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Lyapunov analysis of the ruelle-takens route to chaos in an optical retarded differential system
Volume 72, number 1,2
OPTICS COMMUNICATIONS
1 July 1989
LYAPUNOV ANALYSIS OF T H E RUELLE-TAKENS ROUTE TO CHAOS IN AN OPTICAL RETARDED D I F F E R E N T I A L SYSTEM M. LE BERRE, E. RESSAYRE and A. TALLET
Laboratoire de PhotophysiqueMolbculaire. Bat. 213, UniversitbParis-Sud. 91405 Orsa),, l-'rance Received 10 January. 1989
The Lyapunov analysis of low-dimensional instabilities in retarded differential systems predicts both the Hopf bifurcation thresholds and the fundamental frequencies. A demonstration is made in a saturable thin sample ring cavity.
In many fields it is meaningless to neglect the dependence of the future state of a dynamical system on the past. Such a system generally obeys retarded differential difference equations where the past dependence appears through the observable and not the derivative of the observable. This situation is often encountered in biology and economy; it appears in few areas of physics and can be nicely simulated in optical devices, built in such a way that the light transmitted by a resonant material is sent back on the medium with a delay. When the delayed feedback is nonlinear, these systems may exhibit low-dimensional bifurcations and chaos. The Lyapunov analysis sets up a powerful tool for describing both aperiodic [ 1-4 ] and periodic [ 5 ] behaviors of these retarded differential difference systems for which only poor analytical investigations can be performed. The Lyapunov analysis consists in studying the motion in the tangent space to the state vector X and gives rise to both the Lyapunov spectrum {;tj} and the unit eigenvectors {~Xj/I~Xjl }, or Lyapunov vectors, which define local divergent or contracting directions. The Lyapunov analysis was adapted to the retarded differential Mackey-Glass equation by Farmer who focused on chaos, calculated the {;tj} and pointed out the linear increase of the Lyapunov dimension of the chaotic attractor with the delay [ 1 ]. Further studies of chaotic regimes in optical devices confirmed this proportionality law [2] and more importantly exhibited that the Lyapunov dimension is
equal to the delay divided by the memory time of the autocorrelation function of the feedback [3,4]. Here we point out that the Lyapunov vectors are powerfool tools to analyze low dimensional bifurcations, qualifying the nature of the bifurcation and provide the complex eigenvalues associated with the eigendirections. We apply the method to a RueileTakens route to chaos in a plane-wave saturable passive ring cavity. The coupled equations for the cell input field amplitude, E ( t ) , and the net population difference in the cell, ~a(t), are [6]
E(t+d) =Eo + R E ( t ) exp[ ( - o r ~ 2 ) ( 1 --iA) ~( t) ],
de dt - - ((a- 1 ) - IEI 2 1 - e x p ( - a l e )
(1)
al
In addition to the intracavity input field amplitude Eo and to the round-trip time d, other control parameters are the absorption coefficient, al, the linear wave number increment, ½alA, and the reflectivity factor R. In eqs. (1), time is in unit of the atomic radiative lifetime, 7- '. The input field amplitude is assumed to be in coincidence with a mode of the cavity. The loss of stability of the stationary solution (Es, E*, ¢s) of eqs. ( 1 ) via a H o p f bifurcation is quite generic. On the contrary the next bifurcation is very sensitive to the parameters, especially the value of the cavity mode spacing d - ' with respect to
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123
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7 and to the cavity decay rate K. Here we choose
7d= 1,
(2)
in order that only few cavity modes may be excited, and R=0.95,
od=0.1,
(3)
so that k is very small compared to d - ~. In these conditions the limit cycle is generally found to destabilize via a H o p f bifurcation. Nevertheless the quasiperiodicity (QP2) is structurally stable on a large range of the control parameter Eo, only when the initial phase, ½Aod, is near an even multiple of 2n, together with conditions (2) and (3). These properties are discussed elsewhere [7]. Therefore, here for our demonstration ~zlc¢l is chosen to be equal to 4n. The stationary solutions of the intensity Is= IEsI 2 exhibit multistability. Successive H o p f bifurcations [8] appear on the lowest branch. Time signal l ( t ) displays a first H o p f bifurcation near Eo=0.07 and the second one at Eo = 0.11. Quasi-periodic and frequency locked regimes successively follow until the onset of chaos near Eo=0.35, just above the third Hopf bifurcation at Eo=0.34. Any obtained frequency locking appears on an extremely small range of Eo. This is a result of the condition Kd<< 1, which makes the oscillators weakly coupled even in presence of strong nonlinearities. Therefore ifa circle map models our system, it is expected to be subcritical. Before dealing with a nonlinear analysis of the quasiperiodicity, let us briefly recall the method [ 1,5 ] to apply the Lyapunov analysis to eqs. ( ! ): The time t is divided into intervals of duration d. Choosing an integration step, &T, such as N&t=d with N--,oc, a 3N component state vector X is defined,
X , ={ER(nd+&t), E R ( n d + 2&t) ..... ER(nd+NOt), Et( nd+&t), ..., E~(nd+ N6t), ~o(nd+ ~t ) ..... ~o(nd+ N&t ) } , where ER and E~ stand for Re E and Im E, respectively. The motion in the tangent space is characterized by the set {6X~ } which is the nth iterate of the unit tangent set {&Xko}, k = 1, 2, ..., 3N, through the jacobian derived from the flow defined by eqs. ( 1 ). The orthonormal set {6X~ } is generated by a GramSchmidt procedure which provides Lyapunov ex124
1July 1989
ponents arranged in decreasing order. This fortunate property allows the reduction of the 3N dimensional problem to a much lower dimensional one, the required dimension of which is equal to the number of eigenvectors carrying any relevant information about the state of the system and the next bifurcation. Therefore the Lyapunov analysis should give rise to the complex eigenvalues {exp(d2k+iduk} of the jacobian operator for periodic and quasi-periodic regimes. The line which joins the successive first N points with coordinates (nd+kt~t, ~ E R ( n d + k 6 t ) ) . k = l ..... N, deduced from the Lyapunov vector OX,~ shows the time behavior of a small disturbance of the real part of the electric field, Lk(t)=ek" OER (t) along the eigendirection ek during a roundtrip, nd
l l k = e x p ( a k +iOk) ,
(4)
in case of a periodic solution with period T, with
T=l/ul ak = T2k ,
(21=0),
Ok = +-2 n T vk.
(5)
The usefulness of the Lyapunov analysis straightforwardly appears when the phase-space oscillators are weakly coupled. Any L~(t) is nearly time-periodic and displays the imaginary exponent u~. associated with the vector & xk: Therefore building and diagonalizing the Floquet matrix is not required for analyzing the stability of the periodic regime; if the latter looses its stability through a H o p f bifurcation, the Lyapunov analysis provides a linear stability analysis of the QP2 regime, and so on... The time traces Lk(t) associated with the highest
Volume 72, number 1,2
OPTICS COMMUNICATIONS
six Lyapunov exponents are reported in figs. 1-4 for Eo increasing from 0.05 (fixed point) to 0.23 (frequency locking). At first sight several traces display quite nice regular behaviors directly leading to { vk} whereas others look aperiodic as a result of oscillator coupling (see fig. 4 in a case o f frequency locking). The main idea consists in starting the analysis far below the first Hopfbifurcation where oscillators should be uncoupled and then following the variation o f {kk, vk} as the control parameter increases. Below ht, (fig. 1), the Lyapunov functions Lk(t) appear in pairs, associated with complex conjugate eigenvalues, any real part kk being negative. The frequencies vk can be easily read out from the Lk(t); they are almost in coincidence with three adjacent cavity modes, respectively and the oscillators are all the more damped as their frequency departs from the p u m p field frequency: v, = 0 . 0 8 d - ', v2=d-~, v3 -~ 2 d - t with 0 > 2., > 2.2> 2.3. When increasing Eo, ).~ increases to zero, v, also increases while v2 and v3 remain constant. Just above the first H o p f bifurcation threshold, h, (fig. 2), the Lyapunov function L, (t), associated with the Lyapunov vector 8X(t)
ell i / J ! | I i
1
f ! ~ ~=
L2
AAAAAAA/\ ]UV
:U
,/VV '
L3,4
L5,6
Fig. 1. Time traces of the first six Lyapunov functions in decreasing order with respect to the associated Lyapunov exponent. plotted on 100d for Eo=0.05.2~.2 ~- -2.5× 10-2, ).3.4~ -6.5X 10 -2. )-56~ - 9 × 10-2 (precision _~10-3, 1000 iterations).
1 July 1989
L2
AAAAAIAA, /_
r(VVgVVVVVL~4
L
Fig. 2. As in fig. 1, for Eo=0.07. ,;.,=0, 2.~= -2.8× 10-2, 2~.4= -5.6× 10-2, k5.6=-8× 10-~. tangent to the trajectory in the phase space exhibits the period, T, of the limit cycle. The other function, L2(t), with the same period T, acquires a Lyapunov exponent which is sufficiently negative to insure the stability of the periodic motion. Its Floquet multiplier /~2 is real, positive inside the unit circle. The other four Floquet multipliers are also inside the unit circle with angles which satisfy eq. (5). When Eo increases further, the periodic solutions undergo a H o p f bifurcation at h2 (see fig. 3). The scenario for L3(t) and L4(t) is the same as for L,(t) and L2(t) at the previous H o p f bifurcation. The two Floquet multipliers #3 and # 4 = / ~ approach the unit circle. Just above h2, only one of them has a length equal to unity; the other one has a very small length, leading to a reordering of the Lyapunov functions: Now there are two zero Lyapunov exponents, corresponding to Lm(t) and L2(t) associated with the two directions which are tangent to the trajectory on the torus T 2, respectively. The large period deduced from L t (t) is the period associated with the great circle whereas the period equal to the inverse of the frequency spacing is associated with the small circle. The two oscillators are weakly coupled as expected near threshold. Both fundamental frequencies are directly known 125
Volume 72, n u m b e r 1,2
OPTICS COMMUNICATIONS
I July 1989
L1
L2
L2
ill J, llillla ,ill ill
IW
L,=,5
L5.6
Le
Fig. 3. As in fig. 1, for E 0 = 0 . 1 1 . 2 ~ = 2 2 = 0 , 2 4 . ~ = 5 X I 0 -2.
~.~=-8×10
-3.
without the need for decoding the QP2 power spectrum, and shows the usefulness of the method [9]. The functions L3(t) and L6(t) exhibit the same periods as L, (t) and L2(t), respectively. Only L4(t) and L5 (t) are associated with complex conjugate eigenvalues. As Eo increases further, coupling between the QP2 oscillators increases, and frequency lockings appear. Fig. 4 exhibits the behavior of the Lyapunov functions near a frequency locking (P/Q= 1/4) at Eo=0.23. Its comparison with fig. 3 illustrates the role of the coupling between the QP2 oscillators. It is now difficult to attribute a frequency to the eigendirections e3 and e4 and the strong coupling between L3(t) and L4(t) in fig. 4 foresees chaos. In the last two cases the third oscillator is still uncoupled to the QP2 oscillators as shown by the Lyapunov functions and confirmed by the very negative Lyapunov exponent. Note that the "third" oscillator eigenfrequency at Eo = 0.23, v5.6= 3 / d is quite different from the one at Eo=0.11, v~.5~-2/d. About Eo-~0.12 the 126
Fig. 4. As in fig. 1, for L o = 0 . 2 3 . 2 , =22 = 0 , ;.3 = - 10" 2, 24 = 1.5× 10 -2 , 2 ~ . 6 = - 3 . 6 × 1 0 -2.
Lyapunov vectors with contracting lengths have undergone re-ordering. It follows that a third oscillator emerges with a frequency about 3/d for Eo > 0.12 while the oscillator with frequency 2/d disappears (in terms of its Lyapunov exponent). The situation drastically changes near Eo-~0.31, below the third bifurcation threshold, h3. The QP2 frequencies are still distinguishable but the third oscillator with identical Lyapunov exponents, ().s=26), is strongly coupled to the two others. Actually without knowledge of the QP2 frequencies for lower Eo, it would be difficult to interpret L j (t) and L2 (t). This case emphasizes the predictive character of the method. The h3 threshold appears near Eo=0.34, where a third Lyapunov exponent increases to zero. Coupling between all the oscillators makes reading out the third fundamental frequency of the QP~ regime very difficult. Fortunately, this third frequency was spotted from the steady state. Its estimation at h3 agrees with the emergence of a new peak in the
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OPTICS COMMUNICATIONS
power spectrum of the QP3 signal with respect to those just below ha. The QP~ regime does not have a stable structure and chaos occurs about E o = 0.35. Fig, 5 exhibits the three fundamental frequencies as a function of Eo, in dotted line below threshold ( 2 < 0 ) and in full line above ( 2 = 0 ) . One o f them 1/d is constant; it is the intrinsic frequency of the device. The two others result from driving nonlinearities, and are close to cavity modes. Let us point out that this Lyapunov analysis is quite new for quasiperiodic regimes and appears to be a necessary preliminary to a nonlinear analysis like the circle map analysis [ 6 ]. We perform a stroboscopy o f the temporal solution on the torus T 2. We look at points (E,, E,*,, ~ ) defined at time intervals equal to the period d on the small circle; we define E~ = E c +r~ exp(i2rt0~) ,
l July 1989
ot .Ec h
ols
Fig. 6. Imaginary part Y(t) versus real part X(t) of the electric field E(t) during a round-trip on the large circle of the tore T 2. Eo= 0. l I. Numbers 0 to 7 display stroboscopy of the solution at equal intervals, d.
(6)
17
Y
and build the map 0~+ ~= f ( 0 ~ ) from the numerical solution. The origin Ec is chosen in order that the assumption r~ = constant,
x
.D,~St j '
(7) •
may be available; E~ is determined from the phase space trajectory, Y = f ( X ) , where X(t) and Y(t)are the real and imaginary part of E ( t ) , respectively. Fig. 6 shows, for Eo = 0.11, such a trajectory during a single round-trip o f duration about 7d on the large circle; the eight successive points with coordinates (An, Y, ) are marked and a relevant E~ is given which satisfies eq. (7) well enough. Therefore, we expect that 0, obeys the circle map 0,+ ~= 0, + 1 2 - ( K / 2 n ) sin 2rt On.
(8)
Vd 3
0
o.1
o.a Eo
Fig. 5. Variation of the three fundamental frequencies with respect to Eo. h~, h2, h3 are the successiveHopfbifurcations. Dotted line below threshold• Full line above threshold•
t'
.-'"
t"
,)."
O
Fig. 7. Numerical map 0,+ ~=J(O,) for Eo=0.11. Fig. 7 displays the numerical map 0~+ t =jr(0, ) which agrees fairly well with the circle map (8). The thickness of the line is related to the limit o f validity of eq. (7). As Eo increases further, the amplitude of the small loops on fig. 6 increases, so that all points (r,, 0~) do not obey eq. (7). Nevertheless the map, 0,+1 =f(0~), still displays a large density of points along a line which displays a circle map. Therefore we are able to calculate mean values for the winding number £2 and for K from the curves like fig. 7. The estimate for K gives rise to values always much smaller than unity 0.06 ~
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OPTICS COMMUNICATIONS
well with the "numerical experiment" which does not reveal any frequency locking on a large range of Eo. The absence of frequency locking is the result of the small coupling between the two oscillators on the torus 7"2. A stronger coupling might require an increase of the decay rate. This can be achieved by decreasing the mirror reflectivity R or ( a n d ) increasing the absorption length of the saturable m e d i u m . Neverthcless there is no evidence that the QP2 regime would survive for too large variations of R and al. In conclusion a Ruelle-Takens route to chaos has been analyzed in a saturable ring cavity device: the linear stability analysis has provided the successive Hopf bifurcations and the associated f u n d a m e n t a l frequencies. The n o n l i n e a r analysis of quasiperiodicity has shown that the circle map is generic for a small and good ring cavity device containing saturable m e d i u m . This work was supported by the Centre National de la Recherche Scientifique ( G r a n t ATP USA 84) and NATO ( G r a n t No 8 5 / 0 7 3 4 ) .
128
1 July 1989
References [ I ] J.D. Farmer, Physica 4D (1982) 366. [2] M. Le Berre, E. Ressayre, A. Taller and H.M. Gibbs, Phys. Rev. Lett. 56 (1986) 274. [3] B. Dorizzi, B. Grammaticos, M. Le Berrc, Y. Pomeau, E. Ressayrc and A. Taller, Phys. Rev. A35 (1987) 328. [4 ] M. Lc Berre, E. Ressayre,A. Tallet,H.M. Gibbs, D.L. Kaplan and M.H. Rose, Phys. Rev. A35 (1987) 4020. [ 5 ] M. Le Berre, E. Ressayre and A. Tallet, J. Opt. Soc. Am. B5 (1988) 1051. [6] K. Ikeda, Optics Comm. 30 (1979) 257. [ 7 ] M. Le Berre, E. Ressayre and A. Tallet, in preparation. [8] P. Berge, Y. Pomeau, Ch. Vidal, L'ordre dans le chaos (Hermann, 1984) p. 167. [9] See a typical QP2 power spectrum in Proc. Fourth Intern. Conf. on Optical bistability, eds. W. Firth, N. Peyghambariam and A. Tallet (Editions de Physique. Les Ulis, 1988) p. 389. [ 10] M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A30 (1984) 1960.
OPTICS COMMUNICATIONS
1 July 1989
LYAPUNOV ANALYSIS OF T H E RUELLE-TAKENS ROUTE TO CHAOS IN AN OPTICAL RETARDED D I F F E R E N T I A L SYSTEM M. LE BERRE, E. RESSAYRE and A. TALLET
Laboratoire de PhotophysiqueMolbculaire. Bat. 213, UniversitbParis-Sud. 91405 Orsa),, l-'rance Received 10 January. 1989
The Lyapunov analysis of low-dimensional instabilities in retarded differential systems predicts both the Hopf bifurcation thresholds and the fundamental frequencies. A demonstration is made in a saturable thin sample ring cavity.
In many fields it is meaningless to neglect the dependence of the future state of a dynamical system on the past. Such a system generally obeys retarded differential difference equations where the past dependence appears through the observable and not the derivative of the observable. This situation is often encountered in biology and economy; it appears in few areas of physics and can be nicely simulated in optical devices, built in such a way that the light transmitted by a resonant material is sent back on the medium with a delay. When the delayed feedback is nonlinear, these systems may exhibit low-dimensional bifurcations and chaos. The Lyapunov analysis sets up a powerful tool for describing both aperiodic [ 1-4 ] and periodic [ 5 ] behaviors of these retarded differential difference systems for which only poor analytical investigations can be performed. The Lyapunov analysis consists in studying the motion in the tangent space to the state vector X and gives rise to both the Lyapunov spectrum {;tj} and the unit eigenvectors {~Xj/I~Xjl }, or Lyapunov vectors, which define local divergent or contracting directions. The Lyapunov analysis was adapted to the retarded differential Mackey-Glass equation by Farmer who focused on chaos, calculated the {;tj} and pointed out the linear increase of the Lyapunov dimension of the chaotic attractor with the delay [ 1 ]. Further studies of chaotic regimes in optical devices confirmed this proportionality law [2] and more importantly exhibited that the Lyapunov dimension is
equal to the delay divided by the memory time of the autocorrelation function of the feedback [3,4]. Here we point out that the Lyapunov vectors are powerfool tools to analyze low dimensional bifurcations, qualifying the nature of the bifurcation and provide the complex eigenvalues associated with the eigendirections. We apply the method to a RueileTakens route to chaos in a plane-wave saturable passive ring cavity. The coupled equations for the cell input field amplitude, E ( t ) , and the net population difference in the cell, ~a(t), are [6]
E(t+d) =Eo + R E ( t ) exp[ ( - o r ~ 2 ) ( 1 --iA) ~( t) ],
de dt - - ((a- 1 ) - IEI 2 1 - e x p ( - a l e )
(1)
al
In addition to the intracavity input field amplitude Eo and to the round-trip time d, other control parameters are the absorption coefficient, al, the linear wave number increment, ½alA, and the reflectivity factor R. In eqs. (1), time is in unit of the atomic radiative lifetime, 7- '. The input field amplitude is assumed to be in coincidence with a mode of the cavity. The loss of stability of the stationary solution (Es, E*, ¢s) of eqs. ( 1 ) via a H o p f bifurcation is quite generic. On the contrary the next bifurcation is very sensitive to the parameters, especially the value of the cavity mode spacing d - ' with respect to
0 030-4018/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
123
Volume 72, number 1,2
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7 and to the cavity decay rate K. Here we choose
7d= 1,
(2)
in order that only few cavity modes may be excited, and R=0.95,
od=0.1,
(3)
so that k is very small compared to d - ~. In these conditions the limit cycle is generally found to destabilize via a H o p f bifurcation. Nevertheless the quasiperiodicity (QP2) is structurally stable on a large range of the control parameter Eo, only when the initial phase, ½Aod, is near an even multiple of 2n, together with conditions (2) and (3). These properties are discussed elsewhere [7]. Therefore, here for our demonstration ~zlc¢l is chosen to be equal to 4n. The stationary solutions of the intensity Is= IEsI 2 exhibit multistability. Successive H o p f bifurcations [8] appear on the lowest branch. Time signal l ( t ) displays a first H o p f bifurcation near Eo=0.07 and the second one at Eo = 0.11. Quasi-periodic and frequency locked regimes successively follow until the onset of chaos near Eo=0.35, just above the third Hopf bifurcation at Eo=0.34. Any obtained frequency locking appears on an extremely small range of Eo. This is a result of the condition Kd<< 1, which makes the oscillators weakly coupled even in presence of strong nonlinearities. Therefore ifa circle map models our system, it is expected to be subcritical. Before dealing with a nonlinear analysis of the quasiperiodicity, let us briefly recall the method [ 1,5 ] to apply the Lyapunov analysis to eqs. ( ! ): The time t is divided into intervals of duration d. Choosing an integration step, &T, such as N&t=d with N--,oc, a 3N component state vector X is defined,
X , ={ER(nd+&t), E R ( n d + 2&t) ..... ER(nd+NOt), Et( nd+&t), ..., E~(nd+ N6t), ~o(nd+ ~t ) ..... ~o(nd+ N&t ) } , where ER and E~ stand for Re E and Im E, respectively. The motion in the tangent space is characterized by the set {6X~ } which is the nth iterate of the unit tangent set {&Xko}, k = 1, 2, ..., 3N, through the jacobian derived from the flow defined by eqs. ( 1 ). The orthonormal set {6X~ } is generated by a GramSchmidt procedure which provides Lyapunov ex124
1July 1989
ponents arranged in decreasing order. This fortunate property allows the reduction of the 3N dimensional problem to a much lower dimensional one, the required dimension of which is equal to the number of eigenvectors carrying any relevant information about the state of the system and the next bifurcation. Therefore the Lyapunov analysis should give rise to the complex eigenvalues {exp(d2k+iduk} of the jacobian operator for periodic and quasi-periodic regimes. The line which joins the successive first N points with coordinates (nd+kt~t, ~ E R ( n d + k 6 t ) ) . k = l ..... N, deduced from the Lyapunov vector OX,~ shows the time behavior of a small disturbance of the real part of the electric field, Lk(t)=ek" OER (t) along the eigendirection ek during a roundtrip, nd
l l k = e x p ( a k +iOk) ,
(4)
in case of a periodic solution with period T, with
T=l/ul ak = T2k ,
(21=0),
Ok = +-2 n T vk.
(5)
The usefulness of the Lyapunov analysis straightforwardly appears when the phase-space oscillators are weakly coupled. Any L~(t) is nearly time-periodic and displays the imaginary exponent u~. associated with the vector & xk: Therefore building and diagonalizing the Floquet matrix is not required for analyzing the stability of the periodic regime; if the latter looses its stability through a H o p f bifurcation, the Lyapunov analysis provides a linear stability analysis of the QP2 regime, and so on... The time traces Lk(t) associated with the highest
Volume 72, number 1,2
OPTICS COMMUNICATIONS
six Lyapunov exponents are reported in figs. 1-4 for Eo increasing from 0.05 (fixed point) to 0.23 (frequency locking). At first sight several traces display quite nice regular behaviors directly leading to { vk} whereas others look aperiodic as a result of oscillator coupling (see fig. 4 in a case o f frequency locking). The main idea consists in starting the analysis far below the first Hopfbifurcation where oscillators should be uncoupled and then following the variation o f {kk, vk} as the control parameter increases. Below ht, (fig. 1), the Lyapunov functions Lk(t) appear in pairs, associated with complex conjugate eigenvalues, any real part kk being negative. The frequencies vk can be easily read out from the Lk(t); they are almost in coincidence with three adjacent cavity modes, respectively and the oscillators are all the more damped as their frequency departs from the p u m p field frequency: v, = 0 . 0 8 d - ', v2=d-~, v3 -~ 2 d - t with 0 > 2., > 2.2> 2.3. When increasing Eo, ).~ increases to zero, v, also increases while v2 and v3 remain constant. Just above the first H o p f bifurcation threshold, h, (fig. 2), the Lyapunov function L, (t), associated with the Lyapunov vector 8X(t)
ell i / J ! | I i
1
f ! ~ ~=
L2
AAAAAAA/\ ]UV
:U
,/VV '
L3,4
L5,6
Fig. 1. Time traces of the first six Lyapunov functions in decreasing order with respect to the associated Lyapunov exponent. plotted on 100d for Eo=0.05.2~.2 ~- -2.5× 10-2, ).3.4~ -6.5X 10 -2. )-56~ - 9 × 10-2 (precision _~10-3, 1000 iterations).
1 July 1989
L2
AAAAAIAA, /_
r(VVgVVVVVL~4
L
Fig. 2. As in fig. 1, for Eo=0.07. ,;.,=0, 2.~= -2.8× 10-2, 2~.4= -5.6× 10-2, k5.6=-8× 10-~. tangent to the trajectory in the phase space exhibits the period, T, of the limit cycle. The other function, L2(t), with the same period T, acquires a Lyapunov exponent which is sufficiently negative to insure the stability of the periodic motion. Its Floquet multiplier /~2 is real, positive inside the unit circle. The other four Floquet multipliers are also inside the unit circle with angles which satisfy eq. (5). When Eo increases further, the periodic solutions undergo a H o p f bifurcation at h2 (see fig. 3). The scenario for L3(t) and L4(t) is the same as for L,(t) and L2(t) at the previous H o p f bifurcation. The two Floquet multipliers #3 and # 4 = / ~ approach the unit circle. Just above h2, only one of them has a length equal to unity; the other one has a very small length, leading to a reordering of the Lyapunov functions: Now there are two zero Lyapunov exponents, corresponding to Lm(t) and L2(t) associated with the two directions which are tangent to the trajectory on the torus T 2, respectively. The large period deduced from L t (t) is the period associated with the great circle whereas the period equal to the inverse of the frequency spacing is associated with the small circle. The two oscillators are weakly coupled as expected near threshold. Both fundamental frequencies are directly known 125
Volume 72, n u m b e r 1,2
OPTICS COMMUNICATIONS
I July 1989
L1
L2
L2
ill J, llillla ,ill ill
IW
L,=,5
L5.6
Le
Fig. 3. As in fig. 1, for E 0 = 0 . 1 1 . 2 ~ = 2 2 = 0 , 2 4 . ~ = 5 X I 0 -2.
~.~=-8×10
-3.
without the need for decoding the QP2 power spectrum, and shows the usefulness of the method [9]. The functions L3(t) and L6(t) exhibit the same periods as L, (t) and L2(t), respectively. Only L4(t) and L5 (t) are associated with complex conjugate eigenvalues. As Eo increases further, coupling between the QP2 oscillators increases, and frequency lockings appear. Fig. 4 exhibits the behavior of the Lyapunov functions near a frequency locking (P/Q= 1/4) at Eo=0.23. Its comparison with fig. 3 illustrates the role of the coupling between the QP2 oscillators. It is now difficult to attribute a frequency to the eigendirections e3 and e4 and the strong coupling between L3(t) and L4(t) in fig. 4 foresees chaos. In the last two cases the third oscillator is still uncoupled to the QP2 oscillators as shown by the Lyapunov functions and confirmed by the very negative Lyapunov exponent. Note that the "third" oscillator eigenfrequency at Eo = 0.23, v5.6= 3 / d is quite different from the one at Eo=0.11, v~.5~-2/d. About Eo-~0.12 the 126
Fig. 4. As in fig. 1, for L o = 0 . 2 3 . 2 , =22 = 0 , ;.3 = - 10" 2, 24 = 1.5× 10 -2 , 2 ~ . 6 = - 3 . 6 × 1 0 -2.
Lyapunov vectors with contracting lengths have undergone re-ordering. It follows that a third oscillator emerges with a frequency about 3/d for Eo > 0.12 while the oscillator with frequency 2/d disappears (in terms of its Lyapunov exponent). The situation drastically changes near Eo-~0.31, below the third bifurcation threshold, h3. The QP2 frequencies are still distinguishable but the third oscillator with identical Lyapunov exponents, ().s=26), is strongly coupled to the two others. Actually without knowledge of the QP2 frequencies for lower Eo, it would be difficult to interpret L j (t) and L2 (t). This case emphasizes the predictive character of the method. The h3 threshold appears near Eo=0.34, where a third Lyapunov exponent increases to zero. Coupling between all the oscillators makes reading out the third fundamental frequency of the QP~ regime very difficult. Fortunately, this third frequency was spotted from the steady state. Its estimation at h3 agrees with the emergence of a new peak in the
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power spectrum of the QP3 signal with respect to those just below ha. The QP~ regime does not have a stable structure and chaos occurs about E o = 0.35. Fig, 5 exhibits the three fundamental frequencies as a function of Eo, in dotted line below threshold ( 2 < 0 ) and in full line above ( 2 = 0 ) . One o f them 1/d is constant; it is the intrinsic frequency of the device. The two others result from driving nonlinearities, and are close to cavity modes. Let us point out that this Lyapunov analysis is quite new for quasiperiodic regimes and appears to be a necessary preliminary to a nonlinear analysis like the circle map analysis [ 6 ]. We perform a stroboscopy o f the temporal solution on the torus T 2. We look at points (E,, E,*,, ~ ) defined at time intervals equal to the period d on the small circle; we define E~ = E c +r~ exp(i2rt0~) ,
l July 1989
ot .Ec h
ols
Fig. 6. Imaginary part Y(t) versus real part X(t) of the electric field E(t) during a round-trip on the large circle of the tore T 2. Eo= 0. l I. Numbers 0 to 7 display stroboscopy of the solution at equal intervals, d.
(6)
17
Y
and build the map 0~+ ~= f ( 0 ~ ) from the numerical solution. The origin Ec is chosen in order that the assumption r~ = constant,
x
.D,~St j '
(7) •
may be available; E~ is determined from the phase space trajectory, Y = f ( X ) , where X(t) and Y(t)are the real and imaginary part of E ( t ) , respectively. Fig. 6 shows, for Eo = 0.11, such a trajectory during a single round-trip o f duration about 7d on the large circle; the eight successive points with coordinates (An, Y, ) are marked and a relevant E~ is given which satisfies eq. (7) well enough. Therefore, we expect that 0, obeys the circle map 0,+ ~= 0, + 1 2 - ( K / 2 n ) sin 2rt On.
(8)
Vd 3
0
o.1
o.a Eo
Fig. 5. Variation of the three fundamental frequencies with respect to Eo. h~, h2, h3 are the successiveHopfbifurcations. Dotted line below threshold• Full line above threshold•
t'
.-'"
t"
,)."
O
Fig. 7. Numerical map 0,+ ~=J(O,) for Eo=0.11. Fig. 7 displays the numerical map 0~+ t =jr(0, ) which agrees fairly well with the circle map (8). The thickness of the line is related to the limit o f validity of eq. (7). As Eo increases further, the amplitude of the small loops on fig. 6 increases, so that all points (r,, 0~) do not obey eq. (7). Nevertheless the map, 0,+1 =f(0~), still displays a large density of points along a line which displays a circle map. Therefore we are able to calculate mean values for the winding number £2 and for K from the curves like fig. 7. The estimate for K gives rise to values always much smaller than unity 0.06 ~
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well with the "numerical experiment" which does not reveal any frequency locking on a large range of Eo. The absence of frequency locking is the result of the small coupling between the two oscillators on the torus 7"2. A stronger coupling might require an increase of the decay rate. This can be achieved by decreasing the mirror reflectivity R or ( a n d ) increasing the absorption length of the saturable m e d i u m . Neverthcless there is no evidence that the QP2 regime would survive for too large variations of R and al. In conclusion a Ruelle-Takens route to chaos has been analyzed in a saturable ring cavity device: the linear stability analysis has provided the successive Hopf bifurcations and the associated f u n d a m e n t a l frequencies. The n o n l i n e a r analysis of quasiperiodicity has shown that the circle map is generic for a small and good ring cavity device containing saturable m e d i u m . This work was supported by the Centre National de la Recherche Scientifique ( G r a n t ATP USA 84) and NATO ( G r a n t No 8 5 / 0 7 3 4 ) .
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References [ I ] J.D. Farmer, Physica 4D (1982) 366. [2] M. Le Berre, E. Ressayre, A. Taller and H.M. Gibbs, Phys. Rev. Lett. 56 (1986) 274. [3] B. Dorizzi, B. Grammaticos, M. Le Berrc, Y. Pomeau, E. Ressayrc and A. Taller, Phys. Rev. A35 (1987) 328. [4 ] M. Lc Berre, E. Ressayre,A. Tallet,H.M. Gibbs, D.L. Kaplan and M.H. Rose, Phys. Rev. A35 (1987) 4020. [ 5 ] M. Le Berre, E. Ressayre and A. Tallet, J. Opt. Soc. Am. B5 (1988) 1051. [6] K. Ikeda, Optics Comm. 30 (1979) 257. [ 7 ] M. Le Berre, E. Ressayre and A. Tallet, in preparation. [8] P. Berge, Y. Pomeau, Ch. Vidal, L'ordre dans le chaos (Hermann, 1984) p. 167. [9] See a typical QP2 power spectrum in Proc. Fourth Intern. Conf. on Optical bistability, eds. W. Firth, N. Peyghambariam and A. Tallet (Editions de Physique. Les Ulis, 1988) p. 389. [ 10] M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A30 (1984) 1960.