How does a periodic rotating wave emerge from high-dimensional chaos
in a ring of coupled chaotic oscillators?
Ying Zhang,
1,
* Gang Hu,
2
and Hilda A. Cerdeira
1
1
The Abdus Salam International Centre for Theoretical Physics, P. O. Box 586, 34100Trieste, Italy
2
Department of Physics, Beijing Normal University, Beijing 100875, China
Received 16 February 2001; revised manuscript received 17 May 2001; published 21 August 2001
A route to typical rotating waves from high-dimensional chaos is investigated in diffusively coupled chaotic
Ro ¨ssler oscillators. By increasing the coupling from zero, a high-dimensional spatiotemporal chaos changes
into a coherent state, which is periodic in time and well ordered in space, through consecutive transitions. A
crossover transition from spatially random chaos to spatially ordered chaos with phase locking and orienta-
tional equality for two directions breaking is a crucial step for establishing the typical spatial order of the
rotating wave.
DOI: 10.1103/PhysRevE.64.037203 PACS numbers: 05.45.Xt, 05.45.Jn
The dynamics of networks of coupled oscillators is a fun-
damental problem. In many applications, the oscillators are
identical, dissipative, and the coupling is symmetric. Since
Turing’s seminal work 1 on morphogenesis, rings of
coupled oscillators have been used extensively in physiologi-
cal and biochemical studies 2,3, coupled laser systems, Jo-
sephson junction arrays, electrical circuits, coupled chemical
oscillators, etc. 4–6. In early studies, interest focused on
coupled periodic oscillators 3. During the recent decade,
interest has turned to the study of coupled chaotic oscillators.
Rich behavior of chaotic and regular patterns associated with
various kinds of chaos synchronization has been revealed
7. With weak coupling the ring of the oscillators shows
high-dimensional spatiotemporal chaos. With certain inter-
mediate coupling we can, usually, observe some regular pat-
terns with both spatial and temporal orders. For instance,
rotating waves are typical patterns with these orders. It is
important to understand how spatiotemporal chaos can be
changed to a rotating wave state by continually varying a
certain control parameter, in particular, how the spatial order
of the rotating wave is established in this variation process.
To our knowledge, this problem has not been clearly an-
swered, and this is the central focus of the present paper. We
shall show that the spatial order of the antiphase distribution
of oscillators is established far before the rotating wave ap-
pears. An average-antiphase distribution can occur in a high-
dimensional chaotic state via phase synchronization of
chaos. This synchronization is the root of the spatial order of
the periodic rotating wave.
We take the coupled Ro
¨
ssler oscillators as our model:
x
˙
i
=- y
i
-z
i
+ x
i +1
+x
i -1
-2 x
i
,
y
˙
i
=x
i
+ay
i
+ y
i +1
+ y
i -1
-2 y
i
,
1
z
˙
i
=b + x
i
-c z
i
+ z
i +1
+z
i -1
-2 z
i
,
x
i +n
=x
i
, y
i +n
= y
i
, z
i +n
=z
i
, i =1,2, . . . , n .
For a =0.45, b =2.0, and c =4.0, the single Ro
¨
ssler oscilla-
tor is chaotic. In most of the paper, we fix the system size to
n =6, and an extension to general system size will be briefly
discussed later in this work. For small coupling, 1, the
motion is high-dimensional chaos, and it is chaotic in time
and disordered in space see Fig. 1a. However, for certain
intermediate coupling, the motion becomes regular. For in-
stance, a stable rotating wave solution exists in the range of
0.057 0.090 8. In Fig. 1b, we fix =0.080, where
we can see a rotating wave solution, which has a typical Z
6
spatial symmetry 9, and it is periodic in time and well
ordered in space. In this state, all oscillators take an identical
periodic orbit, but they have equal phase shift T /6 between
each pair of nearest-neighbor oscillators with T being the
period of the motion. In this paper we call this kind of phase
distribution the antiphase distribution 6.
To answer the problem of how the disordered and chaotic
state of Fig. 1a can develop into the regular state of Fig.
1b with both temporal and spatial orders, we first briefly
report on the observations of numerical results by increasing
the coupling strength from zero with a coarse step
=0.001. We find there are several major steps in this varia-
tion. First, the spatially disordered chaos C
D
shown in Fig.
1a transit to a chaotic state with average-antiphase distri-
*Email address: yzhang@ictp.trieste.it
FIG. 1. The orbits of Eqs. 1 in ( x , y ) space. a =0.005 the
disordered chaos state. b =0.080 the rotating wave. The num-
bers i =1,2, . . . ,6 in the figures indicate the positions of the i os-
cillator at an arbitrary instant. The notation same is also used in Fig.
2.
PHYSICAL REVIEW E, VOLUME 64, 037203
1063-651X/2001/643/0372034/$20.00 ©2001 The American Physical Society 64 037203-1