Experimental and Theoretical Investigations of the Influence of the

ISSN 1064�2269, Journal of Communications Technology and Electronics, 2012, Vol. 57, No. 1, pp. 45–53. © Pleiades Publishing, Inc., 2012.Original Russian Text © B.S. Dmitriev, Yu.D. Zharkov, A.A. Koronovskii, A.E. Hramov, V.N. Skorokhodov, 2012, published in Radiotekhnika i Elektronika, 2012, Vol. 57, No. 1,pp. 49–58.

45

INTRODUCTION

Theoretical and experimental investigations of theinfluence of noise on nonlinear dynamic systems ofdifferent nature have aroused considerable interest ofresearchers [1]. Recent investigations demonstratethat noise and fluctuations in dynamic systems mayplay constructive role in many physical [2–4], physio�logical [5], chemical [6], and other systems. In partic�ular, the influence of noise with a certain intensity maylead to the appearance of new dynamic states in thesystems with a small number of degrees of freedom [7–10] and facilitate formation of structures in spatiallydistributed systems [1, 11] as well as to enhance thelevel of secrecy of the information transmitted in com�munication systems with chaotic carrier [12,13].Here, we should also note the phenomenon of noise�induced synchronization [14], when the impact ofnoise on an ensemble of identical chaotic oscillatorsleads to the appearance of the similar states in each ofthe oscillators. At present, such a type of behavior ofnonlinear systems attracts particular interest ofresearchers owing to the fact that the noise�inducedsynchronization becomes apparent at the interface ofthe deterministic and random behavior [15–17] anddemonstrates that the influence of noise on an ensem�ble of identical isolated chaotic systems may facilitateformation of identical behavior of initially uncoordi�nated chaotic systems.

Nevertheless, investigations of the influence ofnoise on generators of chaotic oscillations are mainlyperformed for systems with a small number of degreesof freedom. The number of publications in whichauthors try to analyze the influence of noise on distrib�uted systems and systems with delay (which are char�acterized by the infinite�dimensional phase space) ismuch fewer, and there are practically no publications,in which the influence of noise on distributed systemsis studied experimentally.

Among possible objects for investigations in thisdirection, we would like to specially emphasize elec�tron�wave systems of the microwave electronics. First,these systems can be considered as the referenceexamples of distributed self�oscillating systems in themodern theory of oscillations and waves [18,19]. Sec�ond, as a rule, such systems are widely used as sourcesof the high�power microwave radiation and, as a con�sequence, serve as the basic elements of practically allinformation and communication systems and findwide application in information transmission and pro�cessing as well as in manufacturing processes and sci�entific research. Thus, it is extremely important fromboth the fundamental and practical points of view toreveal main mechanisms of the influence of noise onsuch systems. The scope of practical problems in thisdirection is very large: it includes information trans�mission (including transmission of classified data)with the help of chaotic signals, radar technology withthe use of chaotic signals, problems of electroniccountermeasures, etc.

In this paper, the influence of an external noisesource on generation of regular and chaotic self�oscil�lations in the microwave band is analyzed experimen�tally and theoretically (with the use of numerical mod�eling on the basis of nonlinear nonstationary theory)by the example of a klystron chaotic oscillator withdelayed feedback. This oscillator is one of convenientobjects for the experimental and theoretical analysis ofthe influence of noise on a distributed system of themicrowave band, because various aspects of the auton�omous and non�autonomous dynamics of such a sys�tem are adequately investigated [20–22].

In this paper, we describe the experimental setupand the theoretical model used, the results of experi�mental and numerical investigations of the influenceof noise on dynamics of a klystron self�oscillator withdelayed feedback, and the phenomenon of noise�

DYNAMICS CHAOS IN RADIOPHYSICSAND ELECTRONICS

Experimental and Theoretical Investigations of the Influence of the External Noise on Dynamics of a Klystron Oscillator

B. S. Dmitriev, Yu. D. Zharkov, A. A. Koronovskii, A. E. Hramov, and V. N. SkorokhodovReceived March 4, 2011

Abstract—Experimental and theoretical investigations of the influence of an external noise source on aklystron oscillator with delayed feedback are performed. It is shown that the action of noise leads to suppres�sion of regular, self�modulation, and chaotic signals of the klystron oscillator. The physical mechanism ofgeneration suppression, which is related to the influence of noise on the value of the bunched beam currentin the output klystron cavity is revealed. The possibility of the appearance of synchronization induced bynoise in a klystron chaotic oscillator subjected to an external noise source is shown theoreticallyk.

DOI: 10.1134/S1064226912010056

46

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 57 No. 1 2012

DMITRIEV et al.

induced synchronization in the system under consid�eration.

1. EXPERIMENTAL SETUP

For the experimental investigation of the influenceof noise on the klystron self�oscillator with externalfeedback, we used an oscillator on the basis of aKU�134E five�cavity commercial amplifying klystronoperating in the centimeter wave band. In thisklystron, double�gap resonant cavities operating in theantiphase oscillation mode are used. The feedback isimplemented by introducing the external couplingbetween the input and output cavities. The outputklystron cavity is connected with the input cavity by acoaxial transmission line (the element of the delayedfeedback circuit). The klystron gain in the optimalmode Kg = 35 dB, the frequency bandwidth Δf =30 MHz, and the output power Pout = 20 W.

Accelerating voltage V0 (which is varied from 1200V to 2500 V) and current I0 of the electron beam (var�ied from 10 mA to 75 mA) were used as control param�eters. Different operation modes (stationary genera�tion, self�modulation, and developed chaos [20, 22])can be implemented by choosing these control param�eters in the klystron with delayed feedback.

The circuit diagram of the setup used for the studyof the influence of the external noise on generation inthe klystron self�oscillator is presented in Fig. 1. Amedium�power traveling�wave tube (TWT) with mag�netic periodic focusing system (MPFS) operating inthe absence of the input signal and amplifying theintrinsic noise of the electronic flow is used as a noisegenerator. (The TWT is loaded with a matched loadwith an attenuation factor of 25 dB.).

The shot effect in the TWT is the source of noiseoscillations with a continuous spectrum, which is con�stant in a wide frequency band. The noise signal fromthe TWT is amplified by a semiconductor amplifierand then by klystron amplifier Y (see Figure 1). After�wards, the noise signal passes through polarizationattenuator A installed in the system to ensure continu�ous adjustment of the power of the external noise andthrough tee and enters the feedback circuit of theklystron self�oscillator under investigation. Spectralmeasurements are performed with the help of an Ali�gent E4402B spectrum analyzer operating to the fre�quency band from 10 kHz to 3 GHz.

This experimental setup allows the analysis of: (1)autonomous characteristics of the noise signal fed intothe feedback circuit of the klystron oscillator understudy, (2) the autonomous behavior of the oscillatorunder study, and (3) the characteristics of the klystron

⊕

ME

SW

DCK

A

F1

TWT

A Y

+

DC

F1

Fig. 1. Circuit diagram of the experimental setup for investigations of the influence of noise on dynamics of a klystron oscillatorwith delayed feedback: TWT is the noise source (the TWT in the zero input signal mode), Y is the noise signal amplifier, А are theattenuators, ME is the unit of instruments (Aligent E4402B spectrum analyzer and M3�51 power meter), K is the klystron ampli�fier closed in the feedback loop, Fl are the ferrite isolators, DC is the directional coupler, SW is the switch, and ⊕ is the tee.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 57 No. 1 2012

EXPERIMENTAL AND THEORETICAL INVESTIGATIONS OF THE INFLUENCE 47

output signal under the influence of the external noisesource.

2. THEORETICAL MODEL

For comparison of experimental and theoretical(obtained by means of the numerical simulation)results, we use here the well�known model of a double�cavity klystron oscillator with delayed feedbackdescribed in detail in [20, 21, 23, 24]. It is evident thatthis model is rather rough for detailed description ofdynamics of a real five�cavity klystron oscillator usedin the of experiments. At the same time, it follows fromresults of earlier studies [22, 25] that this numericalmodel describes adequately enough the main pro�cesses and phenomena occurring in the real klystronoscillator and permits one to conduct qualitative (andin some cases even quantitative) comparison of thereal oscillator behavior and the behavior of its simpli�fied numerical model.

The system of lag differential equations describingthe influence of noise on the klystron oscillator isdescribed in the dimensionless form as [20]

(1)

(2)

where and are the slowly varying oscillationamplitudes in the gaps of the input and output resona�tors, respectively; τ is the dimensionless time; Δτ = 1is the dimensionless delay time in the feedback circuit;y = 1 is the damping parameter; α is the resonatorexcitation parameter excitement, which has themeaning of the product of the gain and the feedbackdepth and is proportional to the current of the elec�tronic beam; and is the random delta�correlatedGaussian noise with zero mean:

(3)

where D is the noise intensity (power).

It was supposed in Eq. (1) that the external noise isfed into the feedback circuit in such a manner that isshown in Fig. 1 and affects the speed modulation ofthe electronic beam in the first (input) cavity of theklystron.

For numerical integration of lag stochastic equa�tions (1)–(3) as well as Eqs. (4)–(8) given below, weused the single�step Euler method, which allows cor�rect calculation of a similar problem [1,26]. The stepof integration was chosen equal to Δτ = 10–3.

( )11 2

( )( ) = ( ),

FF F

∂ τ+ γ τ γ τ − Δτ + γξ τ

∂τ

( )2 12 1 1

1

( ) ( )( ) = 2 ( ) ,

( )

F FF J F

F

∂ τ τ+ γ τ α τ

∂τ τ

1( )F τ 2( )F τ

( )ξ τ

' '( ) 0, ( ) ( ) ( ),Dξ τ = ξ τ ξ τ = δ τ − τ

3. RESULTS OF EXPERIMENTALAND NUMERICAL INVESTIGATIONS

OF THE INFLUENCE OF NOISE ON DYNAMICS OF THE KLYSTRON

OSCILLATOR

Let us consider the results of experimental investi�gations of the influence of noise on generation charac�teristics of the klystron oscillator. We will examine theinfluence of the external noise on both regular andmore complicated (self�modulation and chaotic) gen�eration modes of the klystron oscillator. Figure 2shows the evolution of the generation power spectrumof the system with the growth of the power of the exter�nal noise source obtained from the experimentalinvestigation. These investigations were performed ataccelerating voltage V0= 2300 and beam current I0 =20. It is clearly seen that, at small noise power Pn(Pn/Рout < 0.2)Pn (Pn/Pout < 0.2), a noise pedestalappears in the generation spectrum, the height of thispedestal increases as the noise power increases, andthe spectral line generated by the klystron oscillatorgradually widens (Fig. 2b). As the noise power furtherincreases, the generation is suppressed. This suppres�sion is accompanied by a considerable lowering of thepower of generated oscillations and is finished by theindistinguishability of the desired signal against thenoise pedestal (Fig. 2c). The latter effect can be seen atexternal noise power Pn/Pout > 0.2.

The same situation can be seen in the case of theinfluence of an external noise source on generation ofa chaotic signal in the klystron oscillator. Correspond�ing experimental spectra of non�autonomous genera�tion are presented in Fig. 3. Once againg, it is clearlyseen that, in this case, as the external noise powerincreases, generation is suppressed and this suppres�sion is also accompanied by the decrease in the oscil�lation power. At the same time, complete suppressionof the chaotic generation in the klystron oscillatortakes place at essentially higher power levels of thenoise signal than in the case of suppression of genera�tion of a single�frequency signal.

It should be noted that we experimentally observedan interesting effect related to the behavior of theklystron oscillator at the bound of excitation of self�oscillations in the system. Under the influence of thenoise signal, self�oscillations were excited at a smallerbeam current than in the case of the absence of thenoise impact, i.e. the presence of noise slightly reducesthe value of the starting current of the electron beam.Simultaneously, near the value of the starting current,at low noise intensity, we observed small growth of thepower of generated oscillations. However, as the powerof the noise impact further increased, the oscillationpower in the system decreased and generation wassuppressed.

The same results were obtained during the numer�ical analysis of the influence of noise on the klystronoscillator with the help of the model described in Sec�

48


DMITRIEV et al.

tion 2. Typical power spectra at various increasingnoise intensities are shown in Fig. 4 (the mode of sta�tionary generation, the resonator excitation parameterα = 5) and in Fig. 5 (the mode of chaotic self�modu�lation, α = 12).

Let us consider the impact of noise on the mode ofstationary generation. One can see from Fig. 4obtained at the noise intensity D = 0 that, at α = 5, themode of stationary oscillation, which is characterizedby the presence of the single frequency in the power

Mkr1 2.801000 GHz70.29 mVAtten 5 dB

(a)

Ref 100 mVPeakLin

W1 S2S3 FC

AA

Center 2.8 GHzRes BW 300 kHz VBW 300 kHz

Span 50 MHzSweep 4 ms (401 pts)

Atten 5 dB

(b)

Ref 100 mVPeakLin

Mkr1 2.801000 GHz27 mV



W1 S2S3 FC

AA

Atten 5 dB

(c)

Ref 100 mVPeakLin

Mkr1 2.801000 GHz6.276 mV

W1 S2S3 FC

AA



Fig. 2. Experimental power spectra at the output of thefive�cavity klystron oscillator in the case of single�fre�quency autonomous generation for the following intensi�ties of the external noise signal: Pn/Pout = (a) 0, (b) 0.09,and (c) 0.26, where Pout and Pn are the powers of the signaland the noise, respectively.

Mkr1 2.798775GHz25.13 mVAtten 5 dB

(a)

Ref 100 µVPeakLin

W1 S2S3 FC

AA

Center 2.8 GHzRes BW 1 MHz VBW 1 MHz


Mkr1 2.798775GHz5.92 µVAtten 5 dB

(b)

Ref 100 µVPeakLin

W1 S2S3 FC

AA



Mkr1 2.798775GHz1.55 µVAtten 5 dB

(c)

Ref 100 µVPeakLin

W1 S2S3 FC

AA



Fig. 3. Experimental power spectra at the output of thefive�cavity klystron oscillator in the case of autonomouschaotic generation for the following intensities of theexternal noise signal: Pn/Pout = (a) 0, (b) 0.4, and (c) 0.76.



spectrum, arises in the model of the autonomousklystron oscillator. If noise affects the system (D ≠ 0),the dynamics of the klystron oscillator changes. Atsmall intensity D the power spectrum becomes noisybut the fundamental frequency remains clearly pro�nounced (see Fig. 4b). As intensity D of the noiseimpact increases, the power at the main operating fre�quency decreases and, as a result, at a rather high levelof the power of the external noise, we can observecomplete signal suppression at the output of the dou�ble�cavity klystron self�oscillator (Fig. 4c).

Numerical simulation was performed also for othervalues of control parameter α, when the klystron oscil�lator operated in the modes of periodic and chaoticself�modulation of the output signal. In any case, situ�ation similar of complete suppression of the intrinsicgeneration of the klystron oscillator was observed atsufficiently high amplitudes of the external noise. Asan example, Fig. 5 shows the calculation resultsobtained for the impact of the external noise in themodel of the klystron oscillator operating in the cha�otic oscillation mode. The power spectra of the outputsignal are shown at the gradually increased noiseintensity. One can see that, as the power of the noiseimpact increases, the intrinsic deterministic chaotic

dynamics of the double�cavity klystron oscillator issuppressed and, as in the experimental investigations,this dynamics becomes indistinguishable against thenoise pedestal.

It should be noted that, at small values of the cavityexcitation parameter that are close to the starting value(α <3) and at small noise intensity, the signal poweratthe fundamental frequency increases, then (with theincrease in the intensity of the noise impact)decreases, and, at some rather high noise power, issuppressed in the same manner that was observed inthe experimental investigation of the influence ofnoise on the five�cavity klystron oscillator withdelayed feedback.

Thus, we can make the conclusion about goodqualitative correspondence of theoretical and experi�mental results on suppression of self�oscillation (bothregular and chaotic) by the external noise signal in theklystron oscillator with delay.

Let us systematize the results obtained on suppres�sion of self�oscillation in the microwave oscillatorunder consideration subjected to an external noisesource. Figure 6 shows the values of numerically cal�culated intensity D at which we observe suppression ofintrinsic self�oscillation dynamics in the klystron

10–6

10–9

10–12

10–15

10–18

–1,0 0 1,0

(c)

P, dB

f

10–3

10–6

10–9

10–12

10–15

–2.0 –1.0 0 1.0

(a)

P, dB

f

10–3

10–7

10–11

10–15

10–19

–2.0 –1.0 0 1.0

(b)

P, dB

f

Fig. 4. Power spectra at the output of the two�cavity klystron oscillator in the case of stationary generation: Pn/Pout = (a) 0,(b) 0.09, and (c) 0.22.

50


DMITRIEV et al.

oscillator as a function of cavity excitation parameterα (the dimensionless beam current). The oscillationmodes in the autonomous oscillator corresponding toan increase in the current beam are indicated: the sys�tem proceeds from the stationary (single�frequency)generation mode (S) through the periodic self�modu�lation mode (M) to chaotic oscillations (C). One cansee from the figure that, as cavity excitation parameterα increases (up to α ≈ 11.0, which corresponds to peri�odic self�modulation of the output signal of the auton�omous klystron) the noise intensity corresponding tocomplete signal suppression slowly and monotonicallydecreases. When the value of the control parameterreaches α ≈ 11.0, rapid lowering of the level of thenoise power necessary to achieve the mode of oscilla�tion suppression in the oscillator is observed.

The effect of generation suppression by an externalnoise signal can be explained by strong debunching ofthe electron beam in the drift space under the influ�ence of the noise signal modulating electrons in speedin the input cavity and, as a result, by lowering of theamplitude of the first harmonic of the bunched currentin the output cavity occurring as the power of the noisesignal increases. Accordingly, Fig. 7 shows the depen�dence of the first harmonic of the bunched current on

intensity D of the noise impact for different values ofcavity excitation parameter α.

For the value of cavity excitation parameter α =2.5, which corresponds the beam current less than thestarting value (but close to it), the impact of the noisesignal leads to the self�excitation of the oscillator andoscillations appear. Thus, the influence of noise nearthe self�excitation point facilitates lowering of thestarting current. However, as the noise signal furtherincreases (at D ≈ 200) the value of the first harmonic ofthe bunched current reaches a maximum, and thendecreases with further growth of the intensity of thenoise impact. When α exceeds the starting value (α >3) in the klystron oscillator, as noise intensity Dincreases, we can observe lowering of the first har�monic of the bunched current; i.e. the noise impactleads to the noise modulation of the electron velocityand, as a result, facilitates debunching of the electronbeam. Owing to this, at a certain noise power, the valueof the first harmonic in the klystron output cavitybecomes insufficient to maintain oscillations in thesystem, i.e., oscillation suppression is observed. As αincreases, the value of the first harmonic of the elec�tron flow, which penetrates the output cavity,decreases (and, hence, the power of the output signal

10–4

10–6

10–8

10–10

10–12

10–14

–1.0 –0.5 0 0.5

(c)

P, dB

f

10–4

10–6

10–8

10–10

10–12

10–14

–1.0 –0.5 0 0.5

(a)

P, dB

f

10–4

10–6

10–8

10–10

10–12

10–14

–1.0 –0.5 0 0.5

(b)

P, dB

f

Fig. 5. Power spectra at the output of the two�cavity klystron oscillator in the case of chaotic generation: Pn/Pout = (a) 0, (b) 0.37,and (c) 0.71.



of the klystron oscillator also decreases), which leadsto lowering of the required power of the noise impactnecessary for total signal suppression (see Fig. 6).

4. NOISE�INDUCED SYNCHRONIZATION

The mechanism of suppression of chaotic genera�tion by an external high�intensity noise signal allowsus to assume that, in a chaotic klystron oscillator, it ispossible to observe the noise�induced synchronization[14, 16, 27–30]. Let us recall that, speaking about thenoise�induced synchronization, we have in mind amode in which a random signal acting on two chaoticsystems u(t) and with identical control parametersbut starting from different initial conditions( ) leads to synchronization of systems'behavior, i.e., systems, after finishing some transientprocess (which can be very long even in dynamic sys�tems with small dimensionality [31, 32]) begin todemonstrate identical behavior ] [14, 33].The interest in this type of behavior of chaotic oscilla�tors is determined by the fundamental significance ofthe analysis of the influence of noise on dynamic sys�tems. The noise�induced synchronization illustratesthe fact that the impact of noise on an ensemble ofdynamic self�oscillatory systems may facilitate forma�tion of identical behavior of initially uncoordinatedchaotic systems [2, 5, 34–36]. The diagnosing of thepresence of the noise�induced synchronization can beprovided by the direct comparison of the states ofdynamic systems and . It is shown in [14, 16,37] that the main mechanism of formation of thenoise�induced synchronization mode is the suppres�sion of the intrinsic chaotic dynamics in the systemsubjected by noise. Usually, in this case, noise can beconsidered as having the zero mean and exactly thenonzero�mean value of noise causes transition fromthe chaotic generation mode to the mode of regularoscillations or the stationary state [11, 14, 16]. How�ever, in the system analyzed in this paper, suppressionof the chaotic signal is observed under the action of thenoise signal with zero mean and is caused by the beamdebunching at the noise modulation of the flow in theHF gap of the input cavity.

Let us consider numerically the onset of the noise�induced synchronization in a system of two isolatedidentical klystron oscillators demonstrating chaoticdynamics and subjected by the same noise. In thiscase, the mathematical model is

(4)

(5)

( )tv

0 0( ) ( )u t t≠ v

( ) ( )u t t= v

( )u t ( ).tv

( )1

1 111 2

( )( ) = ( ),

FF F


∂τ

( )1 1

1 12 12 1 1 1

1

( ) ( )( ) = 2 ( ) ,

( )

F FF J F

F


∂τ τ

(6)

(7)

( )2

2 211 2

( )( ) = ( ),

FF F


∂τ

( )2 2

2 22 12 1 1 1

1

( ) ( )( ) = 2 ( ) ,

( )

F FF J F

F


∂τ τ

350

300

250

200

150

100

50

03 5 7 9 11

3 5 7 9 11

D

α

S A C

α

Fig. 6. Oscillation suppression bound in the klystron oscil�lator as a function of intensity D of the external noiseimpact and dimensionless beam current α. Under themain figure, the following regions of different oscillationmodes of the autonomous klystron oscillator are shown asfunctions of parameter α: S is the region of stationaryoscillation, M is the region of periodic self�modulation,and C is the region of chaotic self�modulation.

1.0

0.8

0.6

0.4

0.2

0 200 400 600 800

1

2

3

I1

D

Fig. 7. First harmonic I1 of the bunched current vs. noiseintensity D for different values of parameter α: (1) 2.5,(2) 3.5, and (3) 8.5.

52


DMITRIEV et al.

where complex slowly varying amplitudes and correspond to the first and the second klystron oscilla�tor, respectively. The control parameters of these oscil�lators are identical; the dimensionless beam currentα = 12 for both klystrons (which corresponds to thechaotic generation mode) and noise ξ affecting eachoscillator is also the same. The oscillators differ fromeach other only in their initial conditions:

(8)

It is convenient to analyze the onset of the noise�induced synchronization with an increase in noiseintensity D by drawing diagrams in the plane of voltageoscillation amplitudes in the gaps of the first and the

second klystron self�oscillators Theonset of the noise�induced synchronization in theklystron oscillator can be diagnosed by the appearance

of diagonal in amplitude plane which will cor�respond to the identical dynamics of two oscillators.

Corresponding planes are presented in Fig. 8for different noise intensities.

When the external noise source is absent (D = 0),the dynamics of two identical chaotic oscillators start�ing from different initial conditions is different. As aresult, the cloud of points, which occupies practically

all plane is observed (see Fig. 8b). Thisfact indicates that, at each time instant, the dynamics

11,2F 2

1,2F

1 2 1 21 1 2 2( ) ( ), ( ) ( ), ( ,0].F F F Fτ = τ τ = τ τ ∈ −Δτ/ /

1 22 2( ( ), ( )).F Fτ τ

1 22 2( , ),F F

1 22 2( , )F F

1 22 2( ( ), ( ))F Fτ τ

of klystron oscillators is different. As the intensity ofthe noise impact increases (see Fig. 8b drawn at D =450, which corresponds to the ratio of the noise powerto the power of the oscillator signal in the autonomouscase Pn/Pout = 1.1), the dynamics of klystron oscilla�tors becomes more and more synchronized. This factis reflected in the behavior of the points in plane

, which begin to group along the maindiagonal; i.e., oscillators' states gradually cometogether as the intensity of the noise impact increases.Finally, at the noise intensity D = 500 (which corre�sponds to a high power of the noise signal, Pn/Pout =1.2), the dynamics of two klystron oscillators becomeidentical. We can see from Fig. 8c that there is a diag�

onal in plane ( ), which corresponds tothe mode of the noise�induced synchronization. Thephysical mechanism of the initiation of this mode iscompletely determined by the mechanism of suppres�sion of chaotic dynamics by the impact of the externalnoise in the klystron chaos generator described in theprevious section.

6. CONCLUSIONS

In this paper, to the theoretical and experimentalinvestigations of the influence of noise on dynamics ofa klystron oscillator operating in different oscillationmodes (both regular and chaotic) under variations in

1 22 2( ( ), ( ))F Fτ τ

1 22 2( ( ), ( ))F Fτ τ

4

2

0

–2

–4

–6 –4 –2 0 2 4

F22

F21

(c)

8

4

0

–4

–8

–4 –2 0 2 4

F22

F21

(a)

4

0

–4

–8

–8 –6 2 4 6

F22

F21

(b)

–4 –2 0

Fig. 8. Comparison of the dynamics of two klystron oscillators operating under identical noise impacts with an increase in thenoise intensity: D = (a) 0, (b) 450, and (c) 500.



the main control parameters of the system (the inten�sity of the noise impact, parameter α of the excitationof the klystron cavity) have been performed. It hasbeen shown both experimentally and numerically that,in the system under consideration, as the noise inten�sity increases, suppression of the intrinsic dynamics(both in the stationary generation mode and in theself�modulation mode) is observed. The analysis of thetheoretical model has shown that the generation sup�pression effect is explained by the beam debunchingunder the influence of the external noise signal modu�lating the flow in the input cavity and by the decreaseof the amplitude of the first harmonic of the bunchedcurrent in the output cavity with increasing power ofthe external noise. It has been shown that, under theimpact of the same noise on two identical klystronoscillators operating in the chaotic generation mode,both systems begin to demonstrate identical dynamicsat a certain value of the intensity of the noise impact,i.e., the noise�induced synchronization mode occurs.

Theoretical and experimental results of the investi�gation of the influence of an external noise source onoscillations of the klystron oscillator with delayedfeedback are in good qualitative correspondence.

ACKNOWLEDGMENTS

This study was supported by the Council for Grantsof the President of the Russian Federation for theState Support of Leading Scientific Schools (grantNSh�3407.2010.2) and Federal special program “Sci�entific and Pedagogical Staff of Innovational Russia”for 2009–2013.

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