140
5 Arnol’d Diffusion
Fig. 5.4 (a) Black regions are resonance islands and KAM tori of the Arnol’d web. Light regions
are chaos. = 0.001 and the computer run time is t = 1000. (b) Enlargement of a section of (a)
with = 0.001 t = 4000 (from (Froeschle et al. 2000) and reprinted with permission from AAAS)
Arnol’d web can be represented in the two-dimensional plane {I 1 , I 2 }. All
resonances n 1 ω 1 + n 2 ω 2 + n 3 = 0 (with ω i =
dφ i
dt ) appear as straight lines
n 1 I 1 + n 2 I 2 + n 3 = 0 in the I 1 , I 2 plane. The set of all resonances is dense in
the plane I 1 , I 2 .
In Fig. 5.4a, b, = 0.001 and the resonant lines are embedded in large zones
filled with KAM tori. Because the perturbation has a full Fourier spectrum (all
harmonics are present at order ), a large number of resonances are visible at small
and for the computer run time used (t = 1000). As increases, the volume of
invariant tori decreases and the chaotic regions become more evident at the crossing
of resonances. For = 0.04 (Fig. 5.5a, b), the majority of invariant tori have
disappeared because of resonance overlapping, and a chaotically connected region
has replaced large regions of KAM tori.
To obtain the clear differentiation between chaotic and regular orbits in Figs. 5.4
and 5.5, the authors used a version of the Lyapounov exponent called the fast
Lyapounov indicator (FLI) (Froeschle et al. 1997), which allowed them to the
discriminate between regular and chaotic motion more easily than the simpler form
of Lyapounov exponents described in Chap. 2.
5.5 Arnol’d Diffusion in an Optical Lattice
An optical lattice, in one space dimension, is formed when two counter-propagating
laser beams, with the same wavelength and frequency, interfere and form a spatially
periodic polarization pattern. The periodic potential that is formed can trap neutral
two-level atoms. Similarly, an optical lattice, in two space dimensions, can be
5 Arnol’d Diffusion
Fig. 5.4 (a) Black regions are resonance islands and KAM tori of the Arnol’d web. Light regions
are chaos. = 0.001 and the computer run time is t = 1000. (b) Enlargement of a section of (a)
with = 0.001 t = 4000 (from (Froeschle et al. 2000) and reprinted with permission from AAAS)
Arnol’d web can be represented in the two-dimensional plane {I 1 , I 2 }. All
resonances n 1 ω 1 + n 2 ω 2 + n 3 = 0 (with ω i =
dφ i
dt ) appear as straight lines
n 1 I 1 + n 2 I 2 + n 3 = 0 in the I 1 , I 2 plane. The set of all resonances is dense in
the plane I 1 , I 2 .
In Fig. 5.4a, b, = 0.001 and the resonant lines are embedded in large zones
filled with KAM tori. Because the perturbation has a full Fourier spectrum (all
harmonics are present at order ), a large number of resonances are visible at small
and for the computer run time used (t = 1000). As increases, the volume of
invariant tori decreases and the chaotic regions become more evident at the crossing
of resonances. For = 0.04 (Fig. 5.5a, b), the majority of invariant tori have
disappeared because of resonance overlapping, and a chaotically connected region
has replaced large regions of KAM tori.
To obtain the clear differentiation between chaotic and regular orbits in Figs. 5.4
and 5.5, the authors used a version of the Lyapounov exponent called the fast
Lyapounov indicator (FLI) (Froeschle et al. 1997), which allowed them to the
discriminate between regular and chaotic motion more easily than the simpler form
of Lyapounov exponents described in Chap. 2.
5.5 Arnol’d Diffusion in an Optical Lattice
An optical lattice, in one space dimension, is formed when two counter-propagating
laser beams, with the same wavelength and frequency, interfere and form a spatially
periodic polarization pattern. The periodic potential that is formed can trap neutral
two-level atoms. Similarly, an optical lattice, in two space dimensions, can be