Phase synchronization in bidirectionally coupled optothermal devices
R. Herrero
Departament de Fı ´sica i Enginyeria Nuclear, Universitat Polite `cnica de Catalunya, Comte Urgell 187, 08036 Barcelona, Spain
M. Figueras, F. Pi, and G. Orriols
Departament de Fı ´sica, Universitat Auto `noma de Barcelona, 08193 Bellaterra, Spain
Received 21 May 2002; published 27 September 2002
We present the experimental observation of phase synchronization transitions in the bidirectional coupling of
chaotic and nonchaotic oscillators. A variety of transitions are characterized and compared to numerical simu-
lations of a time delayed model. The characteristic 2 phase jumps usually appear during the transitions,
specially in those clearly associated with a saddle-node bifurcation. The study is done with pairs of optothermal
oscillators linearly coupled by heat transfer.
DOI: 10.1103/PhysRevE.66.036223 PACS numbers: 05.45.Xt, 42.50.Ar
I. INTRODUCTION
Synchronization of coupled chaotic oscillators has re-
cently been the object of intensive research and different
types of synchronization have been described 1,2. One of
the relevant behaviors expected for weak couplings is the
phase synchronization PS phenomenon, i.e., the synchroni-
zation of phases while amplitudes have not to be necessarily
correlated. The transition to PS when the coupling is in-
creased was first observed in mutually coupled Ro
¨
ssler mod-
els by Rosenblum et al. 3. A characteristic feature of the
observed transition is the occurrence of intermediate states in
which the phase difference of the oscillators remains almost
fixed, for finite time intervals suddenly interrupted by 2
phase jumps and the mean frequency of such jumps de-
creases with increasing the coupling towards the PS state
3,4. The transition to PS has been numerically studied with
different models considering two or more oscillators and it
has been associated with a variety of dynamical bifurcations
4–10. Phase jumps are almost always observed in the nu-
merically simulated PS transitions but with a variety of scal-
ing properties that seem related to the kind of underlying
bifurcation 3,4,11,9 and the influence of noise 12.
Experimental demonstrations of PS to an external peri-
odic pacing have been reported for a variety of systems ex-
hibiting chaotic evolutions 13–16 and irregular biological
rhythms 17,18. The PS between unidirectionally coupled
chaotic oscillators has been also reported 19,20 and the
concept of phase synchrony has been used for the character-
ization of rather complex oscillatory behaviors such as those
observed in brains 21–23. The first experimental observa-
tion of a transition to PS in bidirectionally coupled oscilla-
tors has been reported very recently 24. In this case the
transition happens via phase jumps occurring upwards or
downwards irregularly, in a similar way as numerically de-
tected in coupled hyperchaotic Ro
¨
ssler oscillators, where PS
has been related to type-II inttermitency 9, and with scaling
properties agreeing well with those observed in coupled
Ro
¨
ssler models 4.
In this work we present a detailed experimental analysis
of a number of synchronization transitions observed in pairs
of bidirectionally coupled oscillators, including intermediate
states with phase jumps and PS states with uncorrelated am-
plitudes. The experiment is done with a kind of optothermal
nonlinear oscillators linearly coupled by heat transfer and we
have used pairs of two- and three-dimensional oscillators
that exhibit periodic and chaotic evolutions when isolated,
respectively. In all of the cases, the coupled elements are
nearly similar but not identical, with slight differences in
both oscillating frequencies and steady-state solutions. Nu-
merical simulations in reasonable agreement with the experi-
mental results indicate that we have observed PS transitions
clearly associated either with a cyclic saddle-node bifurca-
tion or with a secondary Hopf bifurcation. Nevertheless, the
analysis points out a rich variety of PS transitions without a
clear relation with a specific bifurcational process.
II. NONLINEAR DEVICE AND EXPERIMENTAL SETUP
The nonlinear oscillators are based on the so-called opto-
thermal bistability with localized absorption BOITAL and
they have been described in detail elsewhere 25–27.A
BOITAL device consists of a Fabry-Pe
´
rot cavity, where the
input mirror is a partially absorbing film, the rear mirror is a
high-reflection dielectric coating, and the spacer between
mirrors is constituted by N transparent layers with alterna-
tively opposite thermo-optic coefficients. The cavity is illu-
minated with a focalized laser beam and the reflected power
is detected with a photodiode. The light absorption in the
input mirror is affected by the interference effects, as de-
scribed by the Airy function, and it constitutes the nonlinear-
ity of the system. The device presents a multiple stationary
solution associated with the periodicity of the nonlinear
function. The effective dimension of the device dynamics is
N and the system is able to experience up to N -1 different
Hopf bifurcations due to the competition and time delay be-
tween the contributions of the various layers to the light
phase shift within the cavity 25,26.
As shown in Fig. 1, different oscillators separated by a
certain distance can be created by focusing parallel light
beams onto the same transversally extended optical device.
The nonlinear elements are coupled by heat propagation
through the cavity spacer and the separation distance d may
be used to adjust the coupling strength. The oscillators have
PHYSICAL REVIEW E 66, 036223 2002
1063-651X/2002/663/03622310/$20.00 ©2002 The American Physical Society 66 036223-1