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Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback
15 July 1999
Optics Communications 165 Ž1999. 279–292 www.elsevier.comrlocateroptcom
Full length article
Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback E.V. Grigorieva a
a,),1
, H. Haken a , S.A. Kaschenko
b,2
Institute for Theoretical Physics and Synergetics, UniÕersity of Stuttgart, 70550 Stuttgart, Germany b Department of mathematics, YaroslaÕl State UniÕersity, 150000 YaroslaÕl, Russia Received 6 October 1998; received in revised form 10 May 1999; accepted 13 May 1999
Abstract A local dynamics for class B laser systems with optoelectronic and optical time-delayed feedback is considered. In detail, we determine the conditions for Hopf bifurcation and multiple bifurcations. The complicated quasiperiodic dynamics near equilibrium is demonstrated analytically. q 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45.q b; 42.50.Lc; 42.55.Px Keywords: Laser dynamics; Feedback; Bifurcation analysis
1. Introduction A time-delayed feedback control is intensively studied in view of various applications. In laser and optical physics, for example, it has been proposed to suppress relaxation oscillations of ruby lasers w1x, or, oppositely, to produce stable oscillations w2x; to generate low- and high-dimensional chaos w3,4x, or, recently, to stabilize unstable orbits w5,6x. In addition to practical importance, mathematical models of lasers with delayed feedback are certainly attractive from a general point of view as they possess an infinite-dimensional phase space. A theo-
) Corresponding author. E-mail: [email protected] 1 Permanent address: Dept. of Physics, Belarus State University, 220050 Minsk, Belarus. 2 E-mail: [email protected]
retical prediction of developed instabilities in such systems is still an open problem because of considerable difficulties in an analytical description of nonlocal dynamical regimes. One possible way towards this problem is an investigation of multiple bifurcations. The conditions for multiple bifurcations are quite rare in low-dimensional systems, but they typically arise in spatially and time-extended systems. For example, in an Ikeda-like spatially extended optical system w7x such conditions correspond to two purely imaginary pairs of eigenvalues without resonance. However, our investigations have shown that both modes taking part in the dynamics bifurcate supercritically. As a result, a simple dynamical behavior develops in the weakly nonlinear region, namely, a mode-competition phenomenon in which only one mode oscillates periodically while another one is suppressed. That is why a complexity appears relatively far from equilibrium after bifurcations of nonlocal regimes.
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 3 6 - 9
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
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Near equilibria, the conditions for two- Žthree-. dimensional tori creation are fulfilled if one mode bifurcates supercritically but another one does so subcritically w8x. This mathematical requirement looks very hard to be realized in physical systems. Here, we show that it is the case which regularly occurs for lasers with optoelectronic and incoherent optical feedback. Several models of such devices are based on single mode rate equations with a delayed argument w1,2,9–11x du dt dy dt
s Õu Ž y y 1 . y f 1 Ž t ,t . , s q y y y yu y f 2,3 Ž t ,t . ,
Ž 1.
where u and y are proportional to the photon density and carrier inversion, respectively, q is the pumping rate, Õ is the ratio of the photon damping rate in the cavity to the rate of population relaxation, cavity losses are normalized to unity. The feedback function may be approximated as w1,9x f 1 Ž t ,t . s g y Ž t . u Ž t y t .
Ž 2.
in the case of an optoelectronic feedback controlling intracavity loss, or as w2,10x f 2 Ž t ,t . s g u Ž t y t .
Ž 3.
in the case of an optoelectronic feedback controlling pumping rate, or as w11x f 3 Ž t ,t . s g y Ž t . u Ž t y t .
Ž 4.
in the case of an incoherent optical feedback provided by orthogonally polarized reflected light. Here, t is the delay time in the optoelectronic circuit or in the external cavity, g is the coefficient of the feedback. In the case of optoelectronic feedback a positive as well as negative sign is possible but only positive g has a physical meaning for the incoherent optical feedback. The models are normally applied to class B lasers Žincluding semiconductor and CO 2 lasers. which are characterized by a high damping rate of photons in a cavity. Typical dynamical regimes are, as a consequence, relaxation oscillations consisting of sharp radiation pulses. A special nonlocal theory of the corresponding phenomena has been developed in
Refs. w12,13x. It describes, in particular, a hierarchy of isolated periodical solutions of a large amplitude on the basis of pattern complexity on the interval of time delay. Some properties of analogous solutions have been later defined by a perturbation method in Ref. w14x. At the same time, the systems exhibit complicated behavior even in the local vicinity of a stationary state. In fact, quasiperiodic solutions and a corresponding route to chaos have been found numerically as well as experimentally w9,11x in the systems. A similar phenomenon is associated with so-called coherent collapse in semiconductor lasers with external optical feedback w15,16x. Such a dynamic is quite different from that of the well-studied Ikeda system w17x modeling passive optical devices with delayed feedback. This observation motivated us to carry out a local analysis of laser systems Ž1. – Ž4.. In this paper, we first obtain conditions for Hopf bifurcation on different scales of the time delay. Following this hierarchy, we then construct corresponding order parameter equations. They allow us to conclude on the stability of various periodic solutions and on the existence of degenerate points. The next important question being solved concerns these bifurcations of codimension two. Here, we determine conditions for two- and three-dimensional quasiperiodic motion in nonresonant and resonant cases. These conclusions are illustrated numerically. At last, we formulate a parabolic boundary problem of Ginsburg–Landau type as an order parameter equation in the case of a very long time delay. Thus a possibility of high-order complexity near an equilibrium is demonstrated by the analytical investigation.
2. Linear analysis Before starting the local analysis, we note characteristic scales of the system. For class B lasers, Õ ; 10 3 –10 4 is a ‘large’ parameter, while the pumping rate and the feedback coefficient are of the order of unity, normally q ); 1, N g N- 1. The delay time may vary widely from sufficiently small to reasonably large values. However, as balance equations are the result of adiabatic elimination of polarisation dynamics we do not consider a very short t < Õy1 .
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
A very long time delay seems to be principally possible at least for optoelectronic feedback, but a more accurate definition of equations is necessary in the case of a purely optical feedback. As results turn out to be very similar for all these systems, we will refer to the delay function f 2 Ž t,t . only. Also, the positive feedback value will be used everywhere if not especially noted. We consider the local dynamics of the system Ž1.,Ž3. in a small vicinity of the nonzero stationary state y Ž t . s ys s 1, uŽ t . s u s s Ž q y 1.rŽ1 q g .. For our further analysis, it is convenient to introduce a small parameter e s Õy1 r2 instead of large Õ. Also, we put c 0 s q y 1 and c s u s s c 0r 1 q g so that the relaxation frequency of a solitary laser without feedback Žg s 0. becomes v R s ey1 c 0 . Applying a natural scale for the time t 1 s ey1 t and introducing new variables x s u y u s , e z s y y ys we obtain the equations
'
281
(
'
dx d t1 dz d t1
s c 2 z q xz ,
t
ž
s yx y g x t 1 y
e
/
is required to excite the system at delay times shifted by one quarter from the multiple period of the relaxation oscillation of a free laser, namely,
y e Ž c 2 q 1 . z y e xz.
Ž 5. It has already been noted w14x that such a formulation clearly demonstrates common properties of the models Ž1. – Ž4. on the level of a local analysis. Indeed, applying the same change of variables to the models Ž1.,Ž2. and Ž1.,Ž4. one can easily see that the zeroth Ž e 0 . and the first Ž e 1 . approximations almost coincide and only terms of the second order indicate the difference between them. In order parameter equations, it leads, respectively, to a difference only in fifth-order terms. A stability of the stationary state x s 0, z s 0 is determined by the roots l of the characteristic equation for the corresponding linearized problem
l2 q e Ž c 2 q 1 . l q c 2 Ž 1 q g eylt r e . s 0.
Fig. 1. Bifurcation diagram for Õ s10 3 ,q s1.5.
Ž 6.
Following this equation, Fig. 1 presents the bifurcation diagram. It appears that a minimal value of the feedback coefficient q gmin s e q O Ž e 2 . c0
tkse
2p c0
Ž k q 14 . q O Ž e 2 . ,
k s 0,1, . . .
The frequency of such oscillations is slightly shifted from the relaxation frequency
v s c0 y
c 0 gmin 2
q O Ž e 2 . s c0 y e
q 2
qOŽ e 2 . .
When the feedback coefficient increases from the threshold value g ) gmin , two families of bifurcation curves appear in accordance with the frequency becoming far from or close to the relaxation frequency of the free laser. Various instabilities develop beyond this border. In view of the asymptotic analysis it is necessary to distinguish between several regions of parameters leading to different local dynamical properties of the original system: Ø Short time delay t ; e Ži.e. comparable with the period of relaxation oscillations of the solitary laser. and sufficiently high coefficient of feedback N g N- 1. There are critical values tc ,gc for which one or two pairs of roots are purely imagi-
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nary l k s i v k q el0 k ,k s "1," 2 while all others roots have a negative real part l m / k s O Ž1.. Ø Long time delay t ; 1 Žit means the time delay is by ey1 longer than the period of relaxation oscillations. and low coefficient of the feedback g ; e . Then, for every such t ,g an infinite number of the roots given by Ž6. tend to l k s ic q el0 k q . . . , k s "1," 2, . . . , hence, the critical case of infinite dimension is realized with e ™ 0. Ø Within the previous situation, important particular cases occur at the critical values of the time delay tc ; 1 and at the coefficient of feedback gc ; e . For those, one Žor two. pairs of the roots are purely imaginary l k s ic q i ev 0 k ,k s "1," 2 with an accuracy O Ž e 2 . while the other roots a have negative real part of the order of O Ž e .. Ø At very long time delay ; ey1t the critical value of the feedback coefficient tends to the threshold one gmin s e qrc0 q O Ž e 3 .. The characteristic roots may be expressed as l k s ic q i e 2 ŽQ q 2p kty1 . q O Ž e 3 .,k s 0," 1, . . . . Again, we meet the critical case of infinite dimension.
at the critical values of the delay time t s etn" where, respectively,
tq n s
2p n c0
Here we deduce order parameter equations following our hierarchy of time delay scales. As the small parameter e exists in the original system, scales for above critical parameters, time and amplitude scales turn out to be connected with this small parameter. They may be determined by appropriate investigation of the nonlinear problem. In infinite-dimension critical cases, as expected, the procedure results in infinite-dimensional amplitude equations. Finite-dimensional critical cases imply the existence of a local invariant integral manifold of finite dimension in the neighborhood of the stationary state. All solutions with initial conditions from this vicinity approach this manifold in the limit t ™ `. Therefore, the problem to find solutions of infinite dimension Eq. Ž5. is reduced to the investigation of solutions that belong to the manifold which is of two Žfour. dimension for the critical case under consideration. Let us start with short time delay t ; e and sufficiently high feedback level e < g - 1. One pair of eigenvalues is purely imaginary l"s i V "q O Ž e .
p Ž 2 n y 1. c0
(
1qg 1yg
,
Vqs c 0 , Vys c 0'1 y g , n s 1,2, . . .
Ž 7.
As the family of tq n corresponds to the frequency of relaxation oscillations c 0 but the family of ty n to the lower frequency, one can naturally expect different dynamical properties of these modes. Indeed, in the e-neighborhood of the bifurcation point, the local dynamics of system Ž1.,Ž3. is described by the normal form r"
dj dt
s a "j q b "j N j N 2 ,
Ž 8.
where the positive factor r " determines a possible new time scale, the above critical parameters a " depend on deviations from the bifurcation point, and Lyapunov values read as
bqs y 3. Normal form analysis
, ty n s
1
Ž tq g c q 2 i Vq . , 3c 1 y 3g bys Ž ty g c y 2 i Vy . . Ž 3 y 5g . c
Ž 9.
The stable Žunstable. solution of the normal form determines a periodic solution Žof the same stability. of the original system in accordance with
ž xz / s e
1r2
ž
c " y1
iV c
/
j Ž t . eiV
q e 3r2 . . . qc.c.
"
t1
qe
ž
U Ž t ,t 1 . V Ž t ,t 1 .
/
Ž 10 .
where the amplitudes j ,u k ,Õ k depend on ‘slow’ time t s e t 1. Evidently, for t s ety n and g - 1r3 the bifurcation is subcritical; therefore an unstable limit cycle arises in the neighborhood of Re ay- 0 of critical parameters, while for t s etq the bifurcation is n supercritical causing an appearance of the stable limit cycle of the amplitude N x Ns Žye Re aqr Re bq. 1r2 in the vicinity of Re aq) 0 of the critical parameters. This is shown in Fig. 1 with thick and thin lines, respectively. One can also conclude from Eq. Ž9. that the situation is reverse for g - 0, i.e., physically, for a positive feedback. In both cases,
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
however, the stable limit cycle is formed by the mode of a higher frequency. Let us now consider the region of delay times longer than the period of relaxation oscillations, namely, let t ; 1. From the characteristic Eq. Ž6. we can see that critical conditions may be fulfilled only for a sufficiently small coefficient of the feedback g s eg 0 . An infinite number of roots given by Ž6. then tend to "ic with e ™ 0. In this situation, the local dynamics of the system Ž5. is determined by the nonlocal dynamics of the infinite-dimensional quasi-normal form with delayed argument dj dt
s a j q a˜ j Ž t y t . q bj N j N 2 ,
Ž 11 .
values of the feedback coefficient g s eg 0 , if the time delay fulfills
t s tn" s
v 0"
arctan
c2 q 1 2 Ž v 0"q u .
qp Ž2 nyk . ,
Ž 13 . q
where k s Ž0,1. for Žt ,t
y.
, and
(
2
v 0"s yu " 12 g 02 c 2 y Ž 1 q c 2 . .
V 1"s c q e Ž v 0"q u . q O Ž e 2 .
(
ž
c2 q 1 2
,
a˜ s i
g0 c 2
,
bsy
i 6c
qOŽ e 2 . . ,
x s e 1r2 c j Ž t . e iŽ cq eu .t 1 q e . . . qc.c. z i
ž/
Delay Eq. Ž11. being normalized, hence, universal equation, requires a thorough study. We will note only properties which concern physical effects and especially – quasiperiodicity near equilibrium. There evidently exists a set of nonzero solutions j s j n expŽ i fn t .,n s 0,1,2 . . . . Amplitudes of such solutions grow quickly with number n as j n2 ; 4p nrt . This fact clearly indicates the origin of isolated periodic solutions of large amplitude that have been studied in Refs. w12–14x. As this paper is mainly devoted to solutions of small amplitude, below we consider only the stability of the zeroth solution of Eq. Ž11., j s 0, with an accuracy O Ž e . or, equivalently, critical cases of one and two pairs of purely imaginary roots in Eq. Ž5. Žwith an accuracy O Ž e 2 ... One pair of the characteristic roots are purely imaginary with an accuracy O Ž e 2 . while the other roots are imaginary with an accuracy O Ž e . at critical
2
/ Ž 15 .
As before, the frequency Vq 1 tends to the frequency of relaxation oscillations on the bifurcation branches t s tq but on the other branches, t s ty n n , the frequency Vy 1 becomes smaller. In the e 2-neighborhood of the bifurcation point we seek the solutions as series
ž xz / s e ž ci / j Ž t . e 2
Ž 12 .
Ž 14 .
Under these parameters oscillations appear with frequency
s c 0 1 q e 12 yg 0 " g 02 y cy2 Ž 1 q c 2 .
and the parameter u s u Ž e . g wy2prt ,0. completes the value of c ey1 to an integer N multiple 2prt so that u q c ey1 s 2p Nrt . The correspondence between the solutions of the quasi-normal form Ž11. and ones of the system Ž5. is given by
ž /
1
n s 1,2, . . . ,
where a s yi u y
283
i V 1 t1
q e 2 . . . qc.c.
Ž 16 .
with amplitudes j , . . . depending on the slow time t 2 s e 2 t 1. Inserting Ž16. into Ž5. and collecting the terms of the same order, we obtain the normal form r"
dj d t2
s a 1" j q b 1" j N j N 2 .
Ž 17 .
Here, the above critical parameter a 1 depends on deviations from the critical point, the factor r "s 3cwŽ2 q t " Ž c 2 q 1.. 2 q Žt " . 2 Ž v 0 q u . 2 x determines a possible new scale for time, and the Lyapunov values b 1 read as
(
b 1"s .tn" g 02 c 2 y Ž 1 q c 2 . y i Ž 2 q tn" Ž 1 q c 2 . . .
2
Ž 18 .
q y Again, the conditions Re bq 1 - 0 for tn and Re b 1 ) 0 y for tn provide different bifurcations as Fig. 1 illustrates. Thus, along the whole boundary of stability, we observe the change of the bifurcation from the super-
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284
critical type to a subcritical one at the points of codimension-two bifurcation. Specifically, there are two types of such points, the first one corresponds to degenerate points of zeroth Lyapunov values, i.e. gmin . The second type includes points where two modes of nonzero Lyapunov value, stable and unstable, are excited simultaneously. This interesting phenomenon makes dynamics directly near the equilibrium nontrivial.
Codimension-two bifurcations of two pairs of purely imaginary roots occur if the parameters fulfill the condition y tq ˜n , n s 0,1,2, . . . nq 1 s tn s t
A simple approximation for such points is possible at least for relatively short t , 2p Ž n q 1 . c0
, g˜n f
4nq3 2
8 n q 12 n q 5
ž xz / s e ž ci / Ž j e
.
Ž 19 .
For the first point, g˜ 0 f 3r5 the frequency of the stable mode is c 0 q O Ž e . while the frequency of the unstable mode is c 0r2 q O Ž e .. Hence, the strong resonance 1:2 takes place. As n increases only weak resonances are possible. For sufficiently large n, codimension-two bifurcations fall into the region g˜n ; e , t˜n ; 1 with, as it has been shown, V 1"s c q O Ž e .. These cases are still nonresonant ones because the slow time t 2 acting in the normal forms Ž17. is of the order of e 2 . At last, when the delay time t increases up to the order of ey1 , the critical values of the frequencies tend to v s c q O Ž e 2 .. Then delicate resonance phenomena appear and dynamics becomes much more complicated. That has been previously identified with the critical case of infinite dimension and will be studied separately. We start directly with the dynamics in the e 2neighborhood of nonresonant bifurcation points for relatively long time delay t˜n ; 1 as results turn out to be similar for short and long time delays. Note once more that our conclusions are also valid in the e-neighborhood of codimension two bifurcations in quasi-normal form Ž11.. In the region of short time delays t ; e we consider separately only the resonant case as the most special and interesting case.
i Vq 1 t1
y
q h e i V 1 t 1 . q e 2 . . . qc.c.,
Ž 20 . where the amplitudes j ,h , . . . depend on the slow time t 2 s e 2 t 1 , one can find that the local dynamics of the system Ž5. is determined by a nonlocal dynamics of normal form r
4. Codimension two bifurcation
t˜n f e
Taking t˜n ; 1 and introducing the series
r
dj d t2 dh d t2
q 2 2 s aq 1 j q b1 j ŽN j N q 2 N h N . ,
y 2 2 s ay 1 h q b1 h ŽN h N q 2 N j N .
Ž 21 .
where Lyapunov values are given by Ž18. at t s t˜n , y Re bq 1 - 0 and Re b 1 ) 0. A proper investigation of the dynamics of the normal form Ž21. is presented, for example, in Ref. w8x. We extract from this analysis the sector bifurcation diagram and present it in Fig. 2a. In particular, at the parameters y 0 - Re aq 1 - y Ž Re a 1 . r2
there is a stable solution q h s 0,N j Ns Ž yRe aq 1 rRe b 1 .
1r2
which corresponds to a stable limit cycle of the frequency Vq 1 in the original system. If conditions y q y Ž Re aq 1 . r2 - Re a 1 - y Ž Re a 1 .
are valid then there is a stable solution j / 0,h / 0 which corresponds to a two-dimensional torus of the y frequencies Vq 1 , V 1 in the original system. Such a solution may be stable only if the Lyapunov values have opposite signs, i.e. one mode has to bifurcate supercritically but another one does so subcritically. This hard condition is regularly valid in our problem. Moreover, the last solution undergoes secondary y Hopf bifurcation on the line Re aq 1 s yRe a 1 . For this case, the system Ž21. is integrable with the function 2 2 F Ž j ,h . sN j N 2 N h N 2 Ž Re aq 1 q N h N q N j N . Ž 22 .
being constant along solution curves. It opens a possibility for three-dimensional torus formation. To conclude analytically on the stability of the threefrequency quasiperiodic regime, it is necessary to
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
285
Fig. 2. Bifurcation diagram for Ža. normal form Ž21. and Žb. a part of Fig. 1 representing codimension two bifurcation.
take into account higher-order terms in the normal form Ž21.. Rather than giving these somewhat unilluminating formulae here, we present a numerical simulation which indicates very clearly such complicated solutions in the laser model with the feedback controlling the pumping rate. Fig. 2b shows a part of bifurcation diagram near codimension two bifurcation. In this figure, the solid line denotes supercritical and the dotted line subcritical Hopf bifurcation. Parameters chosen for numerical integration of system Ž1. correspond to points
A,B,C,D,E Žpoint D is located between C and E.. The results are demonstrated in Fig. 3a–e, which give the corresponding Poincare´ sections and power spectra of the time series in points A-E, respectively. A simple limit cycle in Fig. 3a is an image of oscillations with frequency Vq 1 , at point A. As g increases an appearance of a two-dimensional torus y of frequencies Vq 1 , V 1 is observed in point B, Fig. 3b. The arising of the third frequency in Fig. 3c Žpoint C. is associated with the Hopf bifurcation of the planar system and, hence, with a three-dimen-
286
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
287
Fig. 4. Poincare´ sections and power spectra for the parameters corresponding to the strong resonance point t s 0.3 g s 0.61Ž a.;0.62Ž b ..
sional torus. As expected, this frequency is sufficiently low. The spectrum, therefore, indicates fre2 y 2 quencies Vq 1 " e v 1 , V 1 " e v 1 and their harmonics. When the system approaches a homoclinic loop Žpoint D., the period of slow amplitude modulation tends to infinity. Fig. 3d demonstrate this tendency. At last, crossing the boundary of the unstable mode creation, the system cannot find itself near the equilibrium. According to Ruelle–Takens scenario, its dynamics becomes essentially nonlocal and chaotic, Fig. 3e. It has been already mentioned that systems Ž1. – Ž4. based on single-mode rate equations have a very similar property up to the terms of the third order. Hence, one can expect two- and three-frequency quasiperiodicity in all such systems. However, the stability of the last complicated regime depends on the fifth-order terms, i.e. on the individual nonlinearity of every system. This, probably, explains the experimental observation of only two-dimensional tori but of no three-dimensional tori in CO 2 laser with delayed feedback controlling intracavity losses w9x.
We end the list of finite-dimensional bifurcation phenomena by the strong resonance 1:2 which takes place at g s gr s 3r5 q O Ž e .,t s tr s e 2prc0 q O Ž e 2 .. In this case leading frequencies of the modes are
vh s v s c 0r2,
vj s 2 vh s c 0 .
With substitutions x s z
ž /
c 4i
0
j Ž t1 . e
'10
i c 0 t1
q
c 2i
0
c0
h Ž t1 . e i 2 t1
'10
q e . . . qc.c.
Ž 23 .
the order parameter equations become ry 1 rq 1
dh d t1 dj d t1
s eay h q b 1 jh ) q b 2 h N j N 2 , s eaq j q b 3h 2 q b4 j N h N 2 q b5 j N j N 2 ,
Ž 24 .
Fig. 3. Poincare´ sections and power spectra for the parameters corresponding to points A,B,C,D,E in Fig. 2b. t s 0.61 g s 0.2Ž a.;0.3Ž b .;0.36Ž c .;0.365Ž d .;0.37Ž e ..
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E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
where coefficients are given by 2
b 1 s i vrc, b 2 s y15i vr Ž 8c . , b 3 s i vrc, b4 s y2 i vr Ž 3c 2 . , b 3 s y3i vr Ž 8c 2 . , 2 q 2 ry 1 s 2 y itr gr c rv , r 1 s 2 q itr gr c r2 v .
It is interesting to emphasize that normal form Ž24. should principally include quadratic as well as cubic terms. ŽIndeed, one cannot neglect h N j N 2 in comparison with jh ) , etc., at any time moment, hence, it is impossible to delete cubic terms by scaling variables.. That is why the dynamics of Eq. Ž24. is quite rich and needs to be separately investigated. In Appendix A, we sketch a way for an analytical research of these equations and give a comparison between numerical solutions of initial equations and ones restored with the normal form. These examples obviously confirm usefulness of Eq. Ž24. for the analysis of mechanisms leading to tori. In particular, there is an attractor h s r 1expŽ i f 1 ., j s r 2 expŽ i f 2 . for which the phase difference Ž f 2 y 2 f 1 . s f t 1 constantly grows in time and the amplitudes r 1 , r 2 are f-periodic. This limit cycle undergoes then secondary Hopf bifurcation so that a two-dimensional torus is formed already in the normal form Ž24.. Such solutions correspond to two- and three-dimensional tori in the original equations near resonance codimension-two bifurcation. Fig. 4 shows two examples of numerical integration of the system. Evidently, mode-locking on a three-dimensional torus, Fig. 4a, as well as very complicated behavior, Fig. 4b, are possible. Contrary to the nonresonant case, a torus destruction results in a nonlocal period doubling cycle.
lems with a long time delay. Recently, similar results have been obtained in connection with the laser problem Ž1.,Ž2. w22,23x. However, as it follows from the present investigation, a correct consideration should take into account the natural large parameter ey2 which strictly determines the scales of other critical parameters of the class-B laser system. It appears that a time delay should be ; ey1 for a space–time representation. Our method also allows us to determine reasonably periodic boundary conditions; such boundary conditions were assumed intuitively in Ref. w22x. Both circumstances – presence of large parameters in the original system and fixation of periodic boundary conditions – lead to distinction between coefficients to the space–time equation obtained here and in Ref. w22x. Quasiperiodic dynamics of the original system may be discussed through bifurcations of travelling waves; this property of the equation was not noted in Ref. w22x. We now demonstrate the main points of such an analysis. Let us consider very long time delay t s ey1t 0 . In this case the critical value of the feedback factor tends to a minimal one, g s e Ž1 q c 2 .rc. Normalizing for convenience the time delay to unity we rewrite the system Ž5. in the form
e 2ty1 0 e 2ty1 0
dx d t3 dz d t3
s c 2 z q xz , s yx y eg 0 x Ž t 3 y 1 . y e Ž c 2 q 1 . z y e xz.
Ž 25 .
where t 3 s trt . For the critical value of g 0 s Ž1 q c 2 .rc the characteristic roots then fulfill an estimation
l k s i v 0 q i2p k q el k1 q e 2l k 2 q O Ž e 3 . , 5. Two-dimensional representation of local dynamics
k s 0," 1, . . . ,
Ž 26 .
where Here we complete the hierarchy of order parameters corresponding to the hierarchy of delay scales. This demonstrates how a transition from finite-dimensional critical cases to infinite-dimensional ones results in infinite-dimensional normalized equation. It has been shown in Refs. w18,21x that the space–time Ginsburg–Landau equation is a quasinormal form for purely temporal dynamical prob-
v 0 s ey2t 0 c q Q ,
v1 s 2 Ž1 q c2 . t 0
y1
,
l k1 s yi v 1 Ž Q q 2p k . , lk 2 s y
v 12 2
2 Ž Q q 2p k . y
q i v 12 Ž Q q 2p k . ,
1 ct 0
Ž Q q 2p k . Ž 27 .
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
and the internal parameter Q g Žy2p ,0x completes the value of ct 0 ey2 to an integer K multiple 2p . With e ™ 0 the critical case of infinite dimension is evidently realized. Under such conditions we have to seek a solution in the form
ž xz / s ž ci / e
` iŽ v 0 y eQ v 1 .t 3
j k Ž t 3 . e i2 p kŽ1y ev 1 .t 3
Ý
Ž 28 .
Inserting these series into the system Ž25. we obtain an infinite system for functions j k djk d t3
s e 3l k 2 j k q c k ,
Ž 29 .
with c k Ž e , j k , j kq1 . . . . taking into account the contribution from the nonlinear terms. We now introduce a time-space variable j Ž t 3 , s . which varies periodically on space, j Ž t 3 , s . s j Ž t 3 , s q 1., and which satisfies the equation
Ej e
E t3
se 3 d
E 2j Es
2
Ej q Ž v 12 q ip .
q Ž a q iQv 12 . j
Es
q b1 j N j N 2 q e b 2 j N j N 2 ,
Ž 30 .
where d s v 12r2,
p s v 12Q q Ž ct 0 .
a s yQ 2v 12r2 y Q Ž ct 0 . b 1 s yi 3c Ž 1 q c 2 .
y1
b 2 s y Ž 1 q 2 i . Ž 9c 2 .
y1
y1
,
,
,
y1
.
It appears that the corresponding infinite system for coefficients of the expansion into eigenfunctions of the periodic boundary problem Ž30. `
j Ž t3 , s . s
one can finally find a parabolic boundary problem
E 2j
Ej E t˜
sd
E s˜
2
Ej q ip
E s˜
q a j q ey1 b 1 j N j N 2
q b2 j N j N 2 q O Ž e . ,
j Ž t˜, s˜. s j Ž t˜, s˜ q 1 .
Ž 33 . Ž 34 .
ksy`
q e . . . qc.c.
e
289
j k Ž t 3 . e i2 p k s
Ý
Ž 31 .
ksy`
coincides Žwith an accuracy of higher order terms. with the system Ž29.. Rescaling additionally the amplitudes, time and comoving space variables with the rules
j ™ ej Ž t˜, s˜. e i e
2
v 12 Q t 3
,
t˜s e 2 t 3 , s˜ s s q e 2v 12 t 3 , Ž 32 .
which plays the role of the quasi-normal form to Eq. Ž25.. The correspondence between solutions to the original system and ones to the normal form Ž33.,Ž34. is given by formulae
ž xz / s e ž ci / e
iŽ v 0 y Q Ž ev 1 y e 2 v 12 .. t 3
=j Ž e 2 t 3 ,t 3 Ž 1 y ev 1 q e 2v 12 . . q c.c.q O Ž e 2 . .
Ž 35 .
Nonlocal steady solutions to this quasi-normal form determine the local dynamics of the original problem. Eq. Ž33. which is of Ginsburg–Landau type may produce, as well-known, a very complicated dynamics including space–time turbulence w24x. One can easily obtain the most simple solutions, traveling waves, corresponding to periodic ones to the original system. To conclude on their stability, however, it is necessary to take into account the terms of the next order that is a consequence of a very strong wave dispersion of the order of ey1 . At the end of our discussion we remark that the space s g w0,1x may be naturally associated with the time delay as well as the number K from the expression ey2 cty1 0 q Q s 2 p K – with a main wave number. In this way periodic boundary conditions Ž34. seem to be very natural for the ‘large space t ’. However, such a transition to a two-dimensional representation is not trivial. It depends on fine peculiarities of an infinite eigenvalue spectrum. For example, there may exist situations leading to special anti-periodic boundary conditions w7,21x which provide inhomogeneous patterns in spatial systems or quasiperiodic motion in systems with a large time delay. It is also important to note a role of the parameters Q and t 0 . Through these parameters the system saves an information on both large parameters Õ and t from the initial problem. That is why despite of ‘large geometrical size t ’, the dynamics of the sys-
290
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
tem may be sensitive to their exact values. As it has been shown w7x, in particular, a two-frequency quasiperiodic attractor Žtorus. is possible for some values of Q resulting in spectacular pattern dynamics.
6. Conclusion We have presented the theory of quasiperiodicity in lasers with delayed optoelectronic feedback. The phenomenon appears due to local nonresonant and resonant interaction of stable and unstable modes. The theory analytically predicts stable two-frequency tori to be a common property of several laser models. Three-dimensional tori have been also demonstrated analytically as well as numerically, that is an additional argument in discussion on an existence of quasiperiodic attractors w19,20x. Break-up of the tori leads to nonlocal chaotic or period-doubling regimes. As a reflection of the infinite-dimensional phase space of the initial problem we have formulated order parameter equations of infinite dimension for sufficiently long time delay. They explain the existence of a set of periodical solutions which results in multistability phenomena in the physical systems. A possibility of high-order complexity near equilibrium has been, in addition, shown by spatio-temporal representation of the local dynamics. It offers lasers with delayed feedback as promising devices for an investigation of spatial pattern formation. We note, however, that the dynamics near equilibria takes a small part of parameter space of the rate equations and coexists often with relaxation oscillations. Therefore, the bifurcation analysis developed here has to be completed by the nonlocal one w12,13x. The obtained results may also illuminate the origin of the coherent collapse in semiconductor lasers in an external cavity. Hence, the construction of order parameter equations seems to be a perspective for the analytical explanation of such an important practical problem.
Acknowledgements This work was supported by the Alexander von Humboldt Foundation.
Appendix A In order to integrate Eq. Ž24. we first specify above critical parameters which are determined by deviations near bifurcation point. Setting g s gr q eg 1 , t s tr q e 2t 1 we then get
ž ž
aqs g 1 y y
a s g1 y
c2
2iv y 1 q gc
2iv
iv
c2 q
1 q gc
2iv
/ /
q t 1 gc c 2 y Ž 1 q c 2 . , y t 1 gc c 2 y Ž 1 q c 2 . .
Introducing variables
h s r 1e i f 1 , j s r 2 e i f 2 , F s f 2 y 2 f 1 one comes to the system
r˙ 1 s e Re A1 r 1 q Ž Re B1cosF y Im B1 sinF . r 1 r 2 y Re B2 r 1 r 22 ,
r˙ 2 s e Re A 2 r 2 q Ž Re B3 cosF q Im B3 sinF . r 12 y Re B4 r 23 y Re B5 r 2 r 12 ,
F˙ s e Ž Im A 2 y 2Im A1 . q Im B3
y Re B3
r 12 r2 r 12 r2
y 2Im B1 r 2 cosF
q 2Re B1 r 2 sinF
y w Im B4 y 2Im B2 x r 22 y Im B5 r 12 ,
Ž A.1 .
with coefficients A1 s ay Ž ry 1 .
y1
B1,2 s b 1,2 Ž ry 1 .
A 2 s aq Ž rq 1 .
,
y1
,
y1
,
B3,4,5 s b 3,4,5 Ž rq 1 .
y1
.
Note peculiarities of the system: first, Lyapunov values again have opposite sings for different modes, Re B1,2 - 0, Re B3,4,5 ) 0; second, only the phase difference F can be determined but every phase f 1,2 cannot. It is our aim now to compare solutions of this system with solutions of the original system given in Fig. 4. With parameters of Eq. Ž5.: Õ s 10 3 , e s 1rÕ , q s 1.5, t s 0.3, g s 0.61Ž a.; 0.62Ž b . bifurcation parameters become gr s 0.6, tr s 0.281;
'
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
291
and Re A1 s y0.4355, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.289
Fig. 5. Time series of the amplitude r 1Ž t1 . Ža., of the variable x Ž t1 . Žb. restored with bifurcation formulae Ž23.,Ž24.,ŽA.1. and of the light intensity Ž uy u s . Žc. obtained from system Ž5.. Parameters correspond to ones given in Fig. 4a.
This little difference does not decrease the significance of the analysis – in natural experiment one can normally fix bifurcation parameters but only estimate small deviations from the last ones. Figs. 5 and 6 present time series of the amplitude r 1Ž t 1 . Ža., of the variable x Ž t 1 . Žb. restored with bifurcation formulae Ž23.,Ž24.,ŽA.1. and of the light intensity Ž u y u s . Žc. obtained from system Ž5.. Fig. 5 connects to Fig. 4a and explains quasiperiodicity through the periodic solution of the normal form, Fig. 5a. This solution then undergoes Hopf bifurcation, Fig. 6a, that means more complicated threefrequency behavior in the original system, Fig. 6c and Fig. 4b. To determine stability of such a regime, again, it is necessary to take into account terms of the higher order. In addition, it would be reasonable to investigate system ŽA.1. at arbitrary parameters, as these equations are normalized, hence, universal ones for reso-
coefficients of normal form ŽA.1. depending only on the bifurcation situation and nonlinearities of the original system become B1 s y0.114 q 0.048i, B2 s y0.381 q 0.162 i, B3 s 0.156 q 0.132 i, B4 s 0.186 q 0.158i, B5 s 0.105 q 0.089i, at last, above critical parameters depending on deviations of the critical point are given, for Ža. and Žb. respectively, as Re A1 s y0.309, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.252 and Re A1 s y0.246, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.305 From numerical simulations of Eq. ŽA.1. we have found that a good agreement can be reached for the slightly shifted parameters, Re A1 s y0.509, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.252
Fig. 6. Time series of the amplitude r 1Ž t1 . Ža., of the variable x Ž t1 . Žb. restored with bifurcation formulae Ž23.,Ž24.,ŽA.1. and of the light intensity Ž uy u s . Žc. obtained from system Ž5.. Parameters correspond to ones given in Fig. 4b.
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E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
nant bifurcation 1:2. We have observed, in particular, complicated chaotic-like regimes.
References w1x H. Statz, G.A.de Mars, D.T. Wilson, C.L. Tang, J. Appl. Phys. 36 Ž1965. 1515. w2x T.L. Paoli, L.E. Ripper, IEEE J. Quant. Electron. QE- 6 Ž1970. 335. w3x K. Ikeda, K. Matsumoto, Physica D 29 Ž1987. 223. w4x M. Wegener, C. Klingshirn, Phys. Rev. A 35 Ž1987. 4247. w5x W. Lu, R.G. Harrison, Optics Comm. 109 Ž1994. 457. w6x L.J. Joneskis, G. Burdge, Tech. Digest on Nonlinear Dynamics in Optical systems, 1990, p. 98rMD22-1-100rMD22-3. w7x E.V. Grigorieva, H. Haken, S.A. Kashchenko, A. Pelster, Physica D Ž1998.. w8x J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, 1983. w9x F.T. Arecchi, G. Giacomelli, A. Lapucci, R. Meucci, Phys. Rev. A 43 Ž1991. 4997.
w10x G. Giacomelli, M. Calzavara, F.T. Arecchi, Optics Comm. 74 Ž1989. 97. w11x J.-L. Chern, K. Otsuka, Optics Comm. 96 Ž1993. 259. w12x E.V. Grigorieva, S.A. Kaschenko, Optics Comm. 102 Ž1993. 183. w13x E.V. Grigorieva, S.A. Kaschenko, Bifur. Chaos 3 Ž1993. 1515. w14x D. Pieroux, T. Erneux, K. Otsuka, Phys. Rev. A 50 Ž1994. 1822. w15x R. Lang, K. Kobayashi, IEEE J. Quant. Elect. QE- 16 Ž1980. 347. w16x J. Mork, B. Tromborg, J. Mark, IEEE J. Quant. Electr. 28 Ž1992. 93. w17x K. Ikeda, H. Daido, O. Akimoto, Phys. Rev. Lett. 45 Ž1980. 709. w18x S.A. Kaschenko, Differ. Uravneniya 25 Ž1989. 1448. w19x H. Haken, Advanced Synergetics, Springer, Berlin, 1983. w20x C. Grebogi, E. Ott, J.A. Yorke, Physica D 15 Ž1985. 354. w21x S.A. Kaschenko, Intern. J. Bifurcation Chaos 6 Ž1996. 1093. w22x G. Giacomelli, A. Politi, Physica D 117 Ž1998. 26. w23x G. Giacomelli, A. Politi, Phys. Rev. Lett. 76 Ž1996. 2687. w24x T.S. Akhromeyeva, S.P. Kurdyumov, G.G. Malinetskii, A.A. Samarskii, Phys. Rep. 176 Ž1989. 189.
Optics Communications 165 Ž1999. 279–292 www.elsevier.comrlocateroptcom
Full length article
Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback E.V. Grigorieva a
a,),1
, H. Haken a , S.A. Kaschenko
b,2
Institute for Theoretical Physics and Synergetics, UniÕersity of Stuttgart, 70550 Stuttgart, Germany b Department of mathematics, YaroslaÕl State UniÕersity, 150000 YaroslaÕl, Russia Received 6 October 1998; received in revised form 10 May 1999; accepted 13 May 1999
Abstract A local dynamics for class B laser systems with optoelectronic and optical time-delayed feedback is considered. In detail, we determine the conditions for Hopf bifurcation and multiple bifurcations. The complicated quasiperiodic dynamics near equilibrium is demonstrated analytically. q 1999 Elsevier Science B.V. All rights reserved. PACS: 05.45.q b; 42.50.Lc; 42.55.Px Keywords: Laser dynamics; Feedback; Bifurcation analysis
1. Introduction A time-delayed feedback control is intensively studied in view of various applications. In laser and optical physics, for example, it has been proposed to suppress relaxation oscillations of ruby lasers w1x, or, oppositely, to produce stable oscillations w2x; to generate low- and high-dimensional chaos w3,4x, or, recently, to stabilize unstable orbits w5,6x. In addition to practical importance, mathematical models of lasers with delayed feedback are certainly attractive from a general point of view as they possess an infinite-dimensional phase space. A theo-
) Corresponding author. E-mail: [email protected] 1 Permanent address: Dept. of Physics, Belarus State University, 220050 Minsk, Belarus. 2 E-mail: [email protected]
retical prediction of developed instabilities in such systems is still an open problem because of considerable difficulties in an analytical description of nonlocal dynamical regimes. One possible way towards this problem is an investigation of multiple bifurcations. The conditions for multiple bifurcations are quite rare in low-dimensional systems, but they typically arise in spatially and time-extended systems. For example, in an Ikeda-like spatially extended optical system w7x such conditions correspond to two purely imaginary pairs of eigenvalues without resonance. However, our investigations have shown that both modes taking part in the dynamics bifurcate supercritically. As a result, a simple dynamical behavior develops in the weakly nonlinear region, namely, a mode-competition phenomenon in which only one mode oscillates periodically while another one is suppressed. That is why a complexity appears relatively far from equilibrium after bifurcations of nonlocal regimes.
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 2 3 6 - 9
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
280
Near equilibria, the conditions for two- Žthree-. dimensional tori creation are fulfilled if one mode bifurcates supercritically but another one does so subcritically w8x. This mathematical requirement looks very hard to be realized in physical systems. Here, we show that it is the case which regularly occurs for lasers with optoelectronic and incoherent optical feedback. Several models of such devices are based on single mode rate equations with a delayed argument w1,2,9–11x du dt dy dt
s Õu Ž y y 1 . y f 1 Ž t ,t . , s q y y y yu y f 2,3 Ž t ,t . ,
Ž 1.
where u and y are proportional to the photon density and carrier inversion, respectively, q is the pumping rate, Õ is the ratio of the photon damping rate in the cavity to the rate of population relaxation, cavity losses are normalized to unity. The feedback function may be approximated as w1,9x f 1 Ž t ,t . s g y Ž t . u Ž t y t .
Ž 2.
in the case of an optoelectronic feedback controlling intracavity loss, or as w2,10x f 2 Ž t ,t . s g u Ž t y t .
Ž 3.
in the case of an optoelectronic feedback controlling pumping rate, or as w11x f 3 Ž t ,t . s g y Ž t . u Ž t y t .
Ž 4.
in the case of an incoherent optical feedback provided by orthogonally polarized reflected light. Here, t is the delay time in the optoelectronic circuit or in the external cavity, g is the coefficient of the feedback. In the case of optoelectronic feedback a positive as well as negative sign is possible but only positive g has a physical meaning for the incoherent optical feedback. The models are normally applied to class B lasers Žincluding semiconductor and CO 2 lasers. which are characterized by a high damping rate of photons in a cavity. Typical dynamical regimes are, as a consequence, relaxation oscillations consisting of sharp radiation pulses. A special nonlocal theory of the corresponding phenomena has been developed in
Refs. w12,13x. It describes, in particular, a hierarchy of isolated periodical solutions of a large amplitude on the basis of pattern complexity on the interval of time delay. Some properties of analogous solutions have been later defined by a perturbation method in Ref. w14x. At the same time, the systems exhibit complicated behavior even in the local vicinity of a stationary state. In fact, quasiperiodic solutions and a corresponding route to chaos have been found numerically as well as experimentally w9,11x in the systems. A similar phenomenon is associated with so-called coherent collapse in semiconductor lasers with external optical feedback w15,16x. Such a dynamic is quite different from that of the well-studied Ikeda system w17x modeling passive optical devices with delayed feedback. This observation motivated us to carry out a local analysis of laser systems Ž1. – Ž4.. In this paper, we first obtain conditions for Hopf bifurcation on different scales of the time delay. Following this hierarchy, we then construct corresponding order parameter equations. They allow us to conclude on the stability of various periodic solutions and on the existence of degenerate points. The next important question being solved concerns these bifurcations of codimension two. Here, we determine conditions for two- and three-dimensional quasiperiodic motion in nonresonant and resonant cases. These conclusions are illustrated numerically. At last, we formulate a parabolic boundary problem of Ginsburg–Landau type as an order parameter equation in the case of a very long time delay. Thus a possibility of high-order complexity near an equilibrium is demonstrated by the analytical investigation.
2. Linear analysis Before starting the local analysis, we note characteristic scales of the system. For class B lasers, Õ ; 10 3 –10 4 is a ‘large’ parameter, while the pumping rate and the feedback coefficient are of the order of unity, normally q ); 1, N g N- 1. The delay time may vary widely from sufficiently small to reasonably large values. However, as balance equations are the result of adiabatic elimination of polarisation dynamics we do not consider a very short t < Õy1 .
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
A very long time delay seems to be principally possible at least for optoelectronic feedback, but a more accurate definition of equations is necessary in the case of a purely optical feedback. As results turn out to be very similar for all these systems, we will refer to the delay function f 2 Ž t,t . only. Also, the positive feedback value will be used everywhere if not especially noted. We consider the local dynamics of the system Ž1.,Ž3. in a small vicinity of the nonzero stationary state y Ž t . s ys s 1, uŽ t . s u s s Ž q y 1.rŽ1 q g .. For our further analysis, it is convenient to introduce a small parameter e s Õy1 r2 instead of large Õ. Also, we put c 0 s q y 1 and c s u s s c 0r 1 q g so that the relaxation frequency of a solitary laser without feedback Žg s 0. becomes v R s ey1 c 0 . Applying a natural scale for the time t 1 s ey1 t and introducing new variables x s u y u s , e z s y y ys we obtain the equations
'
281
(
'
dx d t1 dz d t1
s c 2 z q xz ,
t
ž
s yx y g x t 1 y
e
/
is required to excite the system at delay times shifted by one quarter from the multiple period of the relaxation oscillation of a free laser, namely,
y e Ž c 2 q 1 . z y e xz.
Ž 5. It has already been noted w14x that such a formulation clearly demonstrates common properties of the models Ž1. – Ž4. on the level of a local analysis. Indeed, applying the same change of variables to the models Ž1.,Ž2. and Ž1.,Ž4. one can easily see that the zeroth Ž e 0 . and the first Ž e 1 . approximations almost coincide and only terms of the second order indicate the difference between them. In order parameter equations, it leads, respectively, to a difference only in fifth-order terms. A stability of the stationary state x s 0, z s 0 is determined by the roots l of the characteristic equation for the corresponding linearized problem
l2 q e Ž c 2 q 1 . l q c 2 Ž 1 q g eylt r e . s 0.
Fig. 1. Bifurcation diagram for Õ s10 3 ,q s1.5.
Ž 6.
Following this equation, Fig. 1 presents the bifurcation diagram. It appears that a minimal value of the feedback coefficient q gmin s e q O Ž e 2 . c0
tkse
2p c0
Ž k q 14 . q O Ž e 2 . ,
k s 0,1, . . .
The frequency of such oscillations is slightly shifted from the relaxation frequency
v s c0 y
c 0 gmin 2
q O Ž e 2 . s c0 y e
q 2
qOŽ e 2 . .
When the feedback coefficient increases from the threshold value g ) gmin , two families of bifurcation curves appear in accordance with the frequency becoming far from or close to the relaxation frequency of the free laser. Various instabilities develop beyond this border. In view of the asymptotic analysis it is necessary to distinguish between several regions of parameters leading to different local dynamical properties of the original system: Ø Short time delay t ; e Ži.e. comparable with the period of relaxation oscillations of the solitary laser. and sufficiently high coefficient of feedback N g N- 1. There are critical values tc ,gc for which one or two pairs of roots are purely imagi-
282
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
nary l k s i v k q el0 k ,k s "1," 2 while all others roots have a negative real part l m / k s O Ž1.. Ø Long time delay t ; 1 Žit means the time delay is by ey1 longer than the period of relaxation oscillations. and low coefficient of the feedback g ; e . Then, for every such t ,g an infinite number of the roots given by Ž6. tend to l k s ic q el0 k q . . . , k s "1," 2, . . . , hence, the critical case of infinite dimension is realized with e ™ 0. Ø Within the previous situation, important particular cases occur at the critical values of the time delay tc ; 1 and at the coefficient of feedback gc ; e . For those, one Žor two. pairs of the roots are purely imaginary l k s ic q i ev 0 k ,k s "1," 2 with an accuracy O Ž e 2 . while the other roots a have negative real part of the order of O Ž e .. Ø At very long time delay ; ey1t the critical value of the feedback coefficient tends to the threshold one gmin s e qrc0 q O Ž e 3 .. The characteristic roots may be expressed as l k s ic q i e 2 ŽQ q 2p kty1 . q O Ž e 3 .,k s 0," 1, . . . . Again, we meet the critical case of infinite dimension.
at the critical values of the delay time t s etn" where, respectively,
tq n s
2p n c0
Here we deduce order parameter equations following our hierarchy of time delay scales. As the small parameter e exists in the original system, scales for above critical parameters, time and amplitude scales turn out to be connected with this small parameter. They may be determined by appropriate investigation of the nonlinear problem. In infinite-dimension critical cases, as expected, the procedure results in infinite-dimensional amplitude equations. Finite-dimensional critical cases imply the existence of a local invariant integral manifold of finite dimension in the neighborhood of the stationary state. All solutions with initial conditions from this vicinity approach this manifold in the limit t ™ `. Therefore, the problem to find solutions of infinite dimension Eq. Ž5. is reduced to the investigation of solutions that belong to the manifold which is of two Žfour. dimension for the critical case under consideration. Let us start with short time delay t ; e and sufficiently high feedback level e < g - 1. One pair of eigenvalues is purely imaginary l"s i V "q O Ž e .
p Ž 2 n y 1. c0
(
1qg 1yg
,
Vqs c 0 , Vys c 0'1 y g , n s 1,2, . . .
Ž 7.
As the family of tq n corresponds to the frequency of relaxation oscillations c 0 but the family of ty n to the lower frequency, one can naturally expect different dynamical properties of these modes. Indeed, in the e-neighborhood of the bifurcation point, the local dynamics of system Ž1.,Ž3. is described by the normal form r"
dj dt
s a "j q b "j N j N 2 ,
Ž 8.
where the positive factor r " determines a possible new time scale, the above critical parameters a " depend on deviations from the bifurcation point, and Lyapunov values read as
bqs y 3. Normal form analysis
, ty n s
1
Ž tq g c q 2 i Vq . , 3c 1 y 3g bys Ž ty g c y 2 i Vy . . Ž 3 y 5g . c
Ž 9.
The stable Žunstable. solution of the normal form determines a periodic solution Žof the same stability. of the original system in accordance with
ž xz / s e
1r2
ž
c " y1
iV c
/
j Ž t . eiV
q e 3r2 . . . qc.c.
"
t1
qe
ž
U Ž t ,t 1 . V Ž t ,t 1 .
/
Ž 10 .
where the amplitudes j ,u k ,Õ k depend on ‘slow’ time t s e t 1. Evidently, for t s ety n and g - 1r3 the bifurcation is subcritical; therefore an unstable limit cycle arises in the neighborhood of Re ay- 0 of critical parameters, while for t s etq the bifurcation is n supercritical causing an appearance of the stable limit cycle of the amplitude N x Ns Žye Re aqr Re bq. 1r2 in the vicinity of Re aq) 0 of the critical parameters. This is shown in Fig. 1 with thick and thin lines, respectively. One can also conclude from Eq. Ž9. that the situation is reverse for g - 0, i.e., physically, for a positive feedback. In both cases,
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
however, the stable limit cycle is formed by the mode of a higher frequency. Let us now consider the region of delay times longer than the period of relaxation oscillations, namely, let t ; 1. From the characteristic Eq. Ž6. we can see that critical conditions may be fulfilled only for a sufficiently small coefficient of the feedback g s eg 0 . An infinite number of roots given by Ž6. then tend to "ic with e ™ 0. In this situation, the local dynamics of the system Ž5. is determined by the nonlocal dynamics of the infinite-dimensional quasi-normal form with delayed argument dj dt
s a j q a˜ j Ž t y t . q bj N j N 2 ,
Ž 11 .
values of the feedback coefficient g s eg 0 , if the time delay fulfills
t s tn" s
v 0"
arctan
c2 q 1 2 Ž v 0"q u .
qp Ž2 nyk . ,
Ž 13 . q
where k s Ž0,1. for Žt ,t
y.
, and
(
2
v 0"s yu " 12 g 02 c 2 y Ž 1 q c 2 . .
V 1"s c q e Ž v 0"q u . q O Ž e 2 .
(
ž
c2 q 1 2
,
a˜ s i
g0 c 2
,
bsy
i 6c
qOŽ e 2 . . ,
x s e 1r2 c j Ž t . e iŽ cq eu .t 1 q e . . . qc.c. z i
ž/
Delay Eq. Ž11. being normalized, hence, universal equation, requires a thorough study. We will note only properties which concern physical effects and especially – quasiperiodicity near equilibrium. There evidently exists a set of nonzero solutions j s j n expŽ i fn t .,n s 0,1,2 . . . . Amplitudes of such solutions grow quickly with number n as j n2 ; 4p nrt . This fact clearly indicates the origin of isolated periodic solutions of large amplitude that have been studied in Refs. w12–14x. As this paper is mainly devoted to solutions of small amplitude, below we consider only the stability of the zeroth solution of Eq. Ž11., j s 0, with an accuracy O Ž e . or, equivalently, critical cases of one and two pairs of purely imaginary roots in Eq. Ž5. Žwith an accuracy O Ž e 2 ... One pair of the characteristic roots are purely imaginary with an accuracy O Ž e 2 . while the other roots are imaginary with an accuracy O Ž e . at critical
2
/ Ž 15 .
As before, the frequency Vq 1 tends to the frequency of relaxation oscillations on the bifurcation branches t s tq but on the other branches, t s ty n n , the frequency Vy 1 becomes smaller. In the e 2-neighborhood of the bifurcation point we seek the solutions as series
ž xz / s e ž ci / j Ž t . e 2
Ž 12 .
Ž 14 .
Under these parameters oscillations appear with frequency
s c 0 1 q e 12 yg 0 " g 02 y cy2 Ž 1 q c 2 .
and the parameter u s u Ž e . g wy2prt ,0. completes the value of c ey1 to an integer N multiple 2prt so that u q c ey1 s 2p Nrt . The correspondence between the solutions of the quasi-normal form Ž11. and ones of the system Ž5. is given by
ž /
1
n s 1,2, . . . ,
where a s yi u y
283
i V 1 t1
q e 2 . . . qc.c.
Ž 16 .
with amplitudes j , . . . depending on the slow time t 2 s e 2 t 1. Inserting Ž16. into Ž5. and collecting the terms of the same order, we obtain the normal form r"
dj d t2
s a 1" j q b 1" j N j N 2 .
Ž 17 .
Here, the above critical parameter a 1 depends on deviations from the critical point, the factor r "s 3cwŽ2 q t " Ž c 2 q 1.. 2 q Žt " . 2 Ž v 0 q u . 2 x determines a possible new scale for time, and the Lyapunov values b 1 read as
(
b 1"s .tn" g 02 c 2 y Ž 1 q c 2 . y i Ž 2 q tn" Ž 1 q c 2 . . .
2
Ž 18 .
q y Again, the conditions Re bq 1 - 0 for tn and Re b 1 ) 0 y for tn provide different bifurcations as Fig. 1 illustrates. Thus, along the whole boundary of stability, we observe the change of the bifurcation from the super-
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critical type to a subcritical one at the points of codimension-two bifurcation. Specifically, there are two types of such points, the first one corresponds to degenerate points of zeroth Lyapunov values, i.e. gmin . The second type includes points where two modes of nonzero Lyapunov value, stable and unstable, are excited simultaneously. This interesting phenomenon makes dynamics directly near the equilibrium nontrivial.
Codimension-two bifurcations of two pairs of purely imaginary roots occur if the parameters fulfill the condition y tq ˜n , n s 0,1,2, . . . nq 1 s tn s t
A simple approximation for such points is possible at least for relatively short t , 2p Ž n q 1 . c0
, g˜n f
4nq3 2
8 n q 12 n q 5
ž xz / s e ž ci / Ž j e
.
Ž 19 .
For the first point, g˜ 0 f 3r5 the frequency of the stable mode is c 0 q O Ž e . while the frequency of the unstable mode is c 0r2 q O Ž e .. Hence, the strong resonance 1:2 takes place. As n increases only weak resonances are possible. For sufficiently large n, codimension-two bifurcations fall into the region g˜n ; e , t˜n ; 1 with, as it has been shown, V 1"s c q O Ž e .. These cases are still nonresonant ones because the slow time t 2 acting in the normal forms Ž17. is of the order of e 2 . At last, when the delay time t increases up to the order of ey1 , the critical values of the frequencies tend to v s c q O Ž e 2 .. Then delicate resonance phenomena appear and dynamics becomes much more complicated. That has been previously identified with the critical case of infinite dimension and will be studied separately. We start directly with the dynamics in the e 2neighborhood of nonresonant bifurcation points for relatively long time delay t˜n ; 1 as results turn out to be similar for short and long time delays. Note once more that our conclusions are also valid in the e-neighborhood of codimension two bifurcations in quasi-normal form Ž11.. In the region of short time delays t ; e we consider separately only the resonant case as the most special and interesting case.
i Vq 1 t1
y
q h e i V 1 t 1 . q e 2 . . . qc.c.,
Ž 20 . where the amplitudes j ,h , . . . depend on the slow time t 2 s e 2 t 1 , one can find that the local dynamics of the system Ž5. is determined by a nonlocal dynamics of normal form r
4. Codimension two bifurcation
t˜n f e
Taking t˜n ; 1 and introducing the series
r
dj d t2 dh d t2
q 2 2 s aq 1 j q b1 j ŽN j N q 2 N h N . ,
y 2 2 s ay 1 h q b1 h ŽN h N q 2 N j N .
Ž 21 .
where Lyapunov values are given by Ž18. at t s t˜n , y Re bq 1 - 0 and Re b 1 ) 0. A proper investigation of the dynamics of the normal form Ž21. is presented, for example, in Ref. w8x. We extract from this analysis the sector bifurcation diagram and present it in Fig. 2a. In particular, at the parameters y 0 - Re aq 1 - y Ž Re a 1 . r2
there is a stable solution q h s 0,N j Ns Ž yRe aq 1 rRe b 1 .
1r2
which corresponds to a stable limit cycle of the frequency Vq 1 in the original system. If conditions y q y Ž Re aq 1 . r2 - Re a 1 - y Ž Re a 1 .
are valid then there is a stable solution j / 0,h / 0 which corresponds to a two-dimensional torus of the y frequencies Vq 1 , V 1 in the original system. Such a solution may be stable only if the Lyapunov values have opposite signs, i.e. one mode has to bifurcate supercritically but another one does so subcritically. This hard condition is regularly valid in our problem. Moreover, the last solution undergoes secondary y Hopf bifurcation on the line Re aq 1 s yRe a 1 . For this case, the system Ž21. is integrable with the function 2 2 F Ž j ,h . sN j N 2 N h N 2 Ž Re aq 1 q N h N q N j N . Ž 22 .
being constant along solution curves. It opens a possibility for three-dimensional torus formation. To conclude analytically on the stability of the threefrequency quasiperiodic regime, it is necessary to
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285
Fig. 2. Bifurcation diagram for Ža. normal form Ž21. and Žb. a part of Fig. 1 representing codimension two bifurcation.
take into account higher-order terms in the normal form Ž21.. Rather than giving these somewhat unilluminating formulae here, we present a numerical simulation which indicates very clearly such complicated solutions in the laser model with the feedback controlling the pumping rate. Fig. 2b shows a part of bifurcation diagram near codimension two bifurcation. In this figure, the solid line denotes supercritical and the dotted line subcritical Hopf bifurcation. Parameters chosen for numerical integration of system Ž1. correspond to points
A,B,C,D,E Žpoint D is located between C and E.. The results are demonstrated in Fig. 3a–e, which give the corresponding Poincare´ sections and power spectra of the time series in points A-E, respectively. A simple limit cycle in Fig. 3a is an image of oscillations with frequency Vq 1 , at point A. As g increases an appearance of a two-dimensional torus y of frequencies Vq 1 , V 1 is observed in point B, Fig. 3b. The arising of the third frequency in Fig. 3c Žpoint C. is associated with the Hopf bifurcation of the planar system and, hence, with a three-dimen-
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287
Fig. 4. Poincare´ sections and power spectra for the parameters corresponding to the strong resonance point t s 0.3 g s 0.61Ž a.;0.62Ž b ..
sional torus. As expected, this frequency is sufficiently low. The spectrum, therefore, indicates fre2 y 2 quencies Vq 1 " e v 1 , V 1 " e v 1 and their harmonics. When the system approaches a homoclinic loop Žpoint D., the period of slow amplitude modulation tends to infinity. Fig. 3d demonstrate this tendency. At last, crossing the boundary of the unstable mode creation, the system cannot find itself near the equilibrium. According to Ruelle–Takens scenario, its dynamics becomes essentially nonlocal and chaotic, Fig. 3e. It has been already mentioned that systems Ž1. – Ž4. based on single-mode rate equations have a very similar property up to the terms of the third order. Hence, one can expect two- and three-frequency quasiperiodicity in all such systems. However, the stability of the last complicated regime depends on the fifth-order terms, i.e. on the individual nonlinearity of every system. This, probably, explains the experimental observation of only two-dimensional tori but of no three-dimensional tori in CO 2 laser with delayed feedback controlling intracavity losses w9x.
We end the list of finite-dimensional bifurcation phenomena by the strong resonance 1:2 which takes place at g s gr s 3r5 q O Ž e .,t s tr s e 2prc0 q O Ž e 2 .. In this case leading frequencies of the modes are
vh s v s c 0r2,
vj s 2 vh s c 0 .
With substitutions x s z
ž /
c 4i
0
j Ž t1 . e
'10
i c 0 t1
q
c 2i
0
c0
h Ž t1 . e i 2 t1
'10
q e . . . qc.c.
Ž 23 .
the order parameter equations become ry 1 rq 1
dh d t1 dj d t1
s eay h q b 1 jh ) q b 2 h N j N 2 , s eaq j q b 3h 2 q b4 j N h N 2 q b5 j N j N 2 ,
Ž 24 .
Fig. 3. Poincare´ sections and power spectra for the parameters corresponding to points A,B,C,D,E in Fig. 2b. t s 0.61 g s 0.2Ž a.;0.3Ž b .;0.36Ž c .;0.365Ž d .;0.37Ž e ..
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where coefficients are given by 2
b 1 s i vrc, b 2 s y15i vr Ž 8c . , b 3 s i vrc, b4 s y2 i vr Ž 3c 2 . , b 3 s y3i vr Ž 8c 2 . , 2 q 2 ry 1 s 2 y itr gr c rv , r 1 s 2 q itr gr c r2 v .
It is interesting to emphasize that normal form Ž24. should principally include quadratic as well as cubic terms. ŽIndeed, one cannot neglect h N j N 2 in comparison with jh ) , etc., at any time moment, hence, it is impossible to delete cubic terms by scaling variables.. That is why the dynamics of Eq. Ž24. is quite rich and needs to be separately investigated. In Appendix A, we sketch a way for an analytical research of these equations and give a comparison between numerical solutions of initial equations and ones restored with the normal form. These examples obviously confirm usefulness of Eq. Ž24. for the analysis of mechanisms leading to tori. In particular, there is an attractor h s r 1expŽ i f 1 ., j s r 2 expŽ i f 2 . for which the phase difference Ž f 2 y 2 f 1 . s f t 1 constantly grows in time and the amplitudes r 1 , r 2 are f-periodic. This limit cycle undergoes then secondary Hopf bifurcation so that a two-dimensional torus is formed already in the normal form Ž24.. Such solutions correspond to two- and three-dimensional tori in the original equations near resonance codimension-two bifurcation. Fig. 4 shows two examples of numerical integration of the system. Evidently, mode-locking on a three-dimensional torus, Fig. 4a, as well as very complicated behavior, Fig. 4b, are possible. Contrary to the nonresonant case, a torus destruction results in a nonlocal period doubling cycle.
lems with a long time delay. Recently, similar results have been obtained in connection with the laser problem Ž1.,Ž2. w22,23x. However, as it follows from the present investigation, a correct consideration should take into account the natural large parameter ey2 which strictly determines the scales of other critical parameters of the class-B laser system. It appears that a time delay should be ; ey1 for a space–time representation. Our method also allows us to determine reasonably periodic boundary conditions; such boundary conditions were assumed intuitively in Ref. w22x. Both circumstances – presence of large parameters in the original system and fixation of periodic boundary conditions – lead to distinction between coefficients to the space–time equation obtained here and in Ref. w22x. Quasiperiodic dynamics of the original system may be discussed through bifurcations of travelling waves; this property of the equation was not noted in Ref. w22x. We now demonstrate the main points of such an analysis. Let us consider very long time delay t s ey1t 0 . In this case the critical value of the feedback factor tends to a minimal one, g s e Ž1 q c 2 .rc. Normalizing for convenience the time delay to unity we rewrite the system Ž5. in the form
e 2ty1 0 e 2ty1 0
dx d t3 dz d t3
s c 2 z q xz , s yx y eg 0 x Ž t 3 y 1 . y e Ž c 2 q 1 . z y e xz.
Ž 25 .
where t 3 s trt . For the critical value of g 0 s Ž1 q c 2 .rc the characteristic roots then fulfill an estimation
l k s i v 0 q i2p k q el k1 q e 2l k 2 q O Ž e 3 . , 5. Two-dimensional representation of local dynamics
k s 0," 1, . . . ,
Ž 26 .
where Here we complete the hierarchy of order parameters corresponding to the hierarchy of delay scales. This demonstrates how a transition from finite-dimensional critical cases to infinite-dimensional ones results in infinite-dimensional normalized equation. It has been shown in Refs. w18,21x that the space–time Ginsburg–Landau equation is a quasinormal form for purely temporal dynamical prob-
v 0 s ey2t 0 c q Q ,
v1 s 2 Ž1 q c2 . t 0
y1
,
l k1 s yi v 1 Ž Q q 2p k . , lk 2 s y
v 12 2
2 Ž Q q 2p k . y
q i v 12 Ž Q q 2p k . ,
1 ct 0
Ž Q q 2p k . Ž 27 .
E.V. GrigorieÕa et al.r Optics Communications 165 (1999) 279–292
and the internal parameter Q g Žy2p ,0x completes the value of ct 0 ey2 to an integer K multiple 2p . With e ™ 0 the critical case of infinite dimension is evidently realized. Under such conditions we have to seek a solution in the form
ž xz / s ž ci / e
` iŽ v 0 y eQ v 1 .t 3
j k Ž t 3 . e i2 p kŽ1y ev 1 .t 3
Ý
Ž 28 .
Inserting these series into the system Ž25. we obtain an infinite system for functions j k djk d t3
s e 3l k 2 j k q c k ,
Ž 29 .
with c k Ž e , j k , j kq1 . . . . taking into account the contribution from the nonlinear terms. We now introduce a time-space variable j Ž t 3 , s . which varies periodically on space, j Ž t 3 , s . s j Ž t 3 , s q 1., and which satisfies the equation
Ej e
E t3
se 3 d
E 2j Es
2
Ej q Ž v 12 q ip .
q Ž a q iQv 12 . j
Es
q b1 j N j N 2 q e b 2 j N j N 2 ,
Ž 30 .
where d s v 12r2,
p s v 12Q q Ž ct 0 .
a s yQ 2v 12r2 y Q Ž ct 0 . b 1 s yi 3c Ž 1 q c 2 .
y1
b 2 s y Ž 1 q 2 i . Ž 9c 2 .
y1
y1
,
,
,
y1
.
It appears that the corresponding infinite system for coefficients of the expansion into eigenfunctions of the periodic boundary problem Ž30. `
j Ž t3 , s . s
one can finally find a parabolic boundary problem
E 2j
Ej E t˜
sd
E s˜
2
Ej q ip
E s˜
q a j q ey1 b 1 j N j N 2
q b2 j N j N 2 q O Ž e . ,
j Ž t˜, s˜. s j Ž t˜, s˜ q 1 .
Ž 33 . Ž 34 .
ksy`
q e . . . qc.c.
e
289
j k Ž t 3 . e i2 p k s
Ý
Ž 31 .
ksy`
coincides Žwith an accuracy of higher order terms. with the system Ž29.. Rescaling additionally the amplitudes, time and comoving space variables with the rules
j ™ ej Ž t˜, s˜. e i e
2
v 12 Q t 3
,
t˜s e 2 t 3 , s˜ s s q e 2v 12 t 3 , Ž 32 .
which plays the role of the quasi-normal form to Eq. Ž25.. The correspondence between solutions to the original system and ones to the normal form Ž33.,Ž34. is given by formulae
ž xz / s e ž ci / e
iŽ v 0 y Q Ž ev 1 y e 2 v 12 .. t 3
=j Ž e 2 t 3 ,t 3 Ž 1 y ev 1 q e 2v 12 . . q c.c.q O Ž e 2 . .
Ž 35 .
Nonlocal steady solutions to this quasi-normal form determine the local dynamics of the original problem. Eq. Ž33. which is of Ginsburg–Landau type may produce, as well-known, a very complicated dynamics including space–time turbulence w24x. One can easily obtain the most simple solutions, traveling waves, corresponding to periodic ones to the original system. To conclude on their stability, however, it is necessary to take into account the terms of the next order that is a consequence of a very strong wave dispersion of the order of ey1 . At the end of our discussion we remark that the space s g w0,1x may be naturally associated with the time delay as well as the number K from the expression ey2 cty1 0 q Q s 2 p K – with a main wave number. In this way periodic boundary conditions Ž34. seem to be very natural for the ‘large space t ’. However, such a transition to a two-dimensional representation is not trivial. It depends on fine peculiarities of an infinite eigenvalue spectrum. For example, there may exist situations leading to special anti-periodic boundary conditions w7,21x which provide inhomogeneous patterns in spatial systems or quasiperiodic motion in systems with a large time delay. It is also important to note a role of the parameters Q and t 0 . Through these parameters the system saves an information on both large parameters Õ and t from the initial problem. That is why despite of ‘large geometrical size t ’, the dynamics of the sys-
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tem may be sensitive to their exact values. As it has been shown w7x, in particular, a two-frequency quasiperiodic attractor Žtorus. is possible for some values of Q resulting in spectacular pattern dynamics.
6. Conclusion We have presented the theory of quasiperiodicity in lasers with delayed optoelectronic feedback. The phenomenon appears due to local nonresonant and resonant interaction of stable and unstable modes. The theory analytically predicts stable two-frequency tori to be a common property of several laser models. Three-dimensional tori have been also demonstrated analytically as well as numerically, that is an additional argument in discussion on an existence of quasiperiodic attractors w19,20x. Break-up of the tori leads to nonlocal chaotic or period-doubling regimes. As a reflection of the infinite-dimensional phase space of the initial problem we have formulated order parameter equations of infinite dimension for sufficiently long time delay. They explain the existence of a set of periodical solutions which results in multistability phenomena in the physical systems. A possibility of high-order complexity near equilibrium has been, in addition, shown by spatio-temporal representation of the local dynamics. It offers lasers with delayed feedback as promising devices for an investigation of spatial pattern formation. We note, however, that the dynamics near equilibria takes a small part of parameter space of the rate equations and coexists often with relaxation oscillations. Therefore, the bifurcation analysis developed here has to be completed by the nonlocal one w12,13x. The obtained results may also illuminate the origin of the coherent collapse in semiconductor lasers in an external cavity. Hence, the construction of order parameter equations seems to be a perspective for the analytical explanation of such an important practical problem.
Acknowledgements This work was supported by the Alexander von Humboldt Foundation.
Appendix A In order to integrate Eq. Ž24. we first specify above critical parameters which are determined by deviations near bifurcation point. Setting g s gr q eg 1 , t s tr q e 2t 1 we then get
ž ž
aqs g 1 y y
a s g1 y
c2
2iv y 1 q gc
2iv
iv
c2 q
1 q gc
2iv
/ /
q t 1 gc c 2 y Ž 1 q c 2 . , y t 1 gc c 2 y Ž 1 q c 2 . .
Introducing variables
h s r 1e i f 1 , j s r 2 e i f 2 , F s f 2 y 2 f 1 one comes to the system
r˙ 1 s e Re A1 r 1 q Ž Re B1cosF y Im B1 sinF . r 1 r 2 y Re B2 r 1 r 22 ,
r˙ 2 s e Re A 2 r 2 q Ž Re B3 cosF q Im B3 sinF . r 12 y Re B4 r 23 y Re B5 r 2 r 12 ,
F˙ s e Ž Im A 2 y 2Im A1 . q Im B3
y Re B3
r 12 r2 r 12 r2
y 2Im B1 r 2 cosF
q 2Re B1 r 2 sinF
y w Im B4 y 2Im B2 x r 22 y Im B5 r 12 ,
Ž A.1 .
with coefficients A1 s ay Ž ry 1 .
y1
B1,2 s b 1,2 Ž ry 1 .
A 2 s aq Ž rq 1 .
,
y1
,
y1
,
B3,4,5 s b 3,4,5 Ž rq 1 .
y1
.
Note peculiarities of the system: first, Lyapunov values again have opposite sings for different modes, Re B1,2 - 0, Re B3,4,5 ) 0; second, only the phase difference F can be determined but every phase f 1,2 cannot. It is our aim now to compare solutions of this system with solutions of the original system given in Fig. 4. With parameters of Eq. Ž5.: Õ s 10 3 , e s 1rÕ , q s 1.5, t s 0.3, g s 0.61Ž a.; 0.62Ž b . bifurcation parameters become gr s 0.6, tr s 0.281;
'
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291
and Re A1 s y0.4355, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.289
Fig. 5. Time series of the amplitude r 1Ž t1 . Ža., of the variable x Ž t1 . Žb. restored with bifurcation formulae Ž23.,Ž24.,ŽA.1. and of the light intensity Ž uy u s . Žc. obtained from system Ž5.. Parameters correspond to ones given in Fig. 4a.
This little difference does not decrease the significance of the analysis – in natural experiment one can normally fix bifurcation parameters but only estimate small deviations from the last ones. Figs. 5 and 6 present time series of the amplitude r 1Ž t 1 . Ža., of the variable x Ž t 1 . Žb. restored with bifurcation formulae Ž23.,Ž24.,ŽA.1. and of the light intensity Ž u y u s . Žc. obtained from system Ž5.. Fig. 5 connects to Fig. 4a and explains quasiperiodicity through the periodic solution of the normal form, Fig. 5a. This solution then undergoes Hopf bifurcation, Fig. 6a, that means more complicated threefrequency behavior in the original system, Fig. 6c and Fig. 4b. To determine stability of such a regime, again, it is necessary to take into account terms of the higher order. In addition, it would be reasonable to investigate system ŽA.1. at arbitrary parameters, as these equations are normalized, hence, universal ones for reso-
coefficients of normal form ŽA.1. depending only on the bifurcation situation and nonlinearities of the original system become B1 s y0.114 q 0.048i, B2 s y0.381 q 0.162 i, B3 s 0.156 q 0.132 i, B4 s 0.186 q 0.158i, B5 s 0.105 q 0.089i, at last, above critical parameters depending on deviations of the critical point are given, for Ža. and Žb. respectively, as Re A1 s y0.309, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.252 and Re A1 s y0.246, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.305 From numerical simulations of Eq. ŽA.1. we have found that a good agreement can be reached for the slightly shifted parameters, Re A1 s y0.509, Re A 2 s 0.471,
Ž Im A 2 y 2Im A1 . s 1.252
Fig. 6. Time series of the amplitude r 1Ž t1 . Ža., of the variable x Ž t1 . Žb. restored with bifurcation formulae Ž23.,Ž24.,ŽA.1. and of the light intensity Ž uy u s . Žc. obtained from system Ž5.. Parameters correspond to ones given in Fig. 4b.
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nant bifurcation 1:2. We have observed, in particular, complicated chaotic-like regimes.
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