In a previous paper of one of us [Europhys. Lett.59, 330

336
(2002)] the validity of Greene’s method for determining the critical constant of the standard map (SM) was questioned on the basis of some numerical findings. Here we come back to that analysis and we provide an interpretation of the numerical results, by showing that the conclusions of that paper were wrong as they relied on a plausible but untrue assumption. Hence no contradiction exists with respect to Greene’s method. We show that the previous results, based on the expansion in Lindstedt series, do correspond to the critical constant but for a different map: the semi-standard map (SSM). For such a map no Greene’s method analog is at disposal, so that methods based on Lindstedt series are essentially the only possible ones. Moreover, we study the expansion for two simplified models obtained from the SM and SSM by suppressing the small divisors. We call them the simplified SM and simplified SSM, respectively; the first case turns out to be related to Kepler’s equation after a proper transformation of variables. In both cases we give an analytical solution for the radius of convergence, that represents the singularity in the complex plane closest to the origin. Also here, the radius of convergence of the simplified SM turns out to be lower than that of the simplified SSM. However, despite the absence of small divisors these two radii are lower than those of the true maps (i.e., of the maps with small divisors) when the winding number equals the golden mean. Finally, we study the analyticity domain and, in particular, the critical constant for the two maps without small divisors. The analyticity domain turns out to be a perfect circle for the simplified SSM (as for the SSM itself), while it is stretched along the real axis for the simplified SM, yielding a critical constant which is larger than its radius of convergence.

1.
Abramowitz
,
A. M.
, and
Stegun
,
I. A.
,
Handbook of Mathematical Functions
(
Dover
, New York,
1972
).
2.
Arnol’d
,
V. I.
,
Kozlov
,
V. V.
, and
Neĭshtadt
,
A. I.
,
Dynamical Systems. III. Encyclopaedia of Mathematical Sciences
(
Springer-Verlag
, Berlin,
1988
), Vol.
3
.
3.
Aubry
,
S.
, “
The twist map, the extended Frenkel-Kontorova model and the devil’s staircase
,”
Physica D
7
240
258
(
1983
).
4.
Berretti
,
A.
,
Celletti
,
A.
,
Chierchia
,
L.
, and
Falcolini
,
C.
, “
Natural boundaries for area-preserving twist maps
,”
J. Stat. Phys.
66
,
1613
1630
(
1992
).
5.
Berretti
,
A.
,
Falcolini
,
C.
, and
Gentile
,
G.
, “
The shape of analyticity domains of Lindstedt series: The standard map
,”
Phys. Rev. E
64
,
R015202
(
2001
).
6.
Berretti
,
A.
, and
Gentile
,
G.
, “
Scaling properties for the radius of convergence of Lindstedt series: The standard map
,”
J. Math. Pures Appl.
78
159
176
(
1999
).
7.
Berretti
,
A.
, and
Gentile
,
G.
, “
Scaling properties for the radius of convergence of Lindstedt series: The standard map
,”
J. Math. Pures Appl.
79
691
713
(
2000
).
8.
Berretti
,
A.
, and
Gentile
,
G.
, “
Bryuno function and the standard map
,”
Commun. Math. Phys.
220
,
623
656
(
2001
).
9.
Berretti
,
A.
, and
Gentile
,
G.
, “
Scaling of the critical function for the standard map: Some numerical results
,”
Nonlinearity
17
,
649
670
(
2004
).
10.
Berretti
,
A.
, and
Gentile
,
G.
(private communication).
11.
Carletti
,
T.
, and
Laskar
,
J.
, “
Scaling law in the standard map critical function. Interpolating Hamiltonian and frequency map analysis
,”
Nonlinearity
13
,
2033
2061
(
2000
).
12.
Chirikov
,
B. V.
, “
A universal instability of many-dimensional oscillator systems
,”
Phys. Rep.
52
,
263
379
(
1979
).
13.
Davie
,
A. M.
, “
The critical function for the semistandard map
,”
Nonlinearity
7
,
219
229
(
1994
).
14.
Delshams
,
A.
, and
de la Llave
,
R.
, “
Kam theory and a partial justification of Greene’s criterion for nontwist maps
,”
SIAM J. Math. Anal.
31
,
1235
1269
(
2000
).
15.
Eliasson
,
L. H.
, “
Absolutely convergent series expansions for quasi periodic motions
,”
Math. Phys. Electron. J.
2
, (
1996
), Paper 4,
33
pp
(electronic).
16.
Falcolini
,
C.
, and
de la Llave
,
R.
, “
A rigorous partial justification of Greene’s criterion
,”
J. Stat. Phys.
67
,
609
643
(
1992
).
17.
Falcolini
,
C.
, and
de la Llave
,
R.
, “
Numerical calculation of domains of analyticity for perturbation theories in the presence of small divisors
,”
J. Stat. Phys.
67
,
645
666
(
1992
).
18.
Finch
,
S. R.
,
Mathematical Constants
(
Cambridge University Press
, Cambridge,
2003
).
19.
Floria
,
L. M.
, and
Mazo
,
J. J.
, “
Dissipative dynamics of the Frenkel-Kontorova model
,”
Adv. Phys.
45
,
505
598
(
1996
).
20.
Gallavotti
,
G.
, “
Twistless KAM tori
,”
Commun. Math. Phys.
164
,
145
156
(
1994
).
21.
Gentile
,
G.
,
Diagrammatic Techniques in Perturbations Theory, and Applications
, Symmetry and perturbation theory (Rome, 1998) (
World Scientific
, River Edge, NJ,
1999
), pp.
59
78
.
22.
Gradshteyn
,
I. S.
, and
Ryzhik
,
I. M.
,
Table of Integrals, Series, and Products
(
Academic
, San Diego,
2000
).
23.
Greene
,
J. M.
, “
A method for determining a stochastic transition
,”
J. Math. Phys.
20
,
1183
1201
(
1979
).
24.
Greene
,
J. M.
, and
Percival
,
I. C.
, “
Hamiltonian maps in the complex plane
,”
Physica D
3
,
530
548
(
1981
).
25.
Gyalog
,
T.
, and
Thomas
,
H.
, “
Friction between atomically flat surfaces
,”
Europhys. Lett.
37
,
195
200
(
1997
).
26.
Hardy
,
G. H.
, and
Wright
,
E. M.
,
An Introduction to the Theory of Numbers
, 5th ed. (
Oxford University Press
, New York,
1979
).
27.
Jungreis
,
I.
, “
A method for proving that monotone twist maps have no invariant circles
,”
Ergod. Theory Dyn. Syst.
11
,
79
84
(
1991
).
28.
Katznelson
,
Y.
,
An Introduction to Harmonic Analysis
, Cambridge Mathematical Library, 3rd ed. (
Cambridge University Press
, Cambridge,
2004
).
29.
Kittel
,
C.
,
Introduction to Solid State Physics
, 2nd ed. (
Wiley
, New York,
1953
).
30.
Laskar
,
J.
,
Froeschlé
,
C.
, and
Celletti
,
A.
, “
The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping
,”
Physica D
56
,
253
269
(
1992
).
31.
MacKay
,
R. S.
, “
Greene’s residue criterion
,”
Nonlinearity
5
,
161
187
(
1992
).
32.
MacKay
,
R. S.
,
Renormalisation in Area Preserving Maps
,
Advanced Series in Nonlinear Dynamics Vol. 3
. (
World Scientific
, Singapore,
1993
).
33.
MacKay
,
R. S.
, and
Percival
,
I. C.
, “
Converse KAM: theory and practice
,”
Commun. Math. Phys.
98
,
469
512
(
1985
).
34.
Mather
,
J. N.
, “
Non existence of invariant circles
,”
Ergod. Theory Dyn. Syst.
4
,
301
309
(
1984
).
35.
Percival
,
I. C.
, “
Chaotic boundary of a Hamiltonian map
,”
Physica D
6
,
67
77
(
1982
).
36.
Percival
,
I. C.
, and
Vivaldi
,
F.
, “
Critical dynamics and trees
,”
Physica D
33
,
304
313
(
1988
).
37.
Poincaré
,
H.
,
Les Méthodes Nouvelles de la Mécanique Classique
(
Gauthier-Villars
, Paris,
1899
), Vol.
III
.
38.
Riordan
,
J.
,
Combinatorial Identities
(
Wiley
, New York,
1979
).
39.
Siegel
,
C. L.
, “
Iteration of analytic functions
,”
Ann. Math.
43
,
607
612
(
1942
).
40.
Simon
,
B.
, “
Almost periodic Schrödinger operators. IV. The Maryland model
”,
Ann. Phys. (N.Y.)
159
,
157
183
(
1985
).
41.
Tomlinson
,
G. A.
, “
A molecular theory of friction
,”
Philos. Mag.
7
,
905
939
(
1929
).
42.
van Erp
,
T. S.
, and
Fasolino
,
A.
, “
Aubry transition studied by direct evaluation of the modulation functions of the infinite incommensurate systems
,”
Europhys. Lett.
59
,
330
336
(
2002
).
43.
van Erp
,
T. S.
,
Fasolino
,
A.
,
Radulescu
,
O.
, and
Janssen
,
T.
, “
Pinning and phonon localization in Frenkel-Kontorova models on quasiperiodic substrates
,”
Phys. Rev. B
60
,
6522
6528
(
1999
).
44.
Watson
,
G. N.
,
A Treatise on the Theory of Bessel Functions
(
Cambridge University Press
, Cambridge,
1944
).
45.
Weiss
,
M.
, and
Elmer
,
F. J.
, “
Dry friction in the Frenkel-Kontorova-Tomlinson model: static properties
,”
Phys. Rev. B
53
,
7539
7549
(
1996
).
46.
Weiss
,
M.
, and
Elmer
,
F. J.
, “
Dry friction in the Frenkel-Kontorova-Tomlinson model: dynamical properties
,”
Z. Phys. B: Condens. Matter
104
,
55
69
(
1997
).
47.
Wells
,
D.
,
The Penguin Dictionary of Curious and Interesting Numbers
(
Penguin Books
, Middlesex, England,
1986
).
48.
Whittaker
,
E. T.
, and
Watson
,
G. N.
,
A Course of Modern Analysis
(
Cambridge University Press
, Cambridge,
1997
).
49.
Wilbrink
,
J.
, “
Erratic behavior of invariant circles in standard-like mappings
,”
Physica D
26
,
358
368
(
1987
).
50.
A.
Wintner
,
The Analytic Foundations of Celestial Mechanics
(
Princeton University Press
, Princeton, NJ,
1941
).
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