The nonlinear response of a chaotic system to a chaotic variation in a system parameter is investigated experimentally. Clear experimental evidence of frequency entrainment of the chaotic oscillations is observed. We show that analogous to the frequency locking between coupled periodic oscillations, this effect is generic for coupled chaotic systems.Tang, D. Dykstra, R. ; Hamilton, M. ; Heckenberg, N

C. O. Weiss

D. Gabor

D. Y. Tang

E. Roldán

Emily F. Stone

H. Haken

H. Zeghlache

J. D. Farmer

M. G. Rosenblum

M. W. Hamilton

N. R. Heckenberg

R. Dykstra

Tin Win

Tang, D.

Dykstra, R.

Hamilton, M.

Heckenberg, N.

Adelaide Research &amp; Scholarship

PUBLISHED VERSION  Tang, D. Y.; Dykstra, R.; Hamilton, Murray Wayne; Heckenberg, N. R.  Experimental evidence of frequency entrainment between coupled chaotic oscillations Physical Review E, 1998; 57(3):3649-3651  ©1998 American Physical Society http://link.aps.org/doi/10.1103/PhysRevE.57.3649           http://link.aps.org/doi/10.1103/PhysRevD.62.093023                        http://hdl.handle.net/2440/12766        PERMISSIONS http://publish.aps.org/authors/transfer-of-copyright-agreement   “The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”): [...] 3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.”    29th April 2013  PHYSICAL REVIEW E MARCH 1998VOLUME 57, NUMBER 3Experimental evidence of frequency entrainment between coupled chaotic oscillationsD. Y. Tang, R. Dykstra, M. W. Hamilton,* and N. R. HeckenbergPhysics Department and the Centre for Laser Science, The University of Queensland, Brisbane, Queensland 4072, Australia~Received 3 November 1997!The nonlinear response of a chaotic system to a chaotic variation in a system parameter is investigatedexperimentally. Clear experimental evidence of frequency entrainment of the chaotic oscillations is observed.We show that analogous to the frequency locking between coupled periodic oscillations, this effect is genericfor coupled chaotic systems. @S1063-651X~98!09903-6#PACS number~s!: 05.45.1b, 05.40.1j, 42.65.SfRecently, on the basis of a numerical simulation, Stonehas shown that a phase coherent strange attractor can befrequency entrained to small periodic driving pulses near thepeak frequency in the broadband power spectra of the attrac-tor @1#. Here by the term ‘‘phase coherent attractor’’ shemeans a strange chaotic attractor with the characteristic thatits Fourier power spectrum has a sharp spectral peak, possi-bly together with its harmonics, in a continuous broadbandspectral background. The appearance of a sharp spectral peakin the power spectrum indicates the existence of a predomi-nant frequency in the chaotic oscillation @2#. Therefore,Stone’s numerical result demonstrates that, analogous to theeffect of the frequency locking of a driven periodic oscilla-tions, there exists a similar effect in a driven chaotic oscilla-tion. However, because for a chaotic attractor the phase isintrinsically dynamically unstable, the effect is that the aver-age chaotic pulsation frequency is locked to the frequency ofthe driving periodic pulsations.In the context of chaos synchronization, Rosenblum et al.@3# have also discovered a phenomenon of phase synchroni-zation between weakly coupled self-sustained chaotic oscil-lators. In the phase synchronization regime they found intheir numerical studies that the phases of the coupled chaoticoscillators are locked, while the amplitudes of them varychaotically and are practically uncorrelated. As far as weunderstand, the so-called ‘‘phase synchronization’’ is in facta state of frequency entrainment between the coupled chaoticattractors where their average chaotic pulsation frequenciesare locked.Although the possibility of frequency entrainment of cha-otic attractors has been numerically demonstrated, to the bestof our knowledge no clear experimental evidence of the ef-fect in any physical chaotic systems has been reported so far.To explore the possibility and also to study the interactionbetween coupled chaotic systems, we have conducted an ex-perimental investigation on the dynamics of a chaotic systemsubjected to a chaotic driving force in the form of a chaoticvariation of a system parameter. Experimentally this is mucheasier to achieve than coherently adding to one of the systemvariables, as has been previously considered theoretically.For this reason it is also a case of greater technological sig-nificance. We have experimentally observed the frequency*Visiting from the Department of Physics and MathematicalPhysics, The University of Adelaide, Adelaide, South Australia.571063-651X/98/57~3!/3649~3!/$15.00entrainment of the driven chaotic system to the chaotic driv-ing signal. In this paper we present our experimental results.In particular, we show that the frequency entrainment effectis a generic behavior of coupled chaotic oscillations in thesense that it is insensitive to the coupled chaotic dynamics.Our experiment is carried out on an optically pumped15NH3 single mode ring laser operating chaotically. Detailsabout the laser and its configuration can be found in @4#. Tostudy the effect of frequency entrainment, the pump intensityof the laser is chaotically modulated. This is experimentallydone with the following procedure: Firstly we either record achaotic intensity wave form from the laser with constantpump power or calculate a chaotic wave form with the Lo-renz equations. We then store this chaotic wave form in thememory of an arbitrary function generator ~AFG! ~HewlettPackard 33120A!. Based on the stored chaotic wave form,the AFG produces an analog chaotic signal output whosechaotic wave form is the same as that of the stored signal,but its amplitude and average chaotic pulsation frequencycan be continuously tuned. The chaotic output of the AFG isthen used to amplitude modulate the rf driving signal of anacousto-optic modulator ~AOM!, and the undiffracted beamfrom the AOM is used as the 15NH3 laser pump beam. Ef-fectively the pump intensity obtained in this way consists ofa chaotically varying pump intensity, which acts as the cha-otic driving force in our experiment and a dc pump intensity,which ensures that even without the chaotic driving, the dy-namics of the laser is chaotic. The chaotic intensity output ofthe 15NH3 laser is detected with a Schottky-barrier diode. Tocompare the chaotic laser dynamics with those of the actualdriving signal, the diffracted beam from the AOM is detectedwith a HgTe-CdTe detector. The electrical signals from bothdetectors are low-noise amplified and then displayed simul-taneously on a Tektronix digital storage oscilloscope. A rfspectral analyzer is used to monitor the real-time Fourierpower spectrum of the chaotic laser dynamics. Based on thisinformation the frequency relationship between the chaoticlaser dynamics and the driving chaotic signal is controlled.The chaotic dynamics of the laser with constant pumphave been intensively studied previously and are well knownnow @5,6#. Deterministic chaos such as period-doublingchaos, spiral chaos, and intermittency are typical chaotic dy-namics of the laser. Under suitable conditions it was foundthat the chaotic dynamics of the laser can be well describedby the single mode laser Lorenz-Haken equations @7# or ex-tended Lorenz-Haken equations including the laser cavity3649 © 1998 The American Physical Society3650 57BRIEF REPORTSdetuning @8#. In our present experiment we have operated thelaser in the parameter regime where its chaotic dynamics canbe well understood with the extended laser Lorenz-Hakenequations. For the sake of comparison, we show in Fig. 1 atypical experimentally observed chaotic dynamics of the la-ser without chaotic driving. Figure 1~a! is the chaotic evolu-tion of the laser intensity. Figure 1~b! is the calculated Fou-rier power spectrum of the intensity evolution. As can beclearly identified from the Fourier spectrum, there is a spec-tral peak in the broadband spectrum. As a feature of thedetuned laser Lorenz-Haken equations, when the laser cavitydetuning is reduced to zero, the intensity dynamics of thelaser moves from period-doubling chaos to Lorenz type spi-ral chaos @8#. Experimentally we observed that accompany-ing these dynamics changes, the width of the spectral peakbroadens. The broadening of the spectral peak is a result ofthe increasing chaotic phase fluctuation associated with theLorenz type spiral chaos, which is manifested as the varia-tion in the time intervals between the successive chaotic laserintensity pulsations @1,2#. The position of the spectral peakrepresents the average pulsation frequency of the chaotic os-cillation, and varies as experimental conditions such as thepump laser intensity, gas pressure, and the laser cavity de-tuning are changed. We have studied the chaotic dynamics ofthe laser with a chaotically varying pump by examining howits dynamics changes and the relationship of its dynamics tothat of the driving signal. With a fixed driving signalFIG. 1. A typical chaotic dynamics of the laser without chaoticdriving. Pump intensity is 0.8 W/cm 2. Gas pressure is 19 mbar.Laser cavity tuning is close to the resonance. ~a! Chaotic laser in-tensity evolution. ~b! Fourier power spectrum calculated from thelaser intensity evolution shown in ~a!.strength, we experimentally observed that the separation be-tween average pulsation frequency of the driving signal andthat of the chaotic laser intensity plays an important role inthe chaotic dynamics of the laser. When the separation be-tween the average frequencies is large, under a chaotic pumpthe dynamics of the laser becomes more complicated than itsoriginal single mode dynamics. In this case the chaotic in-tensity evolution of the laser exhibits no obvious relation tothe driving signal. However, when the average frequencyseparation is small, within a certain range that solely dependson the driving signal strength ~with a peak-peak chaoticmodulation of 30% of the average pump intensity, we ob-served a locking range of about 0.15 MHz!, the average pul-sation frequency of the laser locks to that of the drivingsignal and the chaotic intensity pulsations of the laser be-come frequency entrained to that of the driving signal.To demonstrate this frequency entrainment effect, weshow in Fig. 2 a typically observed experimental result. Fig-ure 2~a! is the time evolution of the chaotic laser intensity,Fig. 2~b! is the chaotic driving signal measured by invertingthe intensity of the power diffracted by the AOM. We havemade a part of the modulation signal a dc level so as tocompare the laser dynamics with and without chaotic driv-ing. The original chaotic wave form used to modulate the rfdriving signal of the AOM is a segment of prerecorded cha-otic intensity evolution of the laser without chaotic driving,and the dc level is selected such that it is the mean level ofthe chaotic signal. However, due to the bandwidth limitationof the AOM, the real chaotic wave form of the drivingshown in Fig. 2~b! is significantly distorted from the originalwave form. This drawback shows on the other hand that theobserved effect is not dependent on the detailed driving sig-nal dynamics. Figure 2~a! shows that without the chaoticdriving the intensity dynamics of the laser is a Lorenz spiralchaos. When the chaotic driving is switched on, the dynam-ics of the laser deviates from its original form and normallyafter a transient that depends on the driving strength and theaverage frequency separation, the chaotic intensity pulsationsof the laser become ‘‘synchronized’’ to the chaotic drivingpulses, despite the fact that both the driving signal and theFIG. 2. Chaotic dynamics of the laser under frequency entrain-ment. Average pump intensity is 0.85 W/cm 2. Gas pressure is 19mbar. Laser cavity tuning is almost in resonance. The peak-peakmodulation strength is 20% of the mean pump intensity. ~a! Chaoticlaser intensity evolution. ~b! Evolution of the driving chaotic signal.57 3651BRIEF REPORTSintrinsic chaotic laser dynamics possess strong chaotic phasefluctuations. Once the average frequencies are locked,slightly changing the experimental conditions such as thepump frequency detuning does not change the average pul-sation frequency. When the driving signal is switched off,the chaotic dynamics of the laser returns to its original Lo-renz spiral chaos.We have examined in detail the relationship between thechaotic laser pulsations and the driving signal. While withoutthe chaotic driving, the successive peak intensity pulse returnmap of the laser shows a clear cusp form of structure nor-mally found for spiral chaos @5#, under the frequency entrain-ment shown in Fig. 2, this cusp structure smears out. Bymeasuring the time delay as the time interval between thepeak of the laser intensity pulse to its corresponding drivingpulse, we found that the delay is not fixed but the higher thedriving pulse, the smaller the time delay. Plotting the delaytime return map does not show any relationship betweenthem either. Obviously, in this state the amplitudes of thechaotic laser intensity pulsations are independent of those ofthe driving signal.To quantitatively demonstrate that the average chaoticpulsation frequency of the laser shown in Fig. 2 is locked tothat of the driving signal, we have calculated the instanta-neous phase evolution associated with the chaotic drivingsignal and the chaotic laser intensity evolution based on ana-lytic signal theory @9#. The average overall slope of the cha-otic phase evolution gives the average chaotic pulsation fre-quency. As shown in Fig. 3 these two phase evolutions havethe same average slope. As also evidenced by Fig. 3, theinstantaneous phases vary independently about the averagephase slope. Since the deviation of the instantaneous phasefrom the average phase slope is a result of the chaotic phasemodulation due to the chaotic dynamics, this experimentalresult shows that under the frequency entrainment, the phasedynamics of the driven oscillation can still be chaotic and notrelated to the driving signal.We have also experimentally studied the possibility offrequency entraining the chaotic dynamics of the laser withother chaotic driving signals; e.g., we have used a period-doubling chaos wave form as the driving signal to frequencyentrain the Lorenz-like spiral chaos of the laser and viceversa. We have also used chaotic wave forms calculatedfrom the Lorenz equations as the driving signal and repeatedthe experiment. In all cases we have observed the same ef-fect. Our experimental results suggest that this effect is ageneric feature of coupled chaotic oscillations and it is in-sensitive to the coupled chaotic dynamics.In conclusion, we have experimentally observed the fre-quency entrainment of a chaotic oscillation to a chaotic driv-ing signal. Our experimental results show that although for achaotic oscillation there is no unique natural frequency, un-der coupling it can exhibit an effect similar to the frequencyentrainment of a coupled periodic oscillator. However, in thechaotic case it is the average chaotic pulsation frequenciesthat are locked together. This experimental result suggeststhat in a chaotic oscillation, the average chaotic pulsationfrequency plays the same role as that of the natural frequencyof a periodic oscillation. Because under the frequency en-trainment only the average chaotic pulsation frequency isconstrained and there is no restriction on the phase and am-plitude variations, and for a chaotic oscillation its phase andamplitude are intrinsically dynamically unstable, these resultin that, under the frequency entrainment, the dynamics of thecoupled chaotic systems are still chaotic and irrelevant toeach other. Finally, our experimental results with severalforms of chaos demonstrate that the frequency entrainmenteffect is a generic effect of coupled chaotic oscillations, andit is independent of the particular coupled chaotic dynamics.FIG. 3. Chaotic phase evolution of the laser intensity evolutionand the driving signal. Solid line: phase evolution calculated fromthe laser intensity evolution. Dotted line: phase evolution calculatedfrom the driving signal.@1# Emily F. Stone, Phys. Lett. A 163, 367 ~1992!.@2# J. D. Farmer, Phys. Rev. Lett. 47, 179 ~1981!.@3# M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev.Lett. 76, 1804 ~1996!.@4# Tin Win, M. Y. Li, J. T. Malos, N. R. Heckenberg, and C. O.Weiss, Opt. Commun. 103, 479 ~1993!.@5# C. O. Weiss, R. Vilaseca, N. B. Abraham, R. Cobala´n, E.Rolda´n, G. J. de Valca´rcel, J. Pujol, U. Hu¨nber, and D. Y.Tang, Appl. Phys. B: Laser Opt. 61, 223 ~1995!.@6# E. Rolda´n, G. J. de Valca´rcel, R Vilaseca, R. Cobala´n, V. J.Martı´nez, and R. Gilmore, Quantum Semiclassic. Opt. 9, R1~1997!.@7# H. Haken, Phys. Lett. 53A, 77 ~1975!.@8# H. Zeghlache and P. Mandel, J. Opt. Soc. Am. B 2, 18 ~1985!.@9# D. Gabor, J. IEE London 93, 429 ~1946!.

Experimental evidence of frequency entrainment between coupled chaotic oscillations

Crossref

The nonlinear response of a chaotic system to a chaotic variation in a system parameter is investigated experimentally. Clear experimental evidence of frequency entrainment of the chaotic oscillations is observed. We show that analogous to the frequency locking between coupled periodic oscillations, this effect is generic for coupled chaotic systems

Tang, D. .

Hamilton, M. W.

Heckenberg, N. R.

University of Queensland eSpace

PHYSICAL REVIEW E MARCH 1998VOLUME 57, NUMBER 3Experimental evidence of frequency entrainment between coupled chaotic oscillations
D. Y. Tang, R. Dykstra, M. W. Hamilton,* and N. R. Heckenberg
Physics Department and the Centre for Laser Science, The University of Queensland, Brisbane, Queensland 4072, Australia
~Received 3 November 1997!
The nonlinear response of a chaotic system to a chaotic variation in a system parameter is investigated
experimentally. Clear experimental evidence of frequency entrainment of the chaotic oscillations is observed.
We show that analogous to the frequency locking between coupled periodic oscillations, this effect is generic
for coupled chaotic systems. @S1063-651X~98!09903-6#
PACS number~s!: 05.45.1b, 05.40.1j, 42.65.SfRecently, on the basis of a numerical simulation, Stone
has shown that a phase coherent strange attractor can be
frequency entrained to small periodic driving pulses near the
peak frequency in the broadband power spectra of the attrac-
tor @1#. Here by the term ‘‘phase coherent attractor’’ she
means a strange chaotic attractor with the characteristic that
its Fourier power spectrum has a sharp spectral peak, possi-
bly together with its harmonics, in a continuous broadband
spectral background. The appearance of a sharp spectral peak
in the power spectrum indicates the existence of a predomi-
nant frequency in the chaotic oscillation @2#. Therefore,
Stone’s numerical result demonstrates that, analogous to the
effect of the frequency locking of a driven periodic oscilla-
tions, there exists a similar effect in a driven chaotic oscilla-
tion. However, because for a chaotic attractor the phase is
intrinsically dynamically unstable, the effect is that the aver-
age chaotic pulsation frequency is locked to the frequency of
the driving periodic pulsations.
In the context of chaos synchronization, Rosenblum et al.
@3# have also discovered a phenomenon of phase synchroni-
zation between weakly coupled self-sustained chaotic oscil-
lators. In the phase synchronization regime they found in
their numerical studies that the phases of the coupled chaotic
oscillators are locked, while the amplitudes of them vary
chaotically and are practically uncorrelated. As far as we
understand, the so-called ‘‘phase synchronization’’ is in fact
a state of frequency entrainment between the coupled chaotic
attractors where their average chaotic pulsation frequencies
are locked.
Although the possibility of frequency entrainment of cha-
otic attractors has been numerically demonstrated, to the best
of our knowledge no clear experimental evidence of the ef-
fect in any physical chaotic systems has been reported so far.
To explore the possibility and also to study the interaction
between coupled chaotic systems, we have conducted an ex-
perimental investigation on the dynamics of a chaotic system
subjected to a chaotic driving force in the form of a chaotic
variation of a system parameter. Experimentally this is much
easier to achieve than coherently adding to one of the system
variables, as has been previously considered theoretically.
For this reason it is also a case of greater technological sig-
nificance. We have experimentally observed the frequency
*Visiting from the Department of Physics and Mathematical
Physics, The University of Adelaide, Adelaide, South Australia.571063-651X/98/57~3!/3649~3!/$15.00entrainment of the driven chaotic system to the chaotic driv-
ing signal. In this paper we present our experimental results.
In particular, we show that the frequency entrainment effect
is a generic behavior of coupled chaotic oscillations in the
sense that it is insensitive to the coupled chaotic dynamics.
Our experiment is carried out on an optically pumped
15NH3 single mode ring laser operating chaotically. Details
about the laser and its configuration can be found in @4#. To
study the effect of frequency entrainment, the pump intensity
of the laser is chaotically modulated. This is experimentally
done with the following procedure: Firstly we either record a
chaotic intensity wave form from the laser with constant
pump power or calculate a chaotic wave form with the Lo-
renz equations. We then store this chaotic wave form in the
memory of an arbitrary function generator ~AFG! ~Hewlett
Packard 33120A!. Based on the stored chaotic wave form,
the AFG produces an analog chaotic signal output whose
chaotic wave form is the same as that of the stored signal,
but its amplitude and average chaotic pulsation frequency
can be continuously tuned. The chaotic output of the AFG is
then used to amplitude modulate the rf driving signal of an
acousto-optic modulator ~AOM!, and the undiffracted beam
from the AOM is used as the 15NH3 laser pump beam. Ef-
fectively the pump intensity obtained in this way consists of
a chaotically varying pump intensity, which acts as the cha-
otic driving force in our experiment and a dc pump intensity,
which ensures that even without the chaotic driving, the dy-
namics of the laser is chaotic. The chaotic intensity output of
the 15NH3 laser is detected with a Schottky-barrier diode. To
compare the chaotic laser dynamics with those of the actual
driving signal, the diffracted beam from the AOM is detected
with a HgTe-CdTe detector. The electrical signals from both
detectors are low-noise amplified and then displayed simul-
taneously on a Tektronix digital storage oscilloscope. A rf
spectral analyzer is used to monitor the real-time Fourier
power spectrum of the chaotic laser dynamics. Based on this
information the frequency relationship between the chaotic
laser dynamics and the driving chaotic signal is controlled.
The chaotic dynamics of the laser with constant pump
have been intensively studied previously and are well known
now @5,6#. Deterministic chaos such as period-doubling
chaos, spiral chaos, and intermittency are typical chaotic dy-
namics of the laser. Under suitable conditions it was found
that the chaotic dynamics of the laser can be well described
by the single mode laser Lorenz-Haken equations @7# or ex-
tended Lorenz-Haken equations including the laser cavity3649 © 1998 The American Physical Society
3650 57BRIEF REPORTSdetuning @8#. In our present experiment we have operated the
laser in the parameter regime where its chaotic dynamics can
be well understood with the extended laser Lorenz-Haken
equations. For the sake of comparison, we show in Fig. 1 a
typical experimentally observed chaotic dynamics of the la-
ser without chaotic driving. Figure 1~a! is the chaotic evolu-
tion of the laser intensity. Figure 1~b! is the calculated Fou-
rier power spectrum of the intensity evolution. As can be
clearly identified from the Fourier spectrum, there is a spec-
tral peak in the broadband spectrum. As a feature of the
detuned laser Lorenz-Haken equations, when the laser cavity
detuning is reduced to zero, the intensity dynamics of the
laser moves from period-doubling chaos to Lorenz type spi-
ral chaos @8#. Experimentally we observed that accompany-
ing these dynamics changes, the width of the spectral peak
broadens. The broadening of the spectral peak is a result of
the increasing chaotic phase fluctuation associated with the
Lorenz type spiral chaos, which is manifested as the varia-
tion in the time intervals between the successive chaotic laser
intensity pulsations @1,2#. The position of the spectral peak
represents the average pulsation frequency of the chaotic os-
cillation, and varies as experimental conditions such as the
pump laser intensity, gas pressure, and the laser cavity de-
tuning are changed. We have studied the chaotic dynamics of
the laser with a chaotically varying pump by examining how
its dynamics changes and the relationship of its dynamics to
that of the driving signal. With a fixed driving signal
FIG. 1. A typical chaotic dynamics of the laser without chaotic
driving. Pump intensity is 0.8 W/cm 2. Gas pressure is 19 mbar.
Laser cavity tuning is close to the resonance. ~a! Chaotic laser in-
tensity evolution. ~b! Fourier power spectrum calculated from the
laser intensity evolution shown in ~a!.strength, we experimentally observed that the separation be-
tween average pulsation frequency of the driving signal and
that of the chaotic laser intensity plays an important role in
the chaotic dynamics of the laser. When the separation be-
tween the average frequencies is large, under a chaotic pump
the dynamics of the laser becomes more complicated than its
original single mode dynamics. In this case the chaotic in-
tensity evolution of the laser exhibits no obvious relation to
the driving signal. However, when the average frequency
separation is small, within a certain range that solely depends
on the driving signal strength ~with a peak-peak chaotic
modulation of 30% of the average pump intensity, we ob-
served a locking range of about 0.15 MHz!, the average pul-
sation frequency of the laser locks to that of the driving
signal and the chaotic intensity pulsations of the laser be-
come frequency entrained to that of the driving signal.
To demonstrate this frequency entrainment effect, we
show in Fig. 2 a typically observed experimental result. Fig-
ure 2~a! is the time evolution of the chaotic laser intensity,
Fig. 2~b! is the chaotic driving signal measured by inverting
the intensity of the power diffracted by the AOM. We have
made a part of the modulation signal a dc level so as to
compare the laser dynamics with and without chaotic driv-
ing. The original chaotic wave form used to modulate the rf
driving signal of the AOM is a segment of prerecorded cha-
otic intensity evolution of the laser without chaotic driving,
and the dc level is selected such that it is the mean level of
the chaotic signal. However, due to the bandwidth limitation
of the AOM, the real chaotic wave form of the driving
shown in Fig. 2~b! is significantly distorted from the original
wave form. This drawback shows on the other hand that the
observed effect is not dependent on the detailed driving sig-
nal dynamics. Figure 2~a! shows that without the chaotic
driving the intensity dynamics of the laser is a Lorenz spiral
chaos. When the chaotic driving is switched on, the dynam-
ics of the laser deviates from its original form and normally
after a transient that depends on the driving strength and the
average frequency separation, the chaotic intensity pulsations
of the laser become ‘‘synchronized’’ to the chaotic driving
pulses, despite the fact that both the driving signal and the
FIG. 2. Chaotic dynamics of the laser under frequency entrain-
ment. Average pump intensity is 0.85 W/cm 2. Gas pressure is 19
mbar. Laser cavity tuning is almost in resonance. The peak-peak
modulation strength is 20% of the mean pump intensity. ~a! Chaotic
laser intensity evolution. ~b! Evolution of the driving chaotic signal.
57 3651BRIEF REPORTSintrinsic chaotic laser dynamics possess strong chaotic phase
fluctuations. Once the average frequencies are locked,
slightly changing the experimental conditions such as the
pump frequency detuning does not change the average pul-
sation frequency. When the driving signal is switched off,
the chaotic dynamics of the laser returns to its original Lo-
renz spiral chaos.
We have examined in detail the relationship between the
chaotic laser pulsations and the driving signal. While without
the chaotic driving, the successive peak intensity pulse return
map of the laser shows a clear cusp form of structure nor-
mally found for spiral chaos @5#, under the frequency entrain-
ment shown in Fig. 2, this cusp structure smears out. By
measuring the time delay as the time interval between the
peak of the laser intensity pulse to its corresponding driving
pulse, we found that the delay is not fixed but the higher the
driving pulse, the smaller the time delay. Plotting the delay
time return map does not show any relationship between
them either. Obviously, in this state the amplitudes of the
chaotic laser intensity pulsations are independent of those of
the driving signal.
To quantitatively demonstrate that the average chaotic
pulsation frequency of the laser shown in Fig. 2 is locked to
that of the driving signal, we have calculated the instanta-
neous phase evolution associated with the chaotic driving
signal and the chaotic laser intensity evolution based on ana-
lytic signal theory @9#. The average overall slope of the cha-
otic phase evolution gives the average chaotic pulsation fre-
quency. As shown in Fig. 3 these two phase evolutions have
the same average slope. As also evidenced by Fig. 3, the
instantaneous phases vary independently about the average
phase slope. Since the deviation of the instantaneous phase
from the average phase slope is a result of the chaotic phase
modulation due to the chaotic dynamics, this experimental
result shows that under the frequency entrainment, the phase
dynamics of the driven oscillation can still be chaotic and not
related to the driving signal.
We have also experimentally studied the possibility of
frequency entraining the chaotic dynamics of the laser with
other chaotic driving signals; e.g., we have used a period-
doubling chaos wave form as the driving signal to frequency
entrain the Lorenz-like spiral chaos of the laser and vice
versa. We have also used chaotic wave forms calculated
from the Lorenz equations as the driving signal and repeatedthe experiment. In all cases we have observed the same ef-
fect. Our experimental results suggest that this effect is a
generic feature of coupled chaotic oscillations and it is in-
sensitive to the coupled chaotic dynamics.
In conclusion, we have experimentally observed the fre-
quency entrainment of a chaotic oscillation to a chaotic driv-
ing signal. Our experimental results show that although for a
chaotic oscillation there is no unique natural frequency, un-
der coupling it can exhibit an effect similar to the frequency
entrainment of a coupled periodic oscillator. However, in the
chaotic case it is the average chaotic pulsation frequencies
that are locked together. This experimental result suggests
that in a chaotic oscillation, the average chaotic pulsation
frequency plays the same role as that of the natural frequency
of a periodic oscillation. Because under the frequency en-
trainment only the average chaotic pulsation frequency is
constrained and there is no restriction on the phase and am-
plitude variations, and for a chaotic oscillation its phase and
amplitude are intrinsically dynamically unstable, these result
in that, under the frequency entrainment, the dynamics of the
coupled chaotic systems are still chaotic and irrelevant to
each other. Finally, our experimental results with several
forms of chaos demonstrate that the frequency entrainment
effect is a generic effect of coupled chaotic oscillations, and
it is independent of the particular coupled chaotic dynamics.
FIG. 3. Chaotic phase evolution of the laser intensity evolution
and the driving signal. Solid line: phase evolution calculated from
the laser intensity evolution. Dotted line: phase evolution calculated
from the driving signal.@1# Emily F. Stone, Phys. Lett. A 163, 367 ~1992!.
@2# J. D. Farmer, Phys. Rev. Lett. 47, 179 ~1981!.
@3# M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys. Rev.
Lett. 76, 1804 ~1996!.
@4# Tin Win, M. Y. Li, J. T. Malos, N. R. Heckenberg, and C. O.
Weiss, Opt. Commun. 103, 479 ~1993!.
@5# C. O. Weiss, R. Vilaseca, N. B. Abraham, R. Cobala´n, E.
Rolda´n, G. J. de Valca´rcel, J. Pujol, U. Hu¨nber, and D. Y.Tang, Appl. Phys. B: Laser Opt. 61, 223 ~1995!.
@6# E. Rolda´n, G. J. de Valca´rcel, R Vilaseca, R. Cobala´n, V. J.
Martı´nez, and R. Gilmore, Quantum Semiclassic. Opt. 9, R1
~1997!.
@7# H. Haken, Phys. Lett. 53A, 77 ~1975!.
@8# H. Zeghlache and P. Mandel, J. Opt. Soc. Am. B 2, 18 ~1985!.
@9# D. Gabor, J. IEE London 93, 429 ~1946!.


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Experimental evidence of frequency entrainment between coupled chaotic oscillations

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Adelaide Research & Scholarship

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University of Queensland eSpace

Experimental evidence of frequency entrainment between coupled chaotic oscillations

Abstract

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Full text

Available Versions

Adelaide Research &amp; Scholarship

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University of Queensland eSpace

Adelaide Research & Scholarship