P

7. V . A . Sokolov and E. E. FradMn, Opt. Spektrosk. , 4_~3, No. 3, 555 (1977). 8. M. Mo Nazarenko, I. I. Savel ' ev , S. S. Skulachenko, A. M. Khromykh, and I. I. Yudin, Kvantovaya

Elektron. , 6, No. 8, 1698 (1979). 9. W . W . Chow, J . B. Hamberme, T. J . Hutchings, V. E. Sanders , M. Sargent, and M. O. Scully, IEEE

J . Quantum Elec t ron . , QE-16~ No. 9, 918 (1980). 10. V . E . Sanders , Opt. Commun. , 29, No. 2, 227 (1979). 11. T . A . Dorschner , I. W. Smith, and H. Statz, Proc . IEEE NAECON '78, IEEE, New York (1978), p. 569. 12. J . J . Krebs , W. G. Maisch, G. A. P r inz , and D. W. F o r e s t e r , IEEE Trans . Magnet ics , MAG-16____,

NO. 5, 1179 (1980). 13. Jo B r e s m a n , H. Cook, and D. Hysobey, J . Inst. Navigation, 2_.44, No. 2, 153 (1977).

CHAOS IN A SELF-EXCITED SYSTEM WITH EXTERNAL

HARMONIC FORCING

A. S. Dmitriev UDC 532.517

A se l f -exc i ted sys t em driven by an externa l harmonic force is invest igated numer ica l ly . It i s es tab l i shed on the bas i s of an analys is of t empora l rea l iza t ions , shi f t - type maps , phase po r t r a i t s , and the Lyapunov cha rac t e r i s t i c number s that the solution behaves chaot ical ly for definite i n t e r - va ls of the p a r a m e t e r s . The m e c h a n i s m s of the t rans i t ion f rom de te rmin is t i c to s tochast ic osci l la t ions and of the r e v e r s e t rans i t ions with var ia t ion of the sys t em p a r a m e t e r s a r e studied.

The d i scovery and invest igat ion of the nonl inear s tochast izat ion of osci l la t ions in d iss ipa t ive s y s t e m s with a smal l number of deg rees of f reedom [1, 2] has s t imulated a t tempts to find e l emen ta ry models that exhibit s tochas t ic behavior . One c l a s s of such mode l s c o m p r i s e s nonautonomous s y s t e m s descr ibed by second- o rde r ord inary different ia l equations or a s y s t em of two f i r s t - o r d e r equations with an ex te rna l harmonic force. I t has been shown [3, 4] that even in the well-known and thoroughly invest igated nonl inear osc i l l a to rs descr ibed by the nonautonomous Duffing equation, s tochas t ic osci l la t ions can occur for definite values of the p a r a m e t e r s . Other examples of nonautonomous s y s t e m s exhibiting s tochast ic behavior a r e the p a r a m e t r i c a l l y dr iven osc i l l a to r [5, 6] and the forced B r u s s e l model or " B r u s s e l a t o r " [7].

F or rad iophys ica l appl icat ions i t is of u tmost impor tance to look for s tochast ic i ty and inves t igate the laws governing the t rans i t ion to it in se l f -exc i ted ( "au toosc i l l a to ry ' ) s y s t e m s under the action of an externa l force . Such s y s t e m s can be used to model the in te rac t ion of se l f -exc i ted osc i l l a to rs , and studying the condi- t ions for the onset of s tochast ic osci l la t ions in them will aid in formulat ing c r i t e r i a for the onset of s tochast ie i ty in coupled se l f -exc i ted osc i l la tory sys tems .

The poss ib i l i t i e s fo r the onset of chaos in the e l emen ta ry model of a harmonica l ly dr iven se l f -exc i ted osc i l l a to r desc r ibed by the nonautonomous Van der 1)ol equation

X - - ~., ( I ~ X2) X + X = B cos (ol (1)

a re d i scussed theore t ica l ly in [8, 9]. In pa r t i cu l a r , two potential causes of the onset of chaos a re noted: 1) the poss ib le inception of a homoclinic s t ruc tu re for smal l e and B; 2) the occur rence of s t ra t i f ica t ion of the r ing - in to - r ing mapping under n e a r - r e l a x a t i o n conditions. Weak s tochas t ic i ty has been observed in numer i ca l solutions of (1) for i n t e rmed ia t e values of e [10].

In the p r e sen t a r t i c l e we inves t iga te the s tochast ic r e g i m e s and m e c h a n i s m s of the t rans i t ion to chaos in a ha rmonica l ly dr iven se l f -exc i ted s y s t e m descr ibed by the equation

)i - - ~(1 - -x2)X q- X 3--~B cos ~,,t. (2)

The cubic r eac t ive nonl inear i ty introduced in Eq. (2) is used in a number of c a se s to approx imate the c h a r a c t e r i s t i c s of nonl inear inductances [17].

Inst i tute of Radio Engineer ing and E lec t ron ic s , Academy of Sciences of the USSR. Trans la t ed f rom Izves t iya Vysshikh Uchebnykh Zavedenii , Radiofizika, VoL 26, No. 9, pp. 1081-1086, September , 1983. Original a r t ic le submit ted Decem ber 23, 1982.

0033-8443/83/2609-0795507.50 �9 1984 Plenum Publishing Corpora t ion 795

~ 8=4,20

G =4,~0

0,2I ........

.2

Fig. 1 Fig. 2

To analyze the type of osci l la tory motions in the system, we use tempora l real izat ions of the solution, phase por t ra i t s , shif t- type maps acting in t ime in tervals that are mult iples of the period of the e~ternal force , and the maximum Lyapunov charac te r i s t i c number (LCN). The set of these cha rac t e r i s t i c s affords a reasonably complete descr ipt ion of the p roces se s in the system. We have calculated the real izat ions by the four th -order Runge--Kutta method. The computing step was chosen on the basis of ensuring a re la t ive e r r o r of 10-4-10 -5. The rea l par t of the maximum LCN was computed according to the scheme proposed in [11].

Stochastic Regimes in the System. A numer ica l study of the following sys tem, which is equivalent 1. to (2),

s = Y - - ~ (X 'V3 - - X ) ,

~ = --X ~ + t3 cos ~ot, (3)

indicates the p resence of fa i r ly broad domains of the p a r am e te r s ~o, e ,B in which stochast ic r eg imes exist . As an example we consider the evolution of the behavior of the system as a function of the amplitude B of the driving force for f ixed{w, ~} = ~2.35; 0.818}. F o r small values of B, se l f -exci ted oscil lat ions occur in the system. Then, for B ~ 0.5, competit ion begins to take place between oscil lat ions having two per iods, one of which is the f ree- running period and the other is the per iod of the external force. With a fur ther inc rease in B there are initially narrow zones of synchronizat ion of the sel f -exci ted oscil lat ions with oscil lat ions having the period of subharmonics of the external force , al ternating with bands of chaotic oscil lations, and then there is a broad zone of variat ion of B wherein stochast ic oscillations are observed. F o r large values of B t h e regime of synchronization at the fundamental tone occurs in the system. The t ransi t ion f rom chaos to synchronized oscil lat ions with increasing B begins with the inception of t ra ins of different lengths, where the period of the oscil lat ions within the t ra ins is charac te r i s t i c of the external force. The lengths of the t ra ins gradually increase until finally, at B = 4.25, the p rocess becomes synchronized with the external force (Fig. 1).

The stochastic oscillations cor respond to the domain of posit ive values (2.20 -- B ~ 4.25) of the rea l par t of the maximum LCN (Fig. 2). In the in terva l 1.50 -< B - 2.20 the LCN changes sign severa l t imes as a resul t of al ternation between the stochast ic and regular regimes. A s imi la r pat tern of changing types of oscil lat ions with increasing B is also observed for other fixed values of ~ and e. Typical of this situation is the p resence of severa l t ransi t ions f rom regular to chaotic oscil lat ions and the r ev e r s e t ransi t ions with a monotonic in- c r ea se in the amplitude of the external force.

2. Normal and Anomalous Intermit tency in Trans i t ion f rom Fundamental-Tone SYnChronization to Chaos. The al ternat ion of " laminar" oscillations and "beats" in the t ransi t ion f rom synchronized oscillations to chaos �9 evinces the in termi t tent mechanism of this transit ion.

We denote X n m X(t 0 + nT), where t o is the initial t ime and T is the period of the external force. F igure 3a shows the map Xn+l = f(X n) for the vec tor of parameters~co, ~, ]3} ={2.17, 0.818, 2.70~, for which in termi t tency sets in. The dots of the map are concentrated mainly along a curve whose cent ra l par t is a lmost tangent to the b i sec tor Xn+ ~ = X n. Maps of this type associated with the in termi t tent mechanism of the t ransi t ion to chaos have been observed in the Lorenz model [12, 13], the logistic sys tem [14], and coupled osci l la tors [15].

F or {~, 6} = ~2.35; 0.818 } the t ransi t ion f rom chaos to synchronized oscil lat ions through in te rmi t tency takes place for B = 4.25. The form of the map Xn+ 1 = f(Xn) nea r this t ransi t ion point (Fig. 3b) differs sharply

796

X..4, t

g" 0,818/

e P s" $'4"~

-2 0 X n a

~ = 2,17 5 = 2 3 ~=2,55 13=4.2

2 ": 5

0 2 X~ b

Fig. 3

= 2.55 B = 4,2

:. ~" ~'. d

C

from the preceding case. After severa l chaotic motions, the map point enters a crowding zone, which is s imi lar in form to an acute nngle. The acute angles lie on the b isector Xn+ l = X n. Impinging on the upper side of the angle, the map point rapidly enters the immediate vicinity of the vertex, where it is held through severa l i terat ions, the number of which differs for each entry into the crowding zone. The res idency of the point in the vicinity of the ver tex terminates with a t ransi t ion to the lower branch and depar ture from the crowding zone.

The map Yn+ t = ~P (Yn) for the same values of co and ~ (Fig. 3c) appears more like the maps correspond- ing to "normal" intermittency [12-15]. Here the iteration process enters a "channel" between the bisector

and the map curve at the lower left and moves slowly along it. However, in contrast with normal intermittency, "beats" occur not as a result of departure of the curve from the bisector, but because of the map curve inter-

secting the bisector and the map point moving to the lower branch. Moreover, as B is increased the point of

intersection itself becomes a stable fixed point.

Thus, an analysis of the transition to stoehastieity in the investigated system shows that the local form

of the one-dimensional map corresponding to intermittency is not exhausted by the situation obtained in [12-15]. There are at least two other types of one-dimensional maps corresponding to intermittency. A distinctive feature of both of these types is ambiguity in the vicinity of the intersect ion of the map with the bisector.

The sigrAficance of these maps becomes c lear if we regard them as project ions of a mapping set and bisector f rom four-dimensional space {Xn, Yn, Xn+ 1, Yn+ I} onto the respect ive planes Xn, Xn+ 1 and Yn,

u 1. In the case of intermit tency, the "map curve t' and the bisector in four-dimensional space do pass close by one another, but are not situated in the same plane in the general case, so that new types of one-dimensional maps are induced.

3. Transi t ion from Synchronization at Subharmonics of the External Fo rce to Chaos through In ter - m___.ittency~ After the t ransi t ion f rom harmonic oscillations to chaos for ~ , e, B} = {2.35, 0.818, 4.25) the behavior of the sys tem remains chaotic as B is decreased to 2.20. Then the chaotic oscillations are super- seded by synchronized oscillations having a period 3T, the domain of which extends down to B = 2.10. The domain of synchronized oscillations is followed by a narrow chaos s t r ip and then a zone of synchronized osci l la- tions with a period 7T, another chaos strip, and thereaf ter a zone of synchronized oscillations with a period 13T, etc.

An analysis of time plots and shift-type maps Xn+N = ~(Xn), where N is the subharmonic order , for

values of the components of the vector of pa rame te r s close to the cr i t ica l points (i. e., the points of transit ion between chaotic and determinis t ic oscillations), shows that the mechanism of t ransi t ion to stochast iei ty is invariant for all subharmonic synchronization regimes and represents a special variant of transit ion through intermit ,ency. Near these points, synchronized oscillations with periods 3T, 7T, 13T . . . . are interrupted from time to t ime by "bea ts , " which cause information about the preceding par t of the t r a jec to ry to be lost.

4. Hys teres i s of the Charac ter i s t ic Number. The g raph of the real par t of the ma~dmum LCN as a func- tion of the amplitude of the external force in Fig. 2 was obtained for adiabatic variat ion of the amplitude B. In this case the positions of the c r i t ica l points are pract ica l ly independent of whether these calculations are ca r r i ed out with B decreasing or increasing, i . e . , hys te res i s does not e~dst here. This means that for {co, e} = {2.35, 0.818} there cor responds to each value of B ~ [1.0, 4.5] a unique attracting set in phase space. Depend- ing on the value of B, this set either cor responds to regular oscillations with periods T, 3T, 7T, 1 3 T , . . , or is an attracting set with unstable t ra jec tor ies and cor responds to chaotic oscillations.

797

P,e

0

- o,s!o ! ,,I,~ . ~ _ ~ 1 _ ~ ~

-2 0 X,,.

Fig. 4 Fig. 5

For other values of the parameters , more than one attracting set can correspond to the same set of param- eters , inducing hysteresis effects in the system. This situation ar ises , e.g. , when{~, ~} =~2.35, 0.20}. For these values of co and ~ there is a broad domain of values of B~ [1.30, 2.15] for which two attracting sets exist simultaneously. The upper branch of the graph of the real part of the maximum LCN (Fig. 4) is obtained by calculating the LCN as B increases adiabatically from small to large values. For B ~[1.30, 2.15] it lies above zero and corresponds to chaotic oscillations. The lower branch is obtained by calculating the LCN as B decreases adiabatically. For the same interval of values of B it lies below zero and corresponds to oscilla- tions synchronized at the fundamental tone. Outside this interval the attracting set is unique.

The transitions from stochasticity to synchronized oscillations and from synchronized to stochastic oscillations tend to be rigid. An analysis of a realization of the process at the point of transition from synchronized oscillations to chaos (]3 = 1.36) shows that for this value of the parameter and the appropriate initial conditions the oscillations are still close to being synchronized for a certain period of time, but instability causes them to terminate in chaos. Figure 5 shows a shift-type map for the same values of the parameters and initial conditions. The unstable synchronized oscillations in ]Fig. 5 correspond to a compact cluster of points merging into a single point in the upper right-hand corner. After termination of the synchronized oscillations, the motion of the map point is chaotic, and over a fairly long portion of the t rajectory the synchronized regime is not restored.

For B > 2.15, a strange at tractor does not e~ist in the system. The real part of the maximum LCN changes from ~ 0.2 to ~ --0.3 in transition through this point from left to right.

In conclusion, we note that the presence of hysteresis effects in the investigated system for definite values of the parameters shows up even in the stability analysis of periodic oscillations within the framework of Floquet's theory [16]. However, within the framework of that analysis, which is valid for B << 1, it is assumed that the loss of stability on the part of the periodic oscillations results in the inception of quasiperiodic m oti on.

L I T E R A T U R E C I T E D

1. E.N. Lorenz, J. Atm. Sci., 20, No. I, 130 (1963)o 2. M.I. Rabinovich, Usp. Fiz. Nauk, 125, No. 1, 123 (1978). 3. Y. Ueda, J. Statist. Phys., 2-0, No. 2, 181 (1979). 4. P. Holms, Philos. Trans. R. Soc. London, Ser. A, 298, 419 (1979). 5. I. Ito, Prog. Theor. Phys., 61, 815 (1979). 6. J . B . McLaughlin, J. Statist. Phys. , 24, 375 (1981). 7. K. Tomita and T. Kai, Prog. Theor. Phys. Suppl., 64, 280 (1978). 8. Y. Guckenheimer, IEEE Trans. Circ. Syst., CAS-27~ No. 11, 983 (1980). 9. Y. Guckenheimer, Physica D, 1, 227 (1980).

10. A . S . Dmitriev and V. Ya. Kislov, Radiotekh. Elektron., 2_.77, No. 12, 2454 (1982). Ii. G. Benettin and L. G algani, Phys. Rev. A, I_44, No. 6, 2338 (1976). 12. P. Manneville and Y. Pomeau, Physica D, 1, 219 (1980). 13. Y. Pomeau and P. Manneville, Commun. Math. Phys., 74, 189 (1980). 14. G. Mayer-Kress and H. Haken, Phys. Lett. A, 8__22, No. 4, 151 (1981).

798

15. A . S . Pikovskii and V. I. Sbitiev, 1)reprint of the Leningrad Institute of Nuclear Phys ics No. 641, LIYaF, Leningrad (1981).

16. C. Hayashi, Nonlinear Oscillations in Physical Systems (rev. enl. ed. ), McGraw-Hill , New York (1964). 17. L . A . Bessonov, Nonlinear Elec t r ica l Networks [in Russian], Vysshaya Shkola, Moscow (1977).

PARAMETRIC EXCITATION OF LOW-FREQUENCY WAVE

TURBULENCE IN A SLIGHTLY DISPERSIVE LC-LINE

K. I. Volyak and A. S. Gorshkov UDC 539.18 2

It is shown that within the field crea ted by two narrow-band hi pump waves traveling in opposite direct ions it is possible for dynamic wave turbulence to be excited over a broad spectrum of low frequencies where dispers ion is absent. In order for this to occur, in addition to producing ref lect ions of the waves generated from the boundaries, the medium must also possess both quadrztic and cubic nonlinearity.

It is well known that the mos t charac te r i s t i c nonlinear wave p rocess in slightly dispers ive media is the generat ion of higher harmonics by intense low-frequency signals, leading to t ransfer of energy upward in the spectrum. This is the reason why pa ramet r i c amplification of sound waves is not possible. But, if a high- frequency pump wave is located on the boundary of the strong dispers ion zone, such amplification is, in principle, possible over a wide range of low frequencies, and in the case of marked reflections at the boundaries of the nonlinear medium, pa ramet r i c generat ion of oscil lations at d iscre te frequencies will occur.

It will be shown in the present study that within the field produced by two hf dispersive pump waves traveling in opposite direct ions, excitation of strong wave turbulence may occur within a quite broad range of low frequencies at which dispers ion is absent. In order for this to occur, in addition to producing reflections at the boundaries, the medium (a nonlinear LC-l ine was used e:~perimentally) must be charac te r ized by both quadratic and cubic nonlinearity.

1. Exper iments with the Nonlinear Line. To observe pa ramet r i c excitation of low-frequencyturbulence, a long (113 stage) low-pass f i l ter type LC-l ine was used, with the capaci tances of p--n junction diodes acting as the nonlinear elements. The diode operating point was set by a bias voltage of - -3V applied to the bias bus. Individual stage pa rame te r s were matched to within 1% of nominal values. The line was optimally terminated by active loads, equal to the charac te r i s t i c impedance ot the midpoint of the fi l ter passband, providing a VSWR of 1.05 or less over the major par t of the passband, with a reflection coefficient of 1(~}~. Dispers ion in the line was compensated by introducing positive inductive coupling between adjacent stages with mutual induct~me of 0.1. Up to 4.5 MHz dispers ion was pract ical ly absent, rising' to a significant value only

in the vicinity of the c r i t ica l frequency 6.5 MHz.

In such a sys tem higher harmonics of low-frequency monochromat ic or narrow-band noise signals are eas i ly excited. When hf noise with a sufficiently nar row spect rum is applied to the line input (e. g., with a line width of 50 kHz at a mean frequency of 5 MHz with mean-square amplitude ~ 2 Veff), at the line output s to- chast ic oscil lations over the entire passband can be detected, produced by nonlinear "mizing" of noise com- ponents along the path. Such "mixing" could not be detected in the case of excitation from a standard hf signal

genera tor .

To excite the dynamic turbulent regime, the two line ends were excited through decoupling networks by sinusoidal pump signals at 4-5 MHz. Exper iments were per formed with one common and two separate signal genera tors . At low (compared to the bias) input amplitudes a3(O ) = a3 ( l ) >- 1.5 Vef f (where l is the sys tem length), pa ramet r i c generat ion over a wide frequency range below the pump frequency was observed.

The signal amplitude spect rum at the midpoint of the line, where generat ion is mos t intense, is shown in Fig. ld (the section coordinate x = 59L l , where l l is the effective length of a stage). At f3 = MHz the

- - I ). N. Lebedev Phys ics Institute, Academy of Sciences of the USSR. Transla ted f rom Izvest iya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 26, No. 9, pp. 1087-1091, September, 1983. Original ar t ic le sub- mit ted July 5, 1982; revision submitted Apri l 4, 1983.

0033-8443/83/2609- 0799507.50 �9 1984 Plenum Publishing Corporation 799