Reply to ‘‘Comment on ‘Quantum suppression of chaos in the spin-boson model’ ’’
G. A. Finney
*
and J. Gea-Banacloche
Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701
Received 25 February 1997
We dispute the claim by Bonci et al. Phys. Rev. E 56, 2325 1997 that we have misidentified a chaotic
trajectory, and point out that their new results actually support the main conclusions of our original article.
S1063-651X9701508-0
PACS numbers: 05.45.+b
In the preceding Comment the authors attempt to look
independently at the problem studied by us in a recent pub-
lication 1. They make a number of points, the most impor-
tant of which is that, in their opinion, we have mistakenly
identified as chaotic a trajectory that has, in fact, a vanish-
ingly small Lyapunov exponent.
We did state in our paper 1 that we had found nonvan-
ishing Lyapunov exponents for trajectories in the neighbor-
hood of the + branch, and we stand by that claim. Since
we have not been able to compare our algorithm to the one
used by Bonci et al., we cannot say precisely why we do and
they do not. As we explained in our paper, the chaos is
confined to the neighborhood of the unstable periodic orbit,
and it seems that the truly chaotic trajectories lie in a very
narrow region of phase space around this orbit. This makes
the calculation of Lyapunov exponents for these orbits a little
tricky. If the numerical integration routine takes too large a
step in the wrong direction it may leave the chaotic region
entirely and the Lyapunov exponent will then approach zero
2.
In particular, the trajectory that starts from the Autler-
Townes AT initial conditions that we quote in our paper
( x =0.483, y
0
=0, z
0
=-0.876) may or may not be chaotic
itself—two different algorithms we have tried yield different
results. We need to point out, however, that these are not
exactly the same as the initial conditions for the trajectory
we show and study in our paper Figs. 1a and 1b of 1.
Our approach was typically to take the AT initial conditions
as a starting point in our search for the periodic orbits, which
are always to be found nearby. The initial conditions for the
periodic orbit shown in Fig. 1a of 1 are somewhere in the
range x
0
=0.47380.0002, y
0
=0, z
0
=- 1 -x
0
2
. For n
¯
=81 we find consistently chaotic trajectories in this neigh-
borhood.
In any event, our diagnostic of chaos did not rely solely
on Lyapunov exponents or visual observation of the trajec-
tories. Our paper 1 shows, for instance, a remarkable
change in the spectrum Figs. 2a and 2b of 1—not just
the apparition of many new lines but of actual broadband
structures, characteristic of chaos. On the same page of 1
we give a detailed account of how the periodic orbit becomes
unstable, based on our Floquet analysis of a linearized
model.
Bonci et al. seem to disregard all this evidence and in-
stead use some Poincare
´
sections their Figs. 3 and 4 to
bolster their claim that nothing unusual happens in the neigh-
borhood of the + branch. We agree that those sections do
not show anything special, but since an actual instability
clearly does take place, all their figures prove is that they are
*Present address: USAFA/DFP, U.S. Air Force Academy, Colo-
rado Springs, CO 80840.
FIG. 1. Poincare ´ maps generated by one trajectory in the vicin-
ity of the + periodic orbit and one trajectory in the vicinity of the
- periodic orbit, for n ¯=100 and 81, respectively. For n ¯=100, the
enlargements of what look like single points show the trajectory
winds over a narrow stable torus around the periodic orbit. For n ¯
=81, the - branch is still stable but the + branch has become
unstable: in the Poincare ´ map, the fixed point associated with the
+ periodic orbit has become a hyperbolic, homoclinic point.
Chaos is generic in the vicinity of such points.
PHYSICAL REVIEW E AUGUST 1997 VOLUME 56, NUMBER 2
56 1063-651X/97/562/23292/$10.00 2329 © 1997 The American Physical Society