Aug 3, 1982 - C. Pekeris, Sound Propagation in the Ocean [Russian translation], IL, ... An oscillator with acoustical feedback [5] contains an audio frequency amplifier, ... of such an oscillator as a closed loop consisting of an inertia-free non-.
58. 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73, 74.
75. 76.
V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of the Diffraction of Short Waves [in Russian], Nauka, Moscow (1972). O. E. Rydbeck, Trans. Chalmers Univ. Technol., No. 34 (1944). C. Pekeris, Sound Propagation in the Ocean [Russian translation], IL, Moscow (1951), p. 48. J. R. Wait and A. M. Conda, J. Geophys. Res., 6~6, No. 6, 1725 (1961). B. Burgess and T. B. Jones, Radio Electron. Eng., 45, No. 1/2, 47 (1975). P. H. M. Campbell and T. B. Jones, IEEE Conf. Publ. No. 196, Part II, 47 ~1978). I. N. Zabavina, Problemy Difraktsii i Rasprostraneniya Voln., No. 9, 64 {1969). I. N. Zabavina, Problemy Difraktsii i Rasprostraneniya Voln., No. 15, 121 (1977). G. B]~ntegaard and T. R. Larsen, J. Atmos. Terr. Phys., 37, No. 1 43 (1975). A. B. Orlov and A. E. Pronin, Izv, Vyssh. Uchebn. Zaved., Radiofiz., 18, No. 12, 1786 (1975). S. T. Rybachek and ~. M. Gyunninen, Problemy Difraktsii i Rasprostraneniya Voln., No. 6, 115 (1966). I. Tolstoi and K. S. Klei, Acoustics of the Ocean [in Russian], Mir, Moscow (1969). I. Tolstoi (Tolstoy), J. Geophys. Res., 6_4, No. 7, 815 (1959). I. Tolstoi, Quasioptics [in Russian], Mir, Moscow (1966), p. 63. D. B. Keller and D. S. Papadakis (eds.), The Propagation of Waves and Underwater Acoustics [Russian translation], Mir, Moscow (1980). V. S. Buldyrev and A. I. Lanin, Zh. Vychisl. Mat. Mat. Fiz., 6, No. 1, 90 (1966). V. S. Buldyrev and A. I. Lanin, Numerical Methods for the Solution of Problems of Mathematical Physics [in Russian], Nauka, Moscow (1966), p. 131. A. I. Lanin, Zapiski l~auchnykh Seminarov LOMI, 9, 64 (1968). L. Felsen, Quasioptics [Russian translation], Mir, Moscow (1966), p. 11.
COMPLEX
DYNAMICS
FEEDBACK S. P.
OF
WITH
OSCILLATORS
DELAYED
(REVIEW)
UDC 621.373
Kuznetsov
1. CIRCUIT
INTRODUCTION. OF
GENERALIZED
OSCILLATOR
WITH
DELAYED
FEEDBACK
The generalized circuit of many self-oscillating systems of various physical nature contains an amplifier, the output signal of which is partially applied to its input (Fig. la). We will classify such a device as an oscillator with delayed feedback if the time for signal passage through the closed circuit is sufficiently large (more accurately~ if inequality (1) below is satisfied). Concrete examples of such oscillators are shown schematically in Figs. lc-e. In the traveling wave tube oscillator [1, 2] (Fig. lc) amplification of the electromagnetic wave is accomplished due to its interaction with the electron beam. In the interaction region the phase velocity of the wave is close to that of the beam. Feedback is realized either by creating wave reflections at the ends of the interaction region or with a special external circuit. An oscillator with acoustical feedback [5] contains an audio frequency amplifier, to the input of which is connected a microphone M, with output connected to a loudspeaker LS, located at a d i s t a n c e / f r o m the microphone (Fig. lc); here the delay time is 1/Vs, where vs is the speed of sound. A similar system can be realized in the UHF range by using a solid-state acoustical delay line as one of the elements in the feedback circuit. The nonlinear optical system described in [32] consists of a ring-shaped resonator formed by mirrors 1, 2, and semitransparent m i r r o r 3 (Fig. le). A specimen with nonlinear optical properties is located within the resonator. Excitation is produced by an external coherent radiation source. This fact distinguishes this system from those mentioned previously, although the dynamics of the amplitude of the signal circulating through the feedback loop in this system and in a self-supporting oscillator are similar. Saratov State University, Translated from Izvesti}~a Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 25, No, 12, pp, 1410-1428, December, 1982. Original article submitted November 13, 1981; revision submitted August 3, 1982, 996
0033-8443/82/2512-0996507.50
9 1983 Plenum
]?ublishing Corporation
Retarding system /
To load
~
~a
To load
b
o
L
#
c
Output ' 1 ~ ' ~
d
d Fig. 1
f(A
)o_~/~
A
~
9
"
9
"" a~
0
T
Fig. 2
A~
Fig. 3
OtotoT 2T ?T 4T
T
Fig. 4
O s c i l l a t o r s with delayed feedback w e r e f i r s t studied long ago, the b a s i c studies concerning c e r t a i n m e c h a n i c a l and radio s y s t e m s [4, 5, 6, 10] a p p e a r i n g in the 1930-1950 period. However, it is only r e c e n t l y that study of o s c i l l a t o r y phenomena in o s c i l l a t o r s with delayed feedback has become a subject of p r a c t i c a l i m p o r t a n c e . This is true b e c a u s e , firstly, the c l a s s i c a l studies were p e r f o r m e d before the period of the 1960s and 1970s~ in which m o d e r n concepts of the dynamics of distributed s e l f - o s c i l l a t i n g s y s t e m s were developed and, in p a r t i c u l a r , the p r i n c i p l e s of development of s t o c h a s t i c b e h a v i o r in dynamic s y s t e m s were formulated [21-26]. Secondly, beginning in the 1950s a large n u m b e r of new o s c i l l a t o r s with delayed feedback f i r s t a p p e a r e d and t h e i r unique f e a t u r e s b e c a m e m o r e c l e a r l y known. Thirdly, at the p r e s e n t t i m e t h e r e is wide i n t e r e s t in the dynamics of concrete o s c i l l a t o r s of this type. This is shown by the a p p e a r a n c e of a n u m b e r of studies dedicated to a n a l y s i s of nonstationary p r o c e s s e s in quantum, r e l a t i v i s t i c , and t r a d i t i o n a l UHF e l e c t r o n i c s y s t e m s [2, 3, 15-20, 27-32]. As is well known f r o m the l i t e r a t u r e [4-20, 29-32], depending on circuit p a r a m e t e r s and initial conditions, v a r i o u s s e l f - o s c i l l a t i o n modes can develop in o s c i l l a t o r s with delayed f e e d b a c k - quasiharmonic, s e l f - m o d u l a tion, and stochastic. An obvious n e c e s s i t y is the s y s t e m i z a t i o n of existing r e s u l t s , and distinction between g e n e r a l and s p e c i a l ( c h a r a c t e r i s t i c of a given concrete oscillator) f e a t u r e s of behavior. At the p r e s e n t time, however, this p r o b l e m cannot be completely solved. T h e r e f o r e , the goal of the p r e s e n t review is m o r e limited, and consists of p r e s e n t i n g those p r i n c i p l e s governing the d y n a m i c s of o s c i l l a t o r s with delayed feedback which can be dealt with on the b a s i s of a model of such an o s c i l l a t o r as a closed loop consisting of an i n e r t i a - f r e e nonl i n e a r a m p l i f i e r and a delay tine [4, 5, 14]. This e l e m e n t a r y model p e r m i t s a f a r - r e a c h i n g t h e o r e t i c a l analysis, e n c o m p a s s i n g a w i d e r c l a s s of o s c i l l a t o r s than is possible with complex models. However, as a rule the e l e m e n t a r y model can be relied upon only when studying a definite stage of the t r a n s i e n t p r o c e s s f o r a limited class of initial conditions; conclusions as to the c h a r a c t e r of the s t e a d y - s t a t e r e g i m e of oscillation on the basis of this model must be made with g r e a t c a r e . N e v e r t h e l e s s , the i n f o r m a t i o n provided by the e l e m e n t a r y model a p p r o a c h is in i t s e l f quite detailed, and is useful as a s t a r t i n g point for f u r t h e r study. In connection with this, g r e a t attention will be given to evaluation o f f u n d a m e n t a l p r i n c i p l e s and conditions f o r applicability of the e l e m e n t a r y model. T o g e t h e r with a review of ideas and concepts available at p r e s e n t due to the efforts of v a r i o u s authors [4-20, 29-32], the p r e s e n t study contains a n u m b e r of original r e s u l t s . Due to the v a r i e t y of published m a t e r i a l , with few exceptions citations will not be noted in the course of the exposition, but r e s e r v e d to a special s e p a r a t e section. In Sec. 2 and 4.1 m a t e r i a l f r o m a study by the p r e s e n t author and N. S. Ginzburg, published in the collection , R e l a t i v i s t i c High F r e q u e n c y E l e c t r o n i c s , " is used.
997
2.
INERTIAL
AND NONLINEAR
PROPERTIES.
BASIC
OF ELEMENTARY
CIRCUIT
EQUATION MODEL
Let x(t) and y(t) be functions describing the input and output signals of the opened system (Fig. ib). Depending on the concrete nature of the device studied, x(t) and y(t) m a y characterize the change with time of an electrical voltage, mechanical displacement, or some other physical quantity. If a short pulse is applied to the input of the system, then the output signal will have the approximate form shown in Fig. 2. F r o m the figure one can understand the meaning of twotime intervals which play a fundamental role in the question under study: the pulse at the output appears with a delay T relative to the input and has a duration ~. Now let the input signal be an arbitrary function of time x(t). Then the signal at the output at time t1 is determined by the behavior of the function x(t) in the vicinity of the point tI - T, this vicinity having a width in time of the order of magnitude of ~'. Thus, the quantity ~ is the time scale characterizing the nonlocal nature of the coupling between input and output signals. W e note that this quantity and the width of the frequency band passed by the system obey the well-known uncertainty principle, zA~0 ~2~. For the future we will assume fulfilled the condition T>> 9
(1)
for which delay is significant. We will also a s s u m e that the central frequency of the s y s t e m passband~o 0 is g r e a t e r than o r equal to 7 - i in o r d e r of magnitude. In this case, in place of x(t) and y(t.) it is desirable to Introduce the complex amplitudes z(t) and w(t), such that x(t) =Re[z(t)eia~0tl, y(t) =Re[w (t)el~~ If the complex amplitudes of input and output signals slowly v a r y with a time scale ~', then t h e i r values at the times t - T and t are related algebraically. This relationship should maintain its form for a constant phase shift, i.e., upon the replacement z--ze ict, w-~we i~ Therefore, it has the form w(t) = [6(Izl)z],-~,
(2)
where G( Iz [ ) is the gain of the open s y s t e m at frequency ~o0. To obtain the fundamental equation for the e l e m e n t a r y o s c i l l a t o r model with delayed feedback, we note that for the closed loop s y s t e m the complex amplitudes of the input and output signals must be equal: z(t) -=w(t). Then f r o m Eq. (2) we have
z(0 - [(;(tzl ) z]~_~,
(3)
Equation (3) c o r r e s p o n d s to the concept of the oscillator as a model consisting of a closed loop containing an active clement, the i n e r t i a - f r e e amplitude converter, and the delay line. The nonlinear p r o p e r t i e s of the active element are c h a r a c t e r i z e d by a function f(A) =A [G(A) 1, the typical f o r m of which is shown in Fig. 3. (Systems for which the f o r m of f(A) is qualitatively different will not be considered.) We s t r e s s that in Eq. (3) the time scale ~" does not appear, i.e., in the e l e m e n t a r y model the nonlocal nature of the coupling between input and output circuits is not explicit. Consideration of this nonlocality would imply that the active e l e m e n t poss e s s e s i n e r t i a l p r o p e r t i e s c h a r a c t e r i z e d by the time % As the initial condition for Eq. (3) it is n e c e s s a r y to specify the function z(t) over an initial time segment of duration T, i.e., z (t)10,t 17. Thus, e a c h type of a s y m p t o t i c amplitude behavior c o r r e s p o n d s to a complete s e t of s t e a d y - s t a t e r e g i m e s differing in the values of t h e i r phase invariants m. In e a c h such r e g i m e the oscillation phase changes with t i m e by a linear law $ ~ (t) = (2rrm/T)t +const. The c e n t r a l frequency *We will c o n s i d e r s o m e solution of Eq. (107 A(t) and take B(t) =A(t+0(t)). Here 0(t) is s o m e T - p e r i o d i c function, while 10'(t) I
