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Prog. Theor. Phys. Vol. 109 No. 3 (2003) pp. 313-331
Chaos in Two-Dimensional φ3 Theory with Oscillator Modes
Masanobu Yahiro,1,*
Yasuhito Kaminaga,2,**
Yoshio Saito3,*** and
Shin-Ichi Ohtsubo4,****
1Department of Physics and Earth Sciences, University of the Ryukyus,
Okinawa 903-0213, Japan
2Department of Physics, Gunma National College of Technology,
Maebashi 371-8530, Japan
3Faculty of Medicine, Toho University, Tokyo 143-8540, Japan
4Department of Physics, Kyushu University,
Fukuoka 812-8581, Japan
(Received August 21, 2002)
Abstract:
A classical scalar field in a box with a periodic boundary
is approximately described as a superposition
of the spatially homogeneous mode and the lowest oscillator mode.
This approximation reduces the scalar field theory to
a four-dimensional nonlinear system with three constants,
the total energy E, the “angular momentum” ℓ,
and the wave number k of the oscillator mode.
In (k,ℓ)-space, the parameter combinations which yield
chaos are those for which
(i) 0.3 \lesssimk \lesssim0.9 and 0 ≤ℓ\lesssim0.1,
and those in
(ii) the arm-shaped region that ranges
from (k,ℓ) ≃(0.9, 0.0) to (0.7, 0.4).
Stochasticity is most conspicuous
when E takes its maximum value.
As E decreases, the stochasticity is rapidly lost, and
when E becomes below roughly 60% of the maximum value,
the system behaves deterministically,
for any choice of k,ℓ and the initial conditions.
Stochasticity is lost also in the large ℓ, large k
and small k limits.
There is no (k, ℓ, E) combination that yields chaos
for (almost) all initial conditions.
In the present paper, these results are confirmed numerically.
Some of the types of behavior can be explained
in terms of the curvature of the potential surface,
weak coupling areas, and the shape of the kinetic region.
URL :
http://ptp.ipap.jp/link?PTP/109/313/
DOI : 10.1143/PTP.109.313
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