, ed. S.Jorna, Am.Inst.Phys.conf.proc. 46, 16–120
G.D. Birkhoff, 1913, Proof of Poincaré's Geometric theorem, Trans. Am. Math. Soc. 14, 14–22
G.D. Birkhoff, 1920, Surface transformations and their dynamical applications, Acta Math. 43, 1–119
G.D. Birkhoff, 1927, On the periodic motions of dynamical systems, Acta Math. 50, 359–379
G.D. Birkhoff, 1932, Sur quelques courbes fermeés remarquables, Bull. Soc. Math. de France 60, 1–26
P.L.Boyland and G.R.Hall, 1985, Invariant circles and the order structure of periodic orbits in monotone twist maps, Boston preprint
B.V. Chirikov, 1979, A universal instability of many dimensional oscillator systems, Phys. Reports 52, 263–379
D.F.Escande, 1984, Stochasticity in classical Hamiltonian systems:universal aspects, Austin preprint to appear in Phys. Reports
D.F. Escande and F. Doveil, 1981, Renormalisation method for computing the threshold of the large-scale stochastic instability in two degrees of freedom Hamiltonian systems, J. Stat. Phys. 26, 257–284
J.D.Farmer and I.Satija, 1984, Renormalization of the quasiperiodic transition to chaos for arbitrary winding numbers, Los Alamos preprint
J.M.Finn, 1974, Ph.D. thesis, Univ. Maryland
J.M. Greene, 1968, Two-dimensional measure-preserving mappings, J. Math. Phys. 9, 760–768
J.M. Greene, 1979, A method for determining a stochastic transition, J. Math. Phys. 20, 1183–1201
J.M.Greene, 1980, KAM surfaces computed from the Hénon-Heiles Hamiltonian, in Nonlinear dynamics and the beam-beam interaction, eds M.Month and J.C.Herrera, Am. Inst. Phys. conf. proc. 57
J.M. Greene and R.S. MacKay, 1985, An approximation to the critical commuting pair for breakup of noble tori, Phys. Lett. 107A, 1–4
J.M.Greene, R.S.MacKay and J.Stark, 1985, Boundary circles for area-preserving maps, in preparation
J.M. Greene and I.C. Percival, 1981, Hamiltonian maps in the complex plane, Physica 3D, 530–548
J.D.Hanson, J.R.Cary and J.D.Meiss, 1984, Algebraic decay in self-similar Markov chains, Austin preprint
N.T.A.Haydn, 1985, On invariant circles under renormalisation, Warwick preprint, submitted to Comm. Math. Phys.
R.H.G.Helleman, 1978, Variational solutions of non-integrable systems, in Topics in nonlinear dynamics, ed. S.Jorna, Am. Inst. Phys. conf. proc. 46, 264–285
R.H.G.Helleman, 1980, Self-generated chaotic behaviour in nonlinear mechanics, in Fundamental problems in Statistical Mechanics, vol. 5, ed. E.G.D.Cohen (N.Holland), 165–233; reprinted in Universality in Chaos, ed. P.Cvitanovic (Adam Hilger, 1984), 420–488
M.R.Herman, 1983, Sur les courbes invariantes par les diffeomorphismes de l'anneau, vol. 1, Asterisque 103–104
M.R.Herman, 1984, Sur les courbes invariantes par les diffeomorphismes de l'anneau, vol. 2, to appear
A.Katok, 1983, Periodic and quasiperiodic orbits for twist maps, in dynamical systems and chaos, Sitges 1982, ed. L.Garrido (Springer)
J.C. Keck, 1967, Adv. Chem. Phys. 13, 85–121
R.S. MacKay, 1982, Renormalisation in area-preserving maps, PhD thesis Princeton (Univ. Microfilms Int., Ann Arbor, Michigan)
R.S. MacKay, 1983, A renormalisation approach to invariant circles in area-preserving maps, Physica 7D, 283–300
R.S. MacKay, 1984, Equivariant university classes, Phys. Lett. 106A, 99–100
R.S.MacKay, 1985, Exact results for an approximate renormalisation scheme, in preparation
R.S. MacKay and J.D. Meiss, 1983, Linear stability of periodic orbits in Lagrangian systems, Phys. Lett. 98A, 92–94
R.S. MacKay, J.D. Meiss and I.C. Percival, 1984, Transport in Hamiltonian systems, Physica 13D, 55–81
R.S. MacKay and I.C. Percival, 1985a, Converse KAM: theory and practice, Comm. Math. Phys. 98, 469–512
R.S.MacKay and I.C.Percival, 1985b, Self-similarity of the boundary of Seigel disks for arbitrary rotation number, in preparation
R.S.MacKay and J.Stark, 1985, Lectures on orbits of minimal action for area-preserving maps, Warwick preprint
R.Marcelin, 1915, Ann. Physique 3, 120-
J.N. Mather, 1982a, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology 21, 457–467
J.N. Mather, 1982b, Glancing billairds, Erg. Theory Dyn. Sys. 2, 397–403
J.N.Mather, 1982c, A criterion for non-existence of invariant circles, Princeton preprint, submitted to Publ. Math.I.H.E.S.
J.N. Mather, 1984, Non-existence of invariant circles, Erg. Theory Dyn. Sys. 4, 301–309
B.D.Mestel, 1984, A computer-assisted proof of universality for cubic critical maps of the circle with rotation number the golden mean, in preparation
J.K. Moser, 1962, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, II Math. Phys. Kl. 1, 1–20
J.K.Moser, 1973, Stable and random motions in dynamical systems (Princeton Univ. Press)
V.I. Osceledec, 1968, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, 197–231
S. Ostlund, R. Rand, J. Sethna and E. Siggia, 1983, Universal properties of the transition from quasiperiodicity to chaos in dissipative systems, Physica 8D, 303–342
I.C. Percival, 1980, Variational principles for invariant tori and cantori, in Nonlinear dynamics and the beam-beam interaction, eds M. Month and J.C. Herrera Am. Inst. Phys. conf. proc. 57, 302–310
I.C. Percival, 1982, Chaotic boundary of a Hamiltonian map, Physica 6D, 67–77
Ya.B. Pesin, 1977, Characteristic Lyapunov exponents and smooth ergodic theory, Russ. Math. Surveys 32, 55–114
M. Peyrard and S. Aubry, 1983, Critical behaviour at the transition by breaking of analyticity in the discrete Frenkel-kontorova model, J. Phys. C16, 1593–1608
H. Poincaré, 1885, Sur les courbes definies par les equations differentielles, J. de Math. pure appl., 4eme serie, 1, 167–244
H. Poincaré, 1912, Sur un theoreme de geometrie, Rendicinti del Circolo Mat. dl Palermo 33, 375–407
D.A.Rand, 1984, Universality for critical golden circle maps and the breakdown of dissipative golden invariant tori, submitted to Comm. Math. Phys.
R.C. Robinson, 1970a, Generic properties of conservative systems, Am. J. Math. 92, 562–603
R.C. Robinson, 1970b, Generic properties of conservative systems II, Am. J. Math. 92, 897–906
H. Rüssmann, 1983, On the existence of invariant curves of twist mappings of an annulus, Springer lecture notes in Math. 1007, 677–718
G. Schmidt, 1980, Stochasticity and fixed point transitions, Phys. Rev. 22A, 2849–2854
S. Schmidt and J. Bialek, 1982, Fractal diagrams for Hamiltonian stochasticity, Physica 5D, 397–404
S.J.Shenker and L.P.Kadanoff, 1982, Critical behaviour of a KAM surface: I empirical results, J. Stat. Phys. 27, 631-
R.M. Sinclair, J.C. Hosea and G.V. Sheffield, 1970, A Method for mapping a toroidal magnetic field by storage of phase stabilized electrons, Rev. Sci. Instruments 41, 1552–1559
S. Smale, 1967, Differentiable dynamical systems, Bull. Am. Math. Soc. 73, 747–817
J.Stark, 1984, An exhaustive criterion for the non-existence of invariant circles in area-preserving maps, Warwick preprint, submitted to Erg. Theory Dyn. Sys.
D.K.Umberger and J.D.Farmer, 1984, Fat fractals on the energy surface, Los Alamos preprint, submitted to Phys. Rev. Lett.
E.Wigner, 1937, J. Chem. Phys. 5, 720-
E. Zehnder, 1973, Homoclinic points near elliptic fixed points, Comm. Pure Appl. Math. 26, 131–182
A.Zygmund, 1959, Trigonometric series (2nd edition, Cambridge Univ. Press, 1968) *** DIRECT SUPPORT *** A0124038 00011
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1986 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
MacKay, R.S. (1986). Transition to chaos for area-preserving maps. In: Lecture Notes in Physics. Lecture Notes in Physics, vol 247. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0107356
Download citation
DOI: https://doi.org/10.1007/BFb0107356
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16455-5
Online ISBN: 978-3-540-39792-2
eBook Packages: Springer Book Archive