Literature to the Störmerproblem

Literature to the Stoermer problem

Oliver Knill

Since the undergraduate thesis had been written in a now obsolete text processor SIGNUM, I do not have the entire text. While being a graduate student, a guest of Moser had been interested in the literature of this fascinating topic and I had at that time extracted and cleaned out the literature in ASCII form1. This has survived and is listed below. In the summer of 2010, when we moved within Arlington, I had to get rid of a lot of old stuff. In particular also of thousands of old papers. Being in a wet cellar for 10 years many of my article library had started to rot. A few articles have kept dry and were scanned in a scan orgy, where also a third of my book library was digitized. I could rescue 15 documents on the Stoermer problem. The evanescent paper versions are in "article heaven" now, but their "spirit" will survive electronically here. Of particular value is Mosers 1963 unpublished note on Stoermer problem which I have gotten from him and which remained unpublished up to day. A paper of Truc from 1992 (with letter of Truc) was added while I was still a graduate student in 1992. A Devaney article and an article of Walt and Farley did not make it into the literature list. Back to my undergraduate thesis
a) Stoermerproblem

Ar 1963   V.I. Arnold; Small divisor and stability problems in classical
                       and celestial mechanics, Uspekhi Mat. Nauk.18(6)(1963) pp. 85-191
                       2 pages [PDF]
Br 1970   Martin Braun; Particle Motions in a Magnetic Field; J.Diff.
                        Equ., 8 (1979), pp.294-332 [PDF]
Br 1970a  Martin Braun; Structural Stability and the Stoermer Problem;
                        Indiana Univ. Math. J.,20 (1970),pp. 469-497 [PDF]
Br 1979   Martin Braun; The Stability of the Van Allen Radiation Belt;
                        SIAM J.Appl. Math. 37(3), pp.664-668 [PDF]
Br 1981   Martin Braun; Mathematical Remarks on the Van Allen
                        Radiation Belt; SIAM Rev.,23 (1981) ,pp. 61-94 [PDF]
Br 1981a  Martin Braun; Invariant Curves,Homoclinic Points,and Ergodicity
                        in Area-Preserving Mappings; SIAM J. Math. Anal.,12(4)
Dr 1965   Alex Dragt;   Trapped orbits in a magnetic dipole field; Rev.
                         of Geophys.3(2) (1965), pp.255-298 [PDF]
Dr 1976   Alex Dragt, John M.Finn;  Insolubility of Trapped Particle
                                    Motion in a Magnetic Dipole Field; 
                                    J. Geophys. Res. 81,(1976) pp. 2327 [PDF]
Dr 1978   Alex Dragt, John M.Finn; Normal Form for Mirror Machine
                        Hamiltonians; J.Math. Phys. 20(12) (1979) pp.2649-2660 [PDF]
Ea 1980  Robert Eather; Majestic Lights ;
Fa 1983  Harald Falck-Ytter; Das Polarlicht; Verlag Freies Geistesleben, 
                             Stuttgart 1983
He 1965  W.N.Hess,G.D.Mead and M.P.Nakada; Advances in Particle and
                      Field Research in the Satellite Era; Rev. of Geophys 3(4)
                      (1965) pp. 521-575
Mo 1963  Juergen Moser; Integrals of the Stoermer Problem; unpublished [PDF]
St 1907  Carl Stoermer; Sur les trajectoires des corpuscles electrises
                        dans l'espace sous l'action du magnetisme terrestre;
                        Arch.Sci.Phys.Nat.24, (1907)
St 1955  Carl Stoermer; The Polar Aurora; Clarendon Press, Oxford, 1955
Vo 1954  Rene de Vogelaere; Surface de section dans le proble` me
                        de Stoermer; Acad. Roy. Belg. Bull. Cl. Sci. 40(5)
                        (1954) pp. 705-714 [PDF]
Vo 1958  Rene de Vogelaere; On the structure of symmetric periodic
                   solutions of conservative systems with applications;
                   Ann. of Math. Studies,no 41, Princeton University Press,
                   Princeton N.J.,1958,pp. 53-84 [PDF]

b) Linked Twist maps           

Bo 1977   Rufus Bowen ; On Axiom A Diffeomorphisms ; Proc. C.B.M.S.
                        Regional Conf. Ser. Math.No. 35, Amer. Math. Soc.,
                        Providence R.I.
Be 1984   David Bensimon and Leo Kadanoff; Extended Chaos and Disappearance of 
                        KAM Trajectories ; Physica 13D,(1984) pp.82 -89
Bu 1979   Robert Burton and Robert Easton; Ergodicity of Linked Twist
                        Mappings; Lecture Notes in Math.,No. 819, pp. 35-49
Br 1981a  Martin Braun; Invariant Curves,Homoclinic Points,and
                        Ergodicity in Area-Preserving Mappings; SIAM J. Math.
                        Anal.,12(4) (1981),pp.630-638
De 1978   Robert L. Devaney; Subshifts of Finite Type in Linked Twist
                             Mappings; Proc. of the Amer. Math. Soc.,71(2) 
                             (1978) pp. 334-338
De 1979   R.L. Devaney; Linked Twist Mappings are Almost Anosov;
                        Lecture Notes in Math., No. 819, pp. 121-145
De 1980   R.L. Devaney; Three Area Preserving Mappings Exhibiting
                        Stochastic Behavior; Lect. Notes in Pure and Appl. Math.
                        70, New York 1981
De 1981   R.L. Devaney; The Baker Transformation and a Mapping
                        Associated to the Restricted Three Body Problem; Commun.
                        Math. Phys. 80 (1981) pp. 465-476
Ea 1979   Robert Easton; Chain Transitivity and the Domain of Influence of 
                         an Invariant Set; Lecture Notes in Math. No. 668
                         pp. 95-102
Go 1984  Daniel L.Goroff; Hyperbolic Sets for Twist Maps;  Ergod.
                          Theory and Dyn. Sys. 5 (1985), pp.337-339
Ka 1979  Anatol Katok; Bernoulli Diffeomorphisms on Surfaces ;
                        Annals of Math. 110 (1979) pp. 529-547
Ma 1983  John N. Mather; Nonexistence of Invariant Circles; Ergod.
                      Theory and Dyn. Sys. 3(1983), pp. 301-309
Mc 1983  R.S.Mackay; Renormalisation Approach to Invariant Circles
                     in Area-preserving Maps; Physica 7D,(1983), pp. 283-300
Pr 1983  Feliks Przytycki; Ergodicity of Toral Linked Twist Mappings,
                           Ann. Scient. Ec. Norm. Sup. 4(1) (1983), pp. 345-354
Pr 1982  Feliks Przytycki; Examples  of  Conservative Diffeomorphisms
                      of the twodimensional Torus with Coexistence of Elliptic
                      and Stochastic Behaviour; Ergod. Theory and Dynam. Sys. 2 (1982)
                      pp. 439-463
Pr 1986  Feliks Przytycki; Periodic Points of Linked Twist Mappings;
                           Studia Mathematica, 83 (1986)
Wo 1979  Maciej Wojtkowski ; Linked Twist Mappings have the
                      K-Property; Nonlinear Dynamics, International Conference,
                      New York 1979, pp 66-76
Wo 1981  Maciej Wojtkowski; A Model Problem with Coexistence of
                       Stochastic and Integrable Behaviour; Comm. Math. Phys.
                       80(4) (1981), pp. 453-464
Wo 1982  Maciej Wojtkowski; On the Ergodic Properties of Piecewise Linear
                       Perturbations of the Twist Map; Ergod. Th. and Dynam. Sys.
                       2 (1982); pp. 525-542

c) Lyapunov exponents

Be 1984  Giancarlo Benettin; Power-Law Behaviour of Lyapunov
                      Exponents in some Conservative Systems; Physica D 13
                      (1984) pp. 211-220
Cw 1985  John B.Conway; A Course in Functional Analysis; Graduated
                      Texts in Math.,Springer, NewYork 1985
Co 1984   Eric Cornelis and Maciej Wojtkowski; A Criterion for the
                       Positivity of the Liapunov Characteristic Exponent;
                      Ergod. Th. and Dynam. Sys. 4 (1984) pp. 527-539
Fu 1983  H. Furstenberg and Y. Kifer; Random Matrix Products and
                      Measures on Projective Spaces; Israel J. Of Math. 46(1)
                      (1983) pp.12-36
Pe 1977  Ya.B. Pesin; Lyapunov Characteristic Exponents and Smooth
                      Ergodic Theorie; Russ. Math. Suveys 32(4) (1977),
Sl 1980  Eric V. Slud; Products of Independent Randomly Perturbed
                      Matrices; in  Ergodic Theorie and Dynamical Systems II,
                      Proceedings, Special Year, Maryland 1979-1980,Birkhaeuser
                      pp. 185-192
Wo 1985  Maciej Woitkowski; Invariant Families of Cones and Lyapunov
                      exponents; Ergod. Th. and Dynam. Sys. 5(1985) pp. 145-161
Wo 1976  Maciej Woitkowski; Principles for the Design of Billards
                      with Nonvanishing Lyapunov Exponents; Comm. Math. Phys.
                      105 (1986), pp. 391-414
Yo 1986  L.-S. Young; Random Perturbations of Matrix Cocycles;
                      Ergod. Th. and Dynam. Sys. 6 (1986) pp. 627-637

d) Numerical studies

Bi 1986   Philippe M. Binder and Roderick V. Jensen; Simulating chaotic
                       Behavior with Finite- State Machines; Phys. Rev. A 34(5)
                      (1976) pp. 4460-4463
Br 1971   A. Brahic; Numerical Study of a Simple Dynamical System;
                      Astron. and Astrophys. 12 (1971), pp. 98-110
Fo 1977   Joseph Ford; Relevance of Computer Experiments to the
                      Pure Mathematics of Integrable and Ergodic Dynamical
                      Systems; Soc. Math. de France, Asterisque 49 (1977)
                      pp. 75-92
He1964   Michel Henon and Carl Heiles; The Applicability of the Third
                      Integral of Motion;  Astron. J. 69(1) (1964), pp. 73-80
Ra 1973   F. Rannou; Numerical Study of Discrete Plane Area-
                      preserving Mappings; Astron and Astrophys. 31(1974),
                      pp. 289-301

e) Ergodic theory

Ar 1967   V.I. Arnold and A.Avez; Proble`mes ergodiques de la me' canique
                    classique, Paris 1967
Ka 1975   Anatol B. Katok,Ya. G.Sinai and A.M. Stepin ; Theorie of
                      Dynamical Systems and General Transformation Groups
                      with Invariant Measure;  Math. Anal. 13 (1975) pp. 129-262
Ka 1986   Anatol Katok and Jean-Marie Strelcyn; Invariant Manifolds,
                      Entropy and Billards : Smooth Maps with Singularities;
                      Lecture Notes in Mathematics No. 1222, Springer 1986
Kr 1985   Ulrich Krengel; Ergodic Theorems; New York, De Gruyter
Ma 1987   Ricardo Mane' ; Ergodic Theorie and Differentiable Dynamics;
                      Springer EM, 3.Folge Bd.8, Berlin 1987
Si 1976   Ya.G.Sinai; Introduction to Ergodic Theorie.; Math. Notes
                      18, Princeton University Press 1976;
Wa 1982   Peter Walters; Introduction to Ergodic Theory; Graduated
                      Texts in Math. 79, Springer 1982
(1) Episodes like this (and there are many more since 1987, when this literature list was collected) made me wary of proprietary file formats or formats which need proprietary programs to read them and since (maybe almost obsessively) store everything of value in ASCII text form or in formats in which open source and industry independent conversion programs are available.

This document was added in October 2010