-At94 BBS NONLINEAR DYNAMICS OF COUPLED OSCILLATOR ARRAY(U) - =NAMA RESEARCH LAB MASNINGTON DC D ROSNER 18 KM 88NRL-NR-6i44

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AD-A 1 4 888NRL Memorandum Report 6144

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Nonlinear Dynamics of Coupled Oscillator ArraysDAVID MOSHER I

Plasma Technology BranchPlasma Physics Division

March 18, 1988

DTI

N'Nk\ MAY 04 1988~

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Nonlinear Dynamics of Coupled Oscillator Arrays

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FIELD GROUP SUB-GROUP phase lockingnonlinear oscillatorsmicrowaves

19 ABSTRACT (Continue on reverse if necessary and identify by block number)I.4

The phase-locked dynamics of large oscillator arrays is currently of interest because of possiblemicrowave directed energy applications. Straight-forward integration of the coupled dynamical equations forsuch arrays is computationally costly for the associated multidimensional parameter space, long integrationtimes, various initial conditions and system configurations. Finite difference analogues of the nonlinear dif-ferential equations can reproduce their complex dynamical behavior with a 2 to 3 order-of-magnitudeimprovement in computational time.

Here, the applicability of the finite difference technique is demonstrated by solutions of the dynamical0 equations for 2 coupled oscillators and rings of larger numbers. Parameter studies for these configurations

suggest the values of coupler length and coupling strength required to provide robust phase-locked operation.The finite difference technique can be extended to model large oscillator arrays with other couplinggeometries, amplifier arrays, and additional physical phenomena.

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CONTENTS

'" I. INTRODUCTION .1 . I T O U T O . . . . . . . . . . . . . . . .... .... ...o.o.oo . , , .. . ..o. ..o , . ... .. .. .... .... .. ... ..... .... . . .... .... .... ... I

I1. TW O COUPLED OSCILLATORS .............................................................................. 2

M.. III. EXTENSION TO LARGER ARRAYS ............................................. 4

IV. AREAS OF CONTINUED RESEARCH ....................................................................... 6

, V. REFERENCES .............................................................. 7

SDISTRIBUTION ................................. .................. .......... IS

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IV/

NONLINEAR DYNAMICS OF COUPLED OSCILLATOR ARRAYS

I. INTRODUCTION

Computers have made possible the solution of fundamentalnonlinear dynamical equations in many scientific disciplines.'Along with the numerical work, sophisticated group-theoreticaltechniques have been employed to study similarities of behavior inlarge classes of dynamical equations.2 One highly studied class,driven nonlinear oscillators, includes the equations of coupledoscillators that model sources designed to meet high-power,directed microwave energy objectives. Research in this area ofnonlinear dynamics has demonstrated the following:

* Simple difference equation analogues can faithfullyreproduce the rich spectrum of dynamical behaviorpredicted by the nonlinear differential equations.

* The dynamical behavior (following decay of initialtransients) depends sensitively on key constants calledcontrol parameters.

* Variations in the control parameters can produce abrupt(and frequently counter-intuitive) changes in theoscillator's long-term behavior between free-running,phase-locked, quasiperiodic-subharmonic, and chaoticstates.

* The changes from well-ordered to chaotic behavior occurvia a few paths of universal character; they are similarfor a wide variety of force laws obeying a few generalrequirements.

These characteristics allow one to employ difference equationanalogues to study coupled microwave oscillator arrays.

A general objective of such nonlinear studies is to determine

which portions of the control parameter space are associated withthe different dynamical behaviors. For the work of this report,this objective takes the following form: What values of theexperimental parameters (cavity Q, coupler length, etc.) and whatsystem geometries are required to insure robust phase-lockedbehavior in coupled oscillator arrays relevant to directedmicrowave energy objectives?

The major benefit of the difference equation approach over thedirect numerical solution of the differential equations is a two tothree order-of-magnitude increase in computational speed. Thoughnot required for small oscillator arrays, the increase in speed may

4. be crucial for the economic study of large arrays with theassociated multi-dimensional parameter space, long integration

., times, various initial conditions and configurations. (For

Manuscript approved December 10, 1987.%"

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example, one configuration of 1000 coupled oscillators, integratedfor 100 periods with 100 steps/period, and 100 combinations of 2parameters requires 109 differential equation time steps).

The utility of the difference equation approach is illustratedin the next section, where the method is applied to the two-coupled-oscillator-experiment of Benford and Woo 3 and compared totheir analysis. In Sec. III, the problem is generalized to a ringof oscillators with nearest-neighbor interactions. Calculationswith as many as 16 oscillators are presented and results arediscussed. In Sec. IV, issues associated with large arrays andtechniques to include additional experimental phenomena, such asnoise and pulsed-power variations are briefly discussed.

II. Two Coupled Oscillators

The starting point for this analysis is the differentialequation formulation of Benford and Woo.

d'1 (t) P 1

"" 1(t) " w 1 1 - - sin[# 1 (t) - # 2 (t - )()dt Q}

d* 2 (t) ( 2

dt - 2 1 - -sin[% 2 (t) - 1 (t - Pt)] (2)dt 02

where 2 are the instantaneous phases in oscillators 1 and 2,0)1 .2 are their free-running angular frequencies, Q 1 2 are thecavity quality factors, P., 2 are related to the ratios ofinjected and output powers (in the range of 0.1 to 1), and -r isthe phase transit time through the structure coupling the twooscillators.

The authors have shown that steady, phased-locked behavior

d+1 d* 2

- - - " - constant.dt dt

obeyssi& (3)sinA* - kcos*c

where a +- 2 1, c - aw/2w., 4w o2 - wo ,

WO M (w + W02 )/2 >> w, k - p/Q is the same for the twooscillators, and #€ " wTp. From Eqn.(3), phase lockingis possible when

klcos*cI c £ (4)

F' 2

* If phase-locking does occur

-" (l - kcosAosin*.) (5)

Note that Eqn.(4) is a necessary but insufficient condition forphase locking. The dynamical equations must be solved subject toU. particular initial conditions to determine if locking does occurfor particular values of the control parameters , and k.

The straight-forward way to solve Eqns.(l) and (2) is tochoose a small time increment At << 2n/w o and iterate

+in+' - #i + c 0Ar(l-c)[l - ksin( ,n - +2 n-q)] (6)

- + £%o't(I+c)[I - ksin( , n - *ln-q)] (7)

where 0+2- #1 2 (nAt) and qAt - TV.

-Nonlinear dynamics research has shown that the long termevolutions of such equations preserve their character even whenAt > 2n/w0 . For the problem at hand, the natural choice is At - P

leading to the 2-dimensional iterative map

+ n4 l - #" + 2nW(l - )[l - ksin(+ln - 2n-1)] (8)

42-- - 42 + 2nW(1 + e)[l - ksin(#2 - *z"-)] (9)

where W - wT,/2n is close to the number of wavelengths in thecoupler The equivalent 1-dimensional map, the sine circlemap, is the subject of much analysis.4 -'

When phase-locked, A*" - * 2 n - u is constant, and thedifference equations yield locking conditions identical to Eqns.(3)-(5). One can then be confident that they emulate thedifferential equations even though the iteration time is severalperiods of the oscillation rather than a small fraction of it.It is now shown that the evolution of A* depends sensitively onthe control parameters W and k.

To illustrate the range of long-term dynamical behavior. accessible to a pair of coupled oscillators, consider solutions* of Eqns.(8) and (9) for W - 2, e - .01, and various values of k

(Fig. 1). In the left-hand frames, an initial phase differenceis chosen, and the first 100 iterations of Eqns.(8) and (9) areplotted. The right-hand frames plot the iterations of 8 initialphase differences spanning 2n radians.

0

a...

When k - .0075, the coupling is too weak for locking and 6changes monotonically for the distorted free-running oscillators.This value of k does not satisfy Eqn.(4). When k exceeds .01,all initial conditions eventually lock to the phase differenceI . predicted by Eqn.(3). At about k - .09, phase locking for allinitial conditions is replaced by a stable 3-cycle in which theappearance of a nonlinear subharmonic instability leads tooscillations between three values of 4+. At about k - .11, thethree phases broaden into nearly aperiodic bands. Above k - .14,the bands merge and the evolution looks chaotic. Note that thelarger k value cases satisfy Eqn.(4) but are not phase-locked.

Data of the sort illustrated in Fig. 1 are summarized in

Fig. 2, where 100 successive iterations starting with n - 200 areplotted for 16 initial conditions at each of 400 k values. Inthis way, phase-locking for the ensemble of initial conditionsat fixed k shows as a point, the "trifurcation" as a verticalarray of 3 points, and chaotic evolution as a vertical line.Though the first 200 iterations of each computer experiment mustbe calculated, they are not plotted so that only the long term

}, evolution, free of initial transients, is displayed.

Figure 3 summerizes the dynamic evolution for fixed k andvarying W using the approach described for Fig. 2. In the regionwhere phase-locking occurs, the plots are equivalent to the

* analytic phase difference vs transit time plots of Benford andWoo. The plots show a hashy fill from top to bottom in theregime where Eqn.(4) cannot be satisfied. A new feature of thedynamic solutions is the inability to phase-lock at large delayswhen coupling is strong. These, and other solutions, demonstratethat the onset of subharmonic instabilities produces an upperlimit to coupling strength and coupler length determined from

Wk < .16 (10)

for phase-locking of two oscillators. The equivalent conditionfor large oscillator arrays may represent a stringent limit totheir design.

II1. Extension to Larger Arrays

The simplest way to extend the analysis to larger%JO oscillator arrays is to consider a ring with nearest neighbor* coupling. This geometry minimizes the number of couplings to

each oscillator, and avoids the treatment of free bounderies. Thedifferential equation governing the ith oscillator in a ring of Nis

d_ W.d 1 1 - ksin[ #j(t) - ,#j I(t--,)] ksin[ +j(t) - #j 1 (t-- )]

dt(11)

where all the k factors are assumed identical. It is alsoassumed that all cavity free-running frequencies are

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-' .. . .

the same so that locked phase differences AO between neighboringoscillators will be equal. Adding all the phase differencesaround a phase-locked ring leads to the requirement

NA# 2am ; -- 0, ±1, ±2... (12)IIFor small arrays, operation with m - 0 is probably most effectivefor coherent power amplification. In any case, uncontrolledshot-to-shot variations in phase difference are not desirable, sothat robust phase-locking at a predetermined value of m isrequired.

The difference equation analogue of Eqn.(l1) is used tostudy which phased-locked m occurs for given initial conditionsand control paramter values.

O '€ - +,n + 2W fl - k[sin(o1 " - + _-.1) + sin(#in - *i 1n1)}

- 1 (13)

Here, i - 1 to N, + - *, and *, " *,, and the controlparameters have their previous definition. A large number ofinitial phases are chosen randomly and their long term evolutionis examined. Figure 4 shows the first 100 iterations of twoevolutions for N - 6, k - .04, and W - 1. Each oscillator'shistory is labeled according to its position in the ring. Forabout 90% of the initial conditions, the evolution resemblesFig. 4A, rapid phase locking with m - 0. About 10% of the time,the initial conditions relax to an m - +1 configuration as shownin Fig. 4B.

Computer runs with other N values and integral W valuesshow that m - 0 occurs exclusively for N < 4, that the incidences

" of m - +1 become more frequent as N increases and that m - +2appears occasionally for N in the teens. The phase-lockingcondition equivalent to Eqn.(10) for a ring of oscillators is

Wk < .08 (14)

for all values of N studied. This is reasonable since the two

" couplers for each oscillator have a combined coupling strength of

2k. It is conjectured that the condition for a regular array ofoscillators with n couplings to each oscillator is Wk < .16/n.

":-/ Though not intuitive, nonlinear analysis indicates thatimproved reproducibility and robustness of lock-in may result

from certain noninteger W values.4 -' This improvement isdemonstrated in Fig. 5, constructed in the manner of Fig. 2 with

,.., N - 16. Each of the Ai+ - +i - +i_1 values is plotted each timestep for 10 randomly selected initial conditions and a given k

value. When W - 10, lock in occurs in accordance with Eqn.(14)at m - 0, ±1 and occassionally m - +2. When W - 8.8, all initialconditions collapse to m 0 phase-locking and the critical Wk

Iv,

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product is increased. Careful design of the couplers maytherefore improve reproducibility in the experiment. Initialcalculations indicate that small variations in w0 i betweenoscillators may also improve robustness of phase locking atm -0.

IV. Areas of Continued Research

The finite difference techniques described above can beextended to large arrays with different coupling geometries todetermine which configurations and experimental parameters leadto rapid and robust phase-locking. One possible limitation toperformance in large arrays is described in an analysis of phaseorganization in a large linear array of driven Hamiltonianoscillators.7 There, the system is shown to relax on two timescales: a fast-time in which local regions phase-lock and a muchlonger time (104 iterations) in which these local regionsinteract and merge. One can well imagine local groups of phase-locked microwave oscillators in a large array with chaoticoscillations at group boundaries for the duration of the powerpulse. It is presently unclear if more clever grouping andcoupling of oscillators can alleviate this difficulty. Onepossible solution may involve seed amplifiers to keep localoscillator groups in step.

" Additional physical phenomena can be included in the finitedifference analysis. For example, noise tolerance of frequencylocked dynamics has been studied8 by adding a white noise term tothe sine circle map. A similar procedure can be employed withoscillator arrays to study the effects of experimental noise orstatisti.al effects due to unresolved degrees of freedom. Also,time variations in pulsed power can be studied by applying slowtime variations to the control parameters.

-.. A major improvement in the analysis would be to describeeach oscillator by two difference equations involving the phaseand field amplitude. Descretized versions of the pendulumm-likeequations derived by Fliflet9 or the generalized VanderPol

- - oscillators researched by Walsh, Johnston and Davidson'0 could beemployed. Then, mixed amplifier-oscillator configurations couldbe studied as well as operation with time-varying pulsed power.

6

V. References

1. D. Mosher, "Europe Approaches Chaos with ElectricalCircuits," ONR London Report R-6-84, 1984.

2. M.J. Feigenbaum, J. Stat. Phys. 19, 25(1978).

3. Presentation by J. Benford at the Workshop on High PowerMicrowaves, May 4, 1987, private communication.

4. P. Bak, Physics Today, p44, Dec. 1986.

5. R.V. Jensen, American Scientist 75, pp168-18 1, Mar.-Apr.1987.

6. E.J. Ding, Phys. Rev. Lett. 58, 1059(1987).

7. C. Tang, K. Wiesenfeld, and P. Bak, Phys. Rev. Lett. 58,1161(1987).

8. K. Wiesenfeld and I. Satija, "Noise Tolerance of Frequency-Locked Dynamics," in publication.

9. A. Fliflet, R. Lee, W. Manheimer, and E. Ott, NRL MemorandumReport No. 6064, 1987.

10. J.E. Walsh, G.L. Johnston, and R.C. Davidson, Bull. Am. Phys..3 Soc. 32, 1853(1987).

7

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