ISSN 1064�5624, Doklady Mathematics, 2011, Vol. 84, No. 2, pp. 761–764. © Pleiades Publishing, Ltd., 2011.Original Russian Text © G.A. Leonov, A. A. Fyodorov, 2011, published in Doklady Akademii Nauk, 2011, Vol. 440, No. 4, pp. 459–462.

761

Discrete phase locked loops are widespread inradio electronics and electronic engineering. Theimportance of studying the dynamics of such systemswas indicated in [1–8].

In this paper, we obtain a criterion for the nonexist�ence of cycles of given period in discrete phase lockloops. The proof of the criterion is based on the ideas,methods, and techniques described in [3, 9–12].

At present, few analytical tools are available thatperform effectively on systems with chaotic behavior.Below, we describe one of these tools, namely, a crite�rion for the nonexistence of cycles of fixed period,which is applied to two�dimensional discrete systemswith chaotic behavior.

The equations of a discrete phase lock loop havethe form (see [3, 4])

(1)

Here, ϕ(σ) ∈ C1(�) is a 2π�periodic scalar function, band c are constant n�dimensional vectors, A is a con�stant n × n matrix, xt is the state vector, and r is a con�stant.

Suppose that

(2)

for any σ ∈ �, and let there exist points σ' and σ'' suchthat σ' < σ'' < σ' + 2π, ϕ(σ') = ϕ(σ'') = 0, and

(3)

Without loss of generality, we assume that

xt 1+ Axt bϕ σt( ), x �n, t �0,∈ ∈+=

σt 1+ σt c*xt rϕ σt( ).+ +=

μ1dϕdσ����� μ2≤ ≤

ϕ u( ) ud

σ'

σ''

∫sgn ϕ u( ) u.d

σ''

σ' 2π+

∫sgn–=

The transfer function of system (1) from the input ϕ tothe output –σ is defined as

The cycle of period N is a solution of system (1)such that there exists an integer k for which

Theorem 1. Suppose that there exist ε > 0, T > 0, andG such that

(4)

(5)

for all i = 1, 2 and p = 1, , , …, ,respectively.

Then system (1) has no cycles of period N.Proof. Assume that there exist ε > 0, T > 0, and G

such that inequalities (4) and (5) hold for all i = 1, 2and j = 0, 1, …, N – 1 and that there exists a cycle (xt, σt)of period N satisfying system (1).

Without loss of generality, let G ≤ 0. Since μ1 < 0 < μ2,the fulfillment of inequality (4) for i = 1 implies that itholds for i = 2 as well.

Define the function

This definition yields the estimate

M ϕ u( ) ud

σ'

σ''

∫ ϕ u( ) ud

σ''

σ' 2π+

∫≥ m,= =

ν M m–M m+������������� .=

W p( ) c* A pI–( ) 1– b r.–=

xN x0, σN σ0– 2πk.= =

4T ε 12��Gμi 1 ν+( )–⎝ ⎠

⎛ ⎞ Gν( )2,≥

GReW p( ) ε W p( ) 2– T–

+ Re μ1W p( ) p 1–+( ) μ2W p( ) p 1–+( ) 0≥

e2π iN

�������

e4π iN

�������

e2π N 1–( ) i

N����������������������

F σ( )

2mM m+�������������ϕ σ( ) if σ' σ σ''<≤

2MM m+�������������ϕ σ( ) if σ'' σ σ' 2π.+<≤⎩

⎪⎨⎪⎧

=

Frequency Oscillations Estimatesfor Digital Phase�Locked Loops

Corresponding Member of the RAS G. A. Leonov and A. A. FyodorovPresented by Academician S.K. Korovin June 14, 2011

Received June 8, 2011

DOI: 10.1134/S1064562411060275

Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, Peterhof, St. Petersburg, 198504 Russiae�mail: leonov@math.spbu.ru, fyodorov.spbu@gmail.com

CONTROL THEORY

762

DOKLADY MATHEMATICS Vol. 84 No. 2 2011

LEONOV, FYODOROV

(6)

The function F is 2π�periodic; F(σ') = F(σ'') = 0; and,by virtue of (3),

(7)

The Taylor formula gives

From (7), we obtain the estimate

Therefore, for all t,

(8)

Then, combining (8) with the relation

and inequality (6) and assuming that G ≤ 0, we derivethe estimate

(9)

By virtue of condition (4),

Then, for any t,

G F σt( ) ϕ σt( )–( ) σt 1+ σt–( )

≤ GM m–M m+�������������ϕ σ( ) σt 1+ σt–( ) .

F σ( ) σd

0

2π

∫ 0,=

μ12M

M m+������������� dF

dσ�����.≤

Φ b( ) Φ a( ) Φ' a( ) b a–( ) Φ'' θ( ) b a–( )2

2���������������,+ +=

where Φ x( ) F σ( ) σ, θd

a

x

∫ a b,[ ].∈=

F a( ) b a–( ) μ1M

M m+������������� b a–( )2 F σ( ) σ.d

a

b

∫≤+

F σt( ) σt 1+ σt–( ) μ1M

M m+������������� σt 1+ σt–( )2+

≤ F σ( ) σ.d

σt

σt 1+

∫

F σ( ) σd

σt

σt 1+

∫t 0=

N 1–

∑ F σ( ) σd

σ0

σ0 2πk+

∫ 0,= =

0 G F σt( ) σt 1+ σt–( ) μ1M

M m+������������� σt 1+ σt–( )2+

t 0=

N 1–

∑≤

≤ G ϕ σt( ) σt 1+ σt–( ) μ1M

M m+������������� σt 1+ σt–( )2+

t 0=

N 1–

∑

+ GM m–M m+�������������ϕ σ( ) σt 1+ σt–( ) .

t 0=

N 1–

∑

4T ε Gμ1M

M m+�������������–⎝ ⎠

⎛ ⎞ = 4T ε 12��Gμ1 1 ν+( )–⎝ ⎠

⎛ ⎞ Gν( )2≥

= GM m–M m+�������������⎝ ⎠

⎛ ⎞2

.

(10)

Combining (9) with (10) yields

(11)

Since the function ϕ(σ) is 2π�periodic, thesequence ϕ(σt) is N�periodic. Let the sequence Xj bethe componentwise discrete Fourier transform (DFT)of xt, and let Φj be the DFT of ϕ(σt):

Define ρ = . It is easy to see that

In view of the properties of DFTs, we have Xjρj = AXj +bΦj, whence

(12)

The sequence (σt + 1 – σt) is N�periodic. It followsfrom (1) that

(13)

Combining (12) with (13), we see that the Fouriertransform of the sequence (σt + 1 – σt) is

Parseval’s identity for the right�hand side of (11) gives

(14)

ε Gμ1M

M m+�������������–⎝ ⎠

⎛ ⎞ σt 1+ σt–( )2 T ϕ σt( )( )2+

≥ GM m–M m+������������� σt 1+ σt–( )ϕ σt( ) .

0 Gϕ σt( ) σt 1+ σt–( )–[t 0=

N 1–

∑≥

– ε σt 1+ σt–( )2 T ϕ σt( )( )2 ].–

Xj xte2π iN

������� j t–

, Φj

t 0=

N 1–

∑ ϕ σt( )e2π iN

������� j t–

.

t 0=

N 1–

∑= =

e2π iN

�������

xt 1+ ρ j t–

t 0=

N 1–

∑ A xtρj t–

t 0=

N 1–

∑ ρ j t– ϕ σt( ).

t 0=

N 1–

∑+=

c*Xj c* A ρjI–( )1–bΦj– r– W ρj( )–( )Φj.= =

σt 1+ σt– c*xt rϕ σt( ).+=

c*Xj rΦj+ r– W ρj( )–( )Φj rΦj+ W ρj( )Φj.–= =

0 N Gϕ σt( ) σt 1+ σt–( )– ε σt 1+ σt–( )2–[t 0=

N 1–

∑≥

– T ϕ σt( )( )2 ] N Gϕ σt( ) σt 1+ σt–( )–[t 0=

N 1–

∑=

– ε σt 1+ σt–( ) σt 1+ σt–( ) Tϕ σt( )ϕ σt( ) ]–

= GΦj W ρj( )Φj–( )– ε W ρj( )Φj–( )–(j 0=

N 1–

∑

× W ρj( )Φj–( ) TΦjΦj )– GW ρj( )(j 0=

N 1–

∑=

– ε W ρj( )2

T ) Φj2–

= GReW ρj( ) ε W ρj( )2

– T–( ) Φj2.

j 0=

N 1–

∑

DOKLADY MATHEMATICS Vol. 84 No. 2 2011

FREQUENCY OSCILLATIONS ESTIMATES 763

By virtue of (2), for any t, we have the estimate

Summing this inequality over t from zero to N – 1and applying Parseval’s identity to the resulting sumproduces

(15)

Adding up (14) and (15), we obtain

This sum is nonnegative by virtue of (5). This con�tradiction proves the theorem.

Example. Consider a phase locked loop with a sinephase�detector characteristic and a proportionallyintegrating filter [4]:

(16)

Here, ϕ(σ) = sin(σ + ) – sin , where sin is therelative initial mismatch, b ∈ �, d ∈ �, and ∈

0, . The transfer function of this system is given by

the formula

It is easy to see that

It is straightforward to show that ν is a monotonically

increasing function of . Since ∈ 0, , we con�

clude that ν ∈ [0, 1]. Let

Condition (5) becomes

ϕ σt 1+( ) ϕ σt( )– μ1 σt 1+ σt–( )–( )

× ϕ σt 1+( ) ϕ σt( )– μ2 σt 1+ σt–( )–( ) 0.≤

0 N ϕ σt 1+( ) ϕ σt( )– μ1 σt 1+ σt–( )–( )t 0=

N 1–

∑≥

× ϕ σt 1+( ) ϕ σt( )– μ2 σt 1+ σt–( )–( )

= ρjΦj Φj– μ1 W ρj( )Φj–( )–( )j 0=

N 1–

∑

× ρjΦj Φj– μ2 W ρj( )Φj–( )–( )

= ρj 1– μ1W ρj( )+( ) ρj 1– μ2W ρj( )+( ) Φj2.

j 0=

N 1–

∑

0 GReW ρj( ) ε W ρj( )2

– T–[j 0=

N 1–

∑≥

+ ρj1– μ1W ρj( )+( ) ρj

1– μ2W ρj( )+( ) ] Φj2.

xt 1+ bxt b 1–( )ϕ σt( ), b 1,≠+=

σt 1+ σt 1 d–( )xt dϕ σt( ).–+=

σ̃ σ̃ σ̃σ̃

π2��

W p( ) pd 1 b– d–+p b–

��������������������������� .=

μ1 1, μ2– 1, ν ν σ̃( ) π σ̃sin2 σ̃cos σ̃ σ̃sin+( )�����������������������������������,= = = =

σ̃ σ̃ π2��

ν 13��, G< 2

1 3ν–������������, T ν

1 3ν–������������, ε 2ν 1+

1 3������������.= = =

(17)

Let us apply the theorem to the case of period 3.For j = 0, condition (5) holds automatically. For j = 1, 2,it can be rewritten as

ν ≤ .

Thus, there is no cycle of period 3 in system (16) for

(18)

In what follows, d is assumed to equal –0.5. Figure 1shows the domain Ω1 defined by condition (18), wherethere are no cycles of period 3.

There is a widely known analogue of Theorem 1 [3]that provides a sufficient condition for the globalasymptotic stability of systems of type (1). One of itsconditions is that inequality (5) holds for all complex psuch that |p| = 1. In Fig. 1, the domain defined by thiscondition is denoted by Ω2.

21 3ν–������������ReW ρj( ) 2ν 1+

1 3ν–������������ W ρj( )

2– ν

1 3ν–������������–

+ Re W ρj( )– ρj 1–+( ) W ρj( ) ρj 1–+( ) 0.≥

b2 bd– 2b 2d2– 3d+ +

3b2 bd– 4b d2– d 3+ + +���������������������������������������������������

ν 0; min 13�� b2 bd– 2b 2d2– 3d+ +

3b2 bd– 4b d2– d 3+ + +���������������������������������������������������,

⎩ ⎭⎨ ⎬⎧ ⎫

.∈

0.3

0.2

0.1

0

ν

−5 −4 −3 −2 −1 0 1 2 b

PΩ1

Ω2 ⊂ Ω1 Ω2 ⊂ Ω1

Ω1

Fig. 1. Domain with no cycles of period 3 for d = –0.5.

b3 −2.5 0 b2 1 b1 b

Fig. 2. Bifurcation diagram of system (16).

x

0

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DOKLADY MATHEMATICS Vol. 84 No. 2 2011

LEONOV, FYODOROV

Let us present the results of computer simulation ofsystem (16). The point P = (b, ν) = (–2.5, 0.1) (ν = 0.1corresponds to ≈ 0.0638) is outside Ω1. Numericalcomputation has shown that system (16) with theseparameters and the initial data x0 = σ0 = 0.1 has a cycleof period 3.

Let us describe the behavior of system (16) for dif�ferent values of b with fixed d = –0.5, ν = 0.1, and theinitial data x0 = σ0 = 0.1. For b > b1 ≈ 1.122, xt → ∞.For b ∈ (1, b1), the system has the locally stable equi�librium (x, σ) = (0, 0), to which the solution of system(16) with the indicated initial data tends as t → ∞. Atthe point b = 1 xt coincides with x0 for all t. A stabilityloss bifurcation and the birth of chaos are observed atthis point. For b ∈ (b2, 1), where b2 ≈ 0.718, xt tends ast → ∞ to a chaotic attractor. For b ∈ (b3, b2), whereb3 ≈ –3.122, xt tends as t → ∞ to either a chaoticattractor or a cycle. Finally, for b < b3 xt → ∞. Thebifurcation diagram illustrating the behavior describedabove is shown in Fig. 2.

ACKNOWLEDGMENTS

This work was supported by the Ministry for Educa�tion and Science of the Russian Federation and bySt. Petersburg State University, project no. 6.37.98.2011.

REFERENCES

1. W. C. Lindsey and C. M. Chie, Proc. IEEE, 410–431(1981).

2. H. C. Osborne, IEEE Trans. Commun. 28, 1355–1364(1980).

3. G. A. Leonov and V. B. Smirnova, Mathematical Prob�lems in Phase Synchronization Theory (Nauka,St. Petersburg, 2000) [in Russian].

4. G. A. Leonov and S. M. Seledzhi, Phase Locked Loopsin Analog and Digital Electronics (Nevskii Dialekt,St. Petersburg, 2004) [in Russian].

5. Phase Locked Loops with Discretization Elements, Ed. byV. V. Shakhgil’dyan (Radio i Svyaz’, Moscow, 1989) [inRussian].

6. G. A. Leonov and S. M. Seledzhi, Int. J. BifurcationChaos 15, 1347–1360 (2005).

7. G. A. Leonov and S. M. Seledzhi, Int. J. InnovativeComput. Inf. Control 1, 779–789 (2005).

8. G. A. Leonov and V. B. Smirnova, Math. Nachr. 177,157–181 (1996).

9. G. A. Leonov, Chaotic Dynamics and Classical Theory ofMotion Stability (RKhD, Moscow, 2006) [in Russian].

10. G. A. Leonov, Avtom. Telemekh., No. 10, 47–85(2006).

11. Yu. N. Bakaev and A. A. Guzh, Radiotekh. Elektron.10 (1), 175–196 (1965).

12. G. A. Leonov and A. A. Fyodorov, Dokl. Math. 82,651–654 (2010).

σ̃