Vol. 7 No. 1 ~ g

ACTA MATHEMATICAE APPLICATAE SIN~CA Jan., 1991

EXISTENCE UNIQUENESS AND ASYMPTOTIC STABILITY OF PERIODIC SOLUTIONS OF A

NONLINEAR. EQUATION IN PHASE LOCKED TECHNOLOGY

Jr~ Ju~ (~ ~) (5%anghai Teach~r~ Unive~'sity)

In [1] and [2], %he authors made a deep quali%a~ive analysis of %he equa%ion ~i%h %he character of %angen% de%eo%ed phase and %hey mathema%ically provided a theore%ical basis of why the phase locked loop has no lock-losing poin%. However, according %o many prao%ical express, i% is ra%her diflicul% %o pu% such a phase locked loop in%o practice, %hough i% has fine properties. W. (3. Lindsey [3] made a eireu2% design wi~h the character of detected phase where

( l ÷ ~ ) s i n ~ (0<~<1) . (1) g(~)= l + k cos ~

He pointed cub %ha% such a oiroui~ can be effec~ed practically. One can see bha% in (1)

g(~)-~2 ~g ~

when 2--1, and %his is just %he oharac%er of the Sangent de%ee%ed phase described in [1] and [2]. Moreover, when k=0 , i% becomes %he resu1% in [4] and [5]. So %he study of %he phase locked loop equation with the charao%0r of %he de~ec%ed phase (1) is of practical significance.

In this paper, we consider %he exis%ence, uniqueness and asymptotic s~abili%y of the periodic solutions of the second order phase locked loop equa%ion

d~ +f(~ d~ d~ (I+ ~)sin -dr "~ ' - ~ / - ~ + 1 + k cos q~

(t), (2)

where ~ is a posi%ive parameCer; f ( ~ , ~) is eon%innous and differen%iable with respec% %o %he variables ~, ~; 6(t) is a continuous function of t. I~ is obvious %ha% g(~) = --g(--~) and g/(~) >0 when WI <~'(~" is a roo~ of %he equation 9/(~) =O). The curve de%ermined by F = g(w) is i l lus t ra ted in Fig. 1.

Rewrite equa%ion (2) as

........... Received on January 1~, 1989. Revised on February 20, 1990.

1~0. 1 EXISTENCE, UNIQUENESS AND ASYMPTOTIC STABILITY... 91

~z -~- = - : (~, ~) ~ - ~ ( ~ ) + ~ (~).

(s)

Now, we study the existence and uniqueness of the periodic solutions of system (8) on the cylinder

.[

Fig. i

/~ ' : {-~<~<~; -oo<~< +oo}.. Theorem 1. If system (3) satisfies: 1. I~(~) I < ~ (~>0), ~(~+~)- ~(~) (~>0);

S. /~ is sufficiently small and satisfies

81.~M ~ f~" ~ -~ < ( ~ ( e ) - ~ ) ~ , (4)

where Wo is a root of ~he equation ~/(W)-~M---0 and 0 < ~ o < ~ ' < ~ , then 1. For each solution of (3)," there exists a number ~o (~o depends on the

solution of system (3)), such that the solution satisfies IW(~)I~A1, [~(~)I~B1 when ~ o , where

1

2. (3) has at least one periodic solution with period ~; S. (A1, B1)-*(0, 0), when/~-*0. That is, when ~-*0, the periodic solutions

of (3) approach the singular point (0, 0) of the homogeneous equation of (3). To prove Theorem 1, we first present several lemmas. Consider ~he control

equation of system (3), dW

f . - ~ - z ,

( 5 ) dz - ~ = - ~ - g ( ~ ) + ~ M

( 6 )

and

--~ - - ~ - g (q,) - ~ M .

Lemma 1. The points in the direction field on the upper half cylinder {-~~~, 0~<+oo} determined by (5) and the points in ~he direction field on the lower half cylinder {-~W~,-o~<z~0} determined ~y (6) are symmetric about the origin, and bhey have the same magnitude bu~ opposite directions.

L e m m a 2. Under the condition that

-~ (g(~) ~ ) , ~ ,

92 ACTA MATHEMATICAE APPLICATAE SINICA Vol. ?

the t-rajeotory of system (3) through (go, Zo) does not intersect the straight 1.~e ~ . ~ * .

Let/~ be suftioiently small such ~hat

<v < (g (q,) - ~ M ) aq,.

If

~hen

1

r" <,,,,]": L. J@e

z1~4~ M . 05

~Tve draw the trrajeetory ,~£ of (5) through the point A(~o, ~a). According to Lemma 2, i t must intersect the q~-axis at B(~I, 0), where ~ o ~ 1 ~ * . Draw the

trajectory BG of (6) through the point B. I t intersects the straight line ~'=~o (a<0) at o (q~o, - z,) (See Fig. 2).

I t is obvious that ~B is outwardly convex.

F i g . 2

I B

~ig. 8

T.emma 8. The ordinates of all the points on the are ABG except that of A(~o, zl) satisfy I zl <z~.

L e m m a 4. The ~rajeo~ories of system (3) pass ~hrough the curve

v=.-ff ~ + g ( e ) ~ - o from outside to inside when

I~i~ 2~M I

L e m m a 5. On the upper (lower) half cylinder, all the trajectories of system (3) enter the trajectories of system (5) (~6)) from outside.

The proofs of these lemmas are omitted. Now, we t u r n to the proof of Theorem 1.

In the q~-z plane, we construct a closed curve zr' which surrounds the orlgin, so that the projection of the direction field of (3) on the W-z plane enters /" from outside. The construction o f / ' is as follows.

Dra the trajectory AB of (5) through

No. 1 EXISTENCE, UNIQUENESS AND ASYMPTOTIC STABILITY... 93

I~ inlerseots the w-axis at ~(gx, 0)(See Fig. 3). From Lemma 2, we know

,hat ~o<~1<~' . Draw ,he trajectory 2~G of (6) through 2L

It follows from Lemma 3 that the ordinates of all the points on ABO sa*isfy

1 I<4 . O;

Looa~e a point D on the ex, ended line of AO, so that D/~ = A H . The ooordinates of D are then

Draw ,he ourve

-4V)

, ±(4.M ]= +

whioh passes through A. By symmeiry, this ore-re mQs¢ pass through

4 ~ M ~ D ' ( - ~Oo, T / "

Then, according to +~he symmetry prinoiple about the origin, we may cons*rue*/ ' : ABODA'B'O'D'A (See Fig. 3). We oan show thai on the ~-z plane, the projections of ,he direction field of (3) all en ,or /~ from ou,side. By Lemma 4, it is true on

AD' and DA ~, and by Lemma 5, it is also true on ABO and A'JB~O '. As for OD and O'D~; one only needs to nofdoe tha# on them we have

d~ - -z<0 d$

and

- z > 0 dt

respec*ively. Therefore, ,he solutions of (3) are all bounded and l ~ l < A z = ~ l , ]z]<Bz when ~ t o . From Massora Theorem [6], we know *hat (3) has a, least oue periodio solu,ion wth period o~.

Furthermore, we show tha, (Az, Bz)--~ (0, 0) when/~--~0. Beoause 1

= , ' , - o= +2~o g(~) where 9o sa,is~es g ( ~ ) - p M = 0 , we eonelude ,ha* ~oo-->0, Bz--)0 when /~--)0.

In ,egra , ing (5) along A/~ gives

$

- - " 2 " \ ........ Ol -1 ~

Lo* ~--,0, we then get

94 ACTA MATHEMATICAE APPLICATAE SINICA Vol. 7'

Since

and

0---- -- ~ : az dqa -- f : g (~) dcp.

.j'~ 9'(~,)~zq,>o,

we muss have 9~ = 0. The proof of Theorem 1 is Shus oomple~od. Now, we prov.o ~he uniqueness and asymptotic s~abili~y of ~ e periodi~

solu$ions of sysSem (3). T h e o r e m 2. Suppose Sha~ f (9 , z), e(~) satisfy all oondi~ions of Theorem 1

and ~ha~f(0, 0 ) - / ~ > 0 . If ~ is suffioionfly small, ~hon system (3) has a unique periodic solution wi~h asymp$o~ie s~abili~y.

Proof. In order ~o verify ~he uniqueness of ~he periodio solution of (3), according ~o [7], we only need $o show Sha~ (3) is extremely s~ablo. Suppose ~ha~ ( ~ (0 , z~ (~)) and ( ~ (0, z~ (t)) are ~-~vo arbiSrary solutions of (3), where ]9~ (~)I A~, 1~(0 [~B~ when ~>~to. Then ~hoy should satisfy

r g~l

t ---~- -~a, d~ (7)

:-~ = -f (q,~, ~D ~- g (q,D + ~ (0 and

a~ _ (8),

-'~ -- - - f (9~, z~) z~-- g (9~) .4-. ~e (t).

Sub,racking (8) from (7), we go~

i ,a(q'~- q'~) ...... ~- ~,

d~

d~

Now we ~ry $o simplify ~he righ, hand side of (9). Firs~ we rewrite:

= -f (q'~, ~) ~ +f (q'~, ~) ~-f (~'~, ~) ~ +f (q'~, ~) ~ +g (~) - g (q,D

= -f(~, ~) (~-~D - [f~(~+o~(~-~), ~+o~@~-zD) (q,~--~)]~

- If', (q,~+o~(~-q,D, ~+O~(~-~D) @~- z~)]~

- u' (q'~ + o= (q,~- q,~) ) (~- q,D, where

o<o,<i (~-I, 2). Pu,

• --q'~-~; v=~-~; ~,(0 =q,~(O +o,(~(0-q,~(O), where

( 9 )

NO. 1 EXISTENCE, UNIQUENESS AND ASYMPTOTIC STABILITY... 95

where I Z ( ~ ) l < ~ , .

Then (9) becomes

--~= - [ ~ J ; ( ~ ( ~ ) , ~(~)) +~'(~-(~))]~- [f(e~, ~,) + ~ f ; ( h ~ ( ~ ) , ~ (O) ]v ,

whore 9' (/~. (~)) = (1 +/~) (cos h. (~) --1-- ~)

(~+~~~(~))~ •

Since f (0, 0) =,8 and g~ (0) - 1, we rewrite (10) as gx

f - ~ -~y,

For eonvenienoe, (11) is expressed in matr ix form

(io>

(n>

d~

where

t/ --i -B '

( o o )

I~ is obvious ¢~ha~ the zero solution of %he homogeneous l inear system

du - T u 0 3 ) d$

of (12) is globally asymptotically s~able because all ~he roo%s of ~he oharae~eris~ie equation of (13) have negative real par~s. In addi%ion, when /~ is sufficiently small, B(t) satisfies ~he conditions of Theorem 2 Jn Chapter "2 of [8]. Thus ~he solution of (12) is globally asymp~o~ioa]ly stable. Therefore, system (3) is extremely s~able and i~ has a un/que asymptotically s~able periodio solution. This, establishes Theorem 2.

References [ 1 ] Wang Lian, Wang Muqm, Quali%ative Analysis of a Ty~ of Nonlinear Differential Equations o f

Second Order with Forcing Terms on the Cylinder, ~teta M a t ~ t i v a B~n~a, 23:5 (1980), 763--772. [ 2 ] Wang Muqiu, Zhang Jingyan, Wang Lian, A Phase Locked Loop Equa%ion with Tangen~

Dise.rimina~or and Frequency Modulation :Lupu~ and ICa Popularization, Scienf~a B4rgva, ~, (1980), 554---565.

[ 3 ] W.C. T,indsey, Synchronza¢ion System in Communication and Conkrol, Pren%ice-B:all T,ne Englewood Cliff~, l~ew Tersey, 1972.

[ 4 ] Yie Dawei, Qualif~ive Analysis of a Phase Locked Loop wif~ Sine Discriminaf~r and ~requency.

96 ACTA MATHEM~TICAE APPLICATAE SINICA Vol. 7

Modulation Input, Ke~ T~gbco, 19 (1983), 1162---1165. [ G ] Wang Rongliang, The Existence and Stability of Periodic Solution of a P-L-L Equation with Sine

Chawacteristic and Frequency Modulation Input, Jo~rz~ oJ ~d~g CoZ~ege of Ocec~oZogy, 2, 1987. [ 6 ] ~assera J. L., The Existence of Periodic Solution of Systems of Differential Equation, Duk8 Ma~h.

J., 17 (1950), 457--475. [ 7 ] Lasalle J. L., Lefsche~ S., Stability by Lyapunov's Direct Method with Application, l~ew York,

Academic Press, 1961. [ 8 ] R. Bellman, The Theory of the Solutions of DiffezenUal Equation, Translated by Zhang X/e, Science

~, 1957.