We study the existence of strange nonchaotic attractors (SNA) in the family of Harper maps. We prove that for a set of parameters of positive measure, the map possesses a SNA. However, the set is nowhere dense. By changing the parameter arbitrarily small amounts, the attractor is a smooth curve and not a SNA.
REFERENCES
1.
D.
Ruelle
and F.
Takens
, Commun. Math. Phys.
20
, 167
(1971
).2.
J.-P.
Eckmann
and D.
Ruelle
, Rev. Mod. Phys.
57
, 617
(1985
).3.
C.
Grebogi
, E.
Ott
, S.
Pelikan
, and J. A.
Yorke
, Physica D
13
, 261
(1984
).4.
A.
Prasad
, S. S.
Negi
, and R.
Ramaswamy
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
11
, 291
(2001
).5.
6.
7.
8.
T.
Jäger
, Ergod. Theory Dyn. Syst.
. (to be published).9.
10.
J. A.
Ketoja
and I. I.
Satija
, Physica D
109
, 70
(1997
).11.
J. A.
Ketoja
and I. I.
Satija
, Phys. Rev. Lett.
75
, 2762
(1995
).12.
S. S.
Singh
and R.
Ramaswany
, Phys. Rev. Lett.
83
, 4530
(1999
).13.
S. S.
Singh
and R.
Ramaswany
, Phys. Rev. E
64
, 045204
(2001
).14.
B. D.
Mestel
, A. H.
Osbaldestin
, and B.
Winn
, J. Math. Phys.
41
, 8304
(2000
).15.
A.
Bondeson
, E.
Ott
, and T. M.
Antonsen
, Jr., Phys. Rev. Lett.
55
, 2103
(1985
).16.
M. D.
Choi
, G. A.
Elliott
, and N.
Yui
, Invent. Math.
99
, 225
(1990
).17.
18.
A.
Avila
and S.
Jitomirskaya
, Ann. Math.
(to be published).19.
A.
Haro
and R.
de la Llave
(preprint, 2005
).20.
A.
Haro
and R.
de la Llave
, Chaos
16
, 013120
(2006
).21.
B. D.
Mestel
and A. H.
Osbaldestin
, J. Math. Phys.
45
, 5042
(2004
).22.
B. D.
Mestel
and A. H.
Osbaldestin
, J. Phys. A
37
, 9071
(2004
).23.
24.
J. B.
Sokoloff
, Phys. Rep.
126
, 189
(1985
).25.
K. v.
Klitzing
, G.
Dorda
, and M.
Pepper
, Phys. Rev. Lett.
45
, 494
(1980
).26.
27.
28.
29.
30.
R.
Johnson
and G.
Sell
, J. Differ. Equations
41
, 262
(1981
).31.
R.
Johnson
, J. Differ. Equations
35
, 366
(1980
).32.
M.
Hirsch
, C.
Pugh
, and M.
Shub
, Invariant Manifolds
, Lecture Notes in Mathematics
(Springer-Verlag
, Berlin
, 1977
), Vol. 583
.33.
A.
Haro
and R.
de la Llave
(preprint, 2003
).34.
S.
Datta
, T.
Jäger
, G.
Keller
, and R.
Ramaswamy
, Nonlinearity
17
, 2315
(2004
).35.
36.
37.
38.
39.
S. Y.
Jitomirskaya
and I. V.
Krasovsky
, Math. Res. Lett.
9
, 413
(2002
).40.
41.
B.
Hunt
, and E.
Ott
, Phys. Rev. Lett.
87
, 254101
(2001
).42.
R.
Johnson
and J.
Moser
, Commun. Math. Phys.
84
, 403
(1982
).43.
44.
E.
Coddington
and N.
Levinson
, Theory of Ordinary Differential Equations
(McGraw-Hill
, New York
, 1955
).45.
S.
Jitomirskaya
and B.
Simon
, Commun. Math. Phys.
165
, 201
(1994
).46.
J.
Puig
, Nonlinearity
19
, 355
(2006
).47.
E.
Sorets
and T.
Spencer
, Commun. Math. Phys.
142
, 543
(1991
).48.
J.
Bourgain
, Green’s Function Estimates for Lattice Schrödinger Operators and Applications
, Annals of Mathematics Studies
(Princeton University Press
, Princeton
, 2005
).© 2006 American Institute of Physics.
2006
American Institute of Physics
You do not currently have access to this content.
