phase synchronization: theory and applications

ISSN 0005-1179, Automation and Remote Control, 2006, Vol. 67, No. 10, pp. 1573–1609. c© Pleiades Publishing, Inc., 2006.Original Russian Text c© G.A. Leonov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 10, pp. 47–85.

TOPICAL ISSUE

Phase Synchronization: Theory and Applications1

G. A. Leonov

St. Petersburg State University, St. Petersburg, RussiaReceived February 7, 2006

Abstract—The state-of-the-art of the phase synchronization theory was reviewed. Consider-ation was given to its applications to the synchronous and induction electrical motors, phaselocked loops, and autosynchronization of the unbalanced rotors. The Yakubovich–Kalman fre-quency theorem was widely used to study the phase synchronization systems for global stability.

PACS numbers: 02.30.Yy, 01.60.+q

DOI: 10.1134/S0005117906100031

1. INTRODUCTION

The effect of phase synchronization is frequently encountered. Electrical current is generatedby synchronous electrical generators based on the principle of phase synchronization. Concurrentlywith switching on of the TV set, the special phase synchronization system controlling the time-basegenerator is activated. The floating systems of phase synchronization are currently used to controlthe distributed systems of clock generators in multiprocessor clusters (synchronization of the clocksin a multiprocessor system).

The wide spectrum of applications of the phase synchronization systems is explained by thediversity of their mechanical, electromechanical, and electronic realizations. Yet it is the principleof phase synchronization that is universal: the difference of oscillation phases is transformed in thecontrol action on the generator frequency which results in synchronization of oscillations.

The general theory of phase synchronization was formed in the second half of the last century asthe result of generalization of the methods developed within the framework of the three independenttheories of

(1) synchronous and induction electrical motors,(2) phase locked loops, and(3) autosynchronization of the unbalanced rotors.The present review discusses those divisions of the aforementioned theories within which the

concepts and methods were matured which later developed into the general theory of phase syn-chronization. Described are the three levels of consideration of the phase synchronization problems:(i) mechanical, electromechanical, and electronic realizations, (ii) phase and frequency relations,and (iii) differential, difference, integral, or integro-differential equations. The main notions of thephase synchronization theory are presented. The methods of global study of the phase synchro-nization systems are described such as periodic Lyapunov functions, positive invariant cone grids,and nonlocal reduction.

An attempt is made here to present the basic notions and methods of the phase synchronizationtheory at a possibly simplistic level for as wide as possible circle of the experts in the fields where1 This work was supported by the Russian Foundation for Basic Research, project no. 04-01-00250A, and Program

of the Russia-Netherlands Cooperation NWO-RFFI no. 047.011.2004.004.

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1574 LEONOV

this theory may find application. To this end, the description of the existing methods was basicallyupdated and new, simpler proofs of many results are given. Additionally, interpenetration of themethods and ideas of various particular theories relying on the concepts of phase synchronizationwas demonstrated.

2. SYNCHRONOUS AND INDUCTION ELECTRICAL MOTORS

The pioneering paper of F. Tricomi [1] laid the groundwork for many principles of the generalphase synchronization theory. It dealt with the global analysis of the differential equations of thesynchronous electrical machine. This study was immediately noticed and set forth in the well-knownbook [2]. Later on the Tricomi methods were developed and updated by his numerous followers[3–9]. We consider the simplest mathematical models of synchronous and induction electric motors,discuss them from the point of view of the phase synchronization theory, and describe the mainconcepts of Tricomi.

We describe the following electromechanical model which allows us to consider the dynamics ofthe synchronous and induction electric motors from the general point of view. The main elements ofthese electric motors are the stator and rotor. The former has windings through which alternatingcurrent passes and creates an alternating magnetic field. We assume that the stator windings areconstructed so that as alternating current passes through them the of magnetic field intensity isconstant in magnitude and rotates with a constant angular velocity [10–12] (Fig. 1). This rotatingmagnetic field was first obtained by H. Ferraris and N. Tesla in 1888 [10]. It is clear that the fieldrotation frequency coincides with that of alternating current passing through the stator windings.We assume that the electromagnetic processes in the rotor windings do not affect the parametersof the rotating magnetic field, that is, disregard their impact on the stator currents.

We also assume that two mutually perpendicular windings are situated in the rotor slots asschematized in Fig. 2. A coil of this winding is shown in the figure as a frame with current i1(or i2). In the synchronous machine, one frame—the so-called sampling winding—is closed, andconstant voltage e is applied to the other frame usually through the brushes of the electric motor.This winding is referred to as exciting. In the induction motor, both frames are closed. In thiscase, the windings are said to be short-circuited. Obviously, here e = 0.

We consider motion of the frames in a rotating coordinate system rigidly related with the vectorof magnetic field intensity. In this case, the frame currents i1(t) and i2(t) are determined subject

N

S

i

1

i

2

Fig. 1. Rotating magnetic field. Fig. 2. Diagram of windings.

AUTOMATION AND REMOTE CONTROL Vol. 67 No. 10 2006

PHASE SYNCHRONIZATION 1575

to the law of electromagnetic induction and the Ohm law:

Ldi1(t)dt

+Ri1(t) = e+ SB(sin θ(t))θ(t),

Ldi2(t)dt

+Ri2(t) = SB(cos θ(t))θ(t),(2.1)

where R and L are, respectively, resistance and inductance of each frame which are assumed hereto be identical, B is the magnetic field intensity, S is the area of each frame, and θ(t) is theangle between the plane of the frame with current i1 and the plane perpendicular to the vector ofmagnetic field intensity.

The motion of the rotor with frames relative to the rotating magnetic field is as follows:

Iθ = −βSB(i1(t) sin θ + i2(t) cos θ) −M, (2.2)

where I is the rotor moment of inertia, β is the proportionality coefficient, and M is the momentof the resistance forces (the so-called load moment).

We first assume that L = 0. In this case, by substituting (2.1) in Eq. (2.2), we obtain

Iθ = −βS2B2

Rθ − βSBe

Rsin θ −M. (2.3)

By redefining θ → −θ, this equation can be put down as

θ + αθ + b sin θ = γ, (2.4)

where

α =βS2B2

IR, b =

βSBe

IR, γ =

M

I.

In what follows, for the synchronous machine we assume without loss of generality (e > 0) thatb = 1. By changing time τ = t

√b, Eq. (2.4) can be reduced to an equation of this form.

The equilibrium state of Eq. (2.4)

θ(t) ≡ θ0, θ ≡ 0, (2.5)

where θ0 meets the relations

sin θ0 = γ, cos θ0 > 0,

corresponds to the working mode of the synchronous machine. This mode is possible if γ < 1.In this case, it is locally asymptotically stable. The equilibrium states (2.5) with cos θ0 < 0 areunstable saddle singularities. We note that the stationary sets of system (2.1), (2.2) coincide forL = 0 and L > 0. Local stability and instability of the stationary solutions is also retained atpassing from Eq. (2.4) (case of L = 0) to system (2.1), (2.2) (case of L ≥ 0).

Tricomi’s global analysis of Eq. (2.4) demonstrated that there exists some number α(γ) for whichall solutions of Eq. (2.4) for t→ +∞ tend to the equilibrium states if α > α(γ). If α < α(γ), thenthere are the so-called circular solutions for which θ(t) ≥ ε, ∀ t ≥ t0, where ε > 0 and t0 are somenumbers. In the first case, there exists global pulling in the synchronous mode (singular solutionstending to unstable saddles are physically unobservable). In the second case, disalignment occursunder some initial conditions θ(0), θ(0).

The solution

θ(t) ≡ −s0t+ θ0, θ(t) ≡ −s0,where θ0 is a number, corresponds to the operating mode of the induction electric motor. Thenumber s0 depends on L and, therefore, the formulas for its calculation differ for (2.4) (L = 0) and(2.1), (2.2) (L > 0).


1576 LEONOV

For e = 0, system (2.1), (2.2) can be rearranged in a third-order system. For that, we considerthe following transformations:

θ1 = −θ, s = θ1, x =L

SB(i1 cos θ1 + i2 sin θ1) , y =

L

SB(−i1 sin θ1 + i2 cos θ1) .

In terms of the new variables, system (2.1), (2.2) can be put down as

s = ay + γ,

y = −cy − s− xs,

x = −cx+ ys,

(2.6)

where

a =β(SB)2

IL, c =

R

L.

We note that s0 for Eq. (2.4) is established from s0 = γ/α and for system (2.6), fromacs0c2 + s20

= γ. (2.7)

Therefore, in distinction to the synchronous machine, Eq. (2.4) describes the induction motorinadequately. System (2.6) is the simplest adequate description of the dynamics of the inductionmotor. The function

ϕ(s0) =acs0c2 + s20

is called the static characteristic of the induction motor.The Jacoby matrix in the right-hand side of system (2.6) at the stationary point s = s0, y0 =

−γ/a, x0 = −γs0/ac is as follows:⎛⎜⎝

0 a 0−1 − x0 −c −s0y0 s0 −c

⎞⎟⎠ .

With regard for (2.7), its characteristic polynomial is representable as

p3 + 2cp2 +

(c2 + s20 +

ac2

c2 + s20

)p+ ac

c2 − s20c2 + s20

.

Hence, it follows that the stationary point s0, y0, x0 is locally asymptotically stable for s0 < c andunstable for s0 > c.

In the absence of load (γ = 0), pulling in synchronism occurs for any initial data:

limt→+∞x(t) = lim

t→+∞ y(t) = limt→+∞ s(t) = 0.

This global stability of the zero solution of system (2.6) is easily proved using the Lyapunovfunction

V (x, y, s) = x2 + y2 +s2

a.

Then, the load is reset. This is typical of the induction electric motor used as the drive of a metal-cutting machine. Here, it is again important that in the new transient mode the solution of system(2.6) with nonzero γ and the initial conditions x = y = s = 0 tends to the stationary point x0, y0, s0for t → +∞. A very specific type of synchronism “with subtraction of the frequency components0” is observed here. The conditions where these transients are feasible will be described in whatfollows.



θ

0

– 2

π

η

S

θ

0

θ

Fig. 3. Saddle separatrix.

Here, we present some basic results of Tricomi by passing from Eq. (2.4) to the equivalent system

θ = η,

η = −αη − ϕ(θ),(2.8)

where ϕ(θ) = sin θ − γ.

Theorem 1. Any solution of system (2.8) which is bounded for t ≥ 0 tends to some equilibriumstate for t→ +∞.

This theorem can be readily proved by constructing a Lyapunov type function

V (η, θ) =12η2 +

θ∫

0

ϕ(θ)dθ

and using standard reasoning within the framework of the Lyapunov method [13, 14].For system (2.8), γ < 1 is, obviously, the necessary existence condition for the asymptotically

stable (in the small) equilibrium state corresponding to the stable synchronous operating mode ofthe electric motor. In this case, system (2.8) has the saddle equilibrium state η = 0, θ = θ0 + 2kπ,where θ0 satisfies the conditions

sin θ0 = γ, cos θ0 < 0, θ0 ∈(π

2, π

),

and the asymptotically stable (in the small) equilibrium state η = 0, θ = θ1+2kπ, where θ1 satisfiesthe conditions

sin θ1 = γ, cos θ1 > 0, θ1 ∈(

0,π

2

).

We fix the parameter γ ∈ (0, 1) and vary the parameter α ∈ [0,+∞). For α = 0, system (2.8) isintegrable, and one can easily see that for the separatrix θ(t)+, η(t)+ of the saddle θ = θ0, η = 0,meeting the conditions

limt→+∞ η(t)+ = 0, lim

t→+∞ θ(t)+ = θ0,

η(t) > 0, ∀ t ∈ (T,+∞),

where T is a number, there exists a number τ such that

η(τ)+ = 0, θ+(τ) ∈ (θ0 − 2π, θ0),η(t)+ > 0, ∀ t > τ

(2.9)

(Fig. 3).


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θ

0

– 2

π

η

S

θ

0

θ

Fig. 4. Lower estimate of the saddle separatrix.

Now, we consider the line segment

η = −K(θ − θ0), θ ∈ [θ0 − 2π, θ0].

It is easy to see that on this segment the relations

(η +K(θ − θ0))• = −αη +Kη − sin θ + γ = (θ − θ0)(−K(K − α) +

γ − sin θθ − θ0

)

are satisfied. By using the obvious inequality∣∣∣∣γ − sin θθ − θ0

∣∣∣∣ ≤ 1, ∀ θ = θ0

and assuming that

α > 2,α

2−√α2

4− 1 < K <

α

2+

√α2

4− 1,

for η = −K(θ − θ0), θ ∈ (θ0 − 2π, θ0), we get the estimate

(η +K(θ − θ0))• < 0 (2.10)

from which it follows (Fig. 4) that in the band {η, θ | θ ∈ (θ0 − 2π, θ0)} the separatrix θ(t)+, η(t)+

will lies above the line segment {η, θ | η = −K(θ − θ0), θ ∈ (θ0 − 2π, θ0)}. Hence, for α > 2 thereexists no number τ for which relations (2.9) are satisfied.

It is notorious that a piece of the trajectory S : {η(t)+, θ(t)+ | t ≥ τ} is continuously dependenton the parameter α from which and the above considerations the following well-known result isobtained.

Theorem 2 ([1]). For any γ > 0, there exists a number α(γ) ∈ (0, 2] such that system (2.8)with such parameters γ and α(γ) has in the cylindrical phase space {η, θ mod 2π} the homoclinictrajectory

limt→+∞ η(t) = lim

t→−∞ η(t) = 0,

limt→+∞ θ(t) = θ0, lim

t→−∞ θ(t) = θ0 − 2π.

It is also easy to demonstrate that α(γ) is defined uniquely from γ and is a monotonicallyincreasing function. The following arrangement of the trajectories in the phase space of system(2.8) ensues from Theorems 1 and 2.

(1) For α > α(γ), the separatrices tending to the saddle equilibria of system (2.8) are theboundaries of the attraction domains of the asymptotically stable equilibrium states for t → +∞.It can be shown [9] that the entire phase space is decomposed into such attraction domains (Fig. 5).



θ

0

– 4

π

η

θ

1

θθ

1

– 2

π θ

0

– 2

π θ

0

θ

1

+ 2

π θ

0

+ 2

π

Fig. 5. Global stability.

θ

0

– 4

π

η

θ

1

θθ

1

– 2

π θ

0

– 2

π θ

0

θ

1

+ 2

π θ

0

+ 2

π

Fig. 6. Homoclinic bifurcation

θ

0

– 4

π

η

θ

1

θθ

0

– 2

π θ

0

θ

1

– 2

π

Fig. 7. Case of α < α(γ).

(2) For α = α(γ), the attraction domains of the stable equilibrium states are also bounded bysuch separatrices. However, these domains do not fill the entire phase space any more.

In the domain lying over the homoclinic orbits, all trajectories tend to infinity for t→ +∞ andapproach the set consisting of the homoclinic orbits (Fig. 6).


1580 LEONOV

(3) For α < α(γ), the instability corridors appear between the separatrices (Fig. 7).In these corridors all trajectories tend to infinity for t → +∞. The entire phase space is

decomposed into such corridors and the attraction domains of the stable equilibrium states.Therefore, case 1 corresponds to global stability: all trajectories tend to the equilibrium states

for t → +∞. For the synchronous machine, this corresponds to damping of rotor hunting in anytransient mode, and as the result the machine enters the synchronous mode.

Case 3 is possible both as a transition to the synchronous mode and as machine fall out ofsynchronism, which depends on the initial conditions θ(0), η(0). We present here the best existinganalytical estimates of α(γ).

Theorem 3 ([5, 6]).

α(γ) < 2 sin(

12θ1

).

We note that for small γ this estimate becomes

α(γ) < γ. (2.11)

Theorem 4 ([6]).

α(γ)2 >√

3(cos θ1)2 + 1 − 2 cos θ1.

Calculation of α(γ) by numerical methods is discussed in [15, 16]. Generalization of the methodsand results presented above to more complex models of synchronous machines is described in[14, 17]. Other bifurcations in the phase synchronization systems such as period duplication, An-dronov–Hopf, or semistable cycle were considered and discussed in [7, 8, 15–24].

3. PHASE LOCKED LOOPS

The phase locked loops (PLL) are widely spread in radio engineering and telecommunications.After their advent in the 1930’s–1940’s, both the theory and practice of the PLL systems wereactively developed [15–19, 25–36]. In the recent decade, the PLL systems found wide use in thecontrol of the clock generators of the digital signal processors, multiprocessor clusters, and otherfacilities for digital information processing [16, 24, 37–44].

We describe here the main principles of design and analysis of the phase locked loops. Since the1960’s it became a tradition in this field to use three levels of PLL description:

(1) level one—description of the electronic realization of the system,(2) level two—description of the block diagram depicting the phase relations,(3) level three—description in terms of the differential, integral, or difference equations.We present here a now-classical description of the PLL with generators of harmonic oscillations

[25, 28]. We consider the PLL at the first level (Fig. 8) where the master oscillator (MO) and the

Fig. 8. Electronic circuit of the PLL.



Fig. 9. Multiplier and filter.

θ

θ

Fig. 10. Phase detector and filter.

slave oscillator (SO) produce “almost harmonic high-frequency oscillations”

fj(t) = Aj sin(ωj(t)t+ ψj).

The unit × multiplies the oscillations f1(t) and f2(t) and outputs f1(t)f2(t). The input, ξ(t),and output, σ(t), of the linear filter are related as follows:

σ(t) = α0(t) +t∫

0

γ(t− τ)ξ(τ) dτ,

where γ(t) is the pulse transient function of the filter and α0(t) is an exponentially decaying functionwhich depends on the initial state of the filter at t = 0. The electronic realizations of generators,multipliers, and filters can be found in [37, 45].

Now, we reformulate the high-frequency property of the oscillations fj(t) in the following con-dition. We consider a long time interval [0, T ] which can be decomposed into smaller intervals like[τ, τ + δ], (τ ∈ [0, T ]) where the following relations are satisfied:

|γ(t) − γ(τ)| ≤ Cδ, |ωj(t) − ωj(τ | ≤ Cδ,

∀ t ∈ [τ, τ + δ], ∀ τ ∈ [0, T ],|ω1(τ) − ω2(τ)| ≤ C1, ∀ τ ∈ [0, T ],

ωj(t) ≥ R, ∀ t ∈ [0, T ].

(3.1)

It is assumed here that δ is sufficiently small as compared with the fixed numbers T,C,C1, and thenumber R is sufficiently great as compared with the number δ. The latter fact implies that oversmall time intervals [τ, τ+δ] the functions γ(t) and ωj(t) are almost constant and the functions fj(t)oscillate on them rapidly as harmonic functions. It is also clear that these conditions take place forhigh-frequency oscillations.

Now, we consider two block diagrams shown in Figs. 9 and 10. Here, θj(t) = ωj(t)t+ ψj whichare called the phases of oscillations fj(t) and PD is the so-called phase detector (discriminator),a nonlinear unit with the characteristic ϕ(θ). The phases θj(t) are fed into the PD input, and itsoutput is the function ϕ(θ1(t) − θ2(t)).

The signals f1(t)f2(t) and ϕ(θ1(t) − θ2(t)) arrive to the identical filters with the identical pulsetransient functions γ(t). The filters output the functions g(t) and G(t), respectively. The classicaldesign of the PLL’s is based on the well-known result.


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θ

θ

Fig. 11. PLL block diagram at the level of phase relations.

Theorem 5. If conditions (3.1) and

ϕ(θ) =12A1A2 cos θ

are satisfied, then for the same initial states of the filter, the relation

|G(t) − g(t)| ≤ C3δ, ∀ t ∈ [0, T ]

is true, where C3 is some number independent of δ.

Therefore, the outputs g(t) and G(t) of the two block diagrams in Figs. 9 and 10 differ littlefrom each other, and one may pass (from the point of view of asymptotics in δ) to the next levelof description, that of phase relations 2. In this case, the block diagram in Fig. 8 transforms intothe block diagram in Fig. 11.

Proof of Theorem 5. For t ∈ [0, T ], obviously,

g(t)−G(t)=t∫

0

γ(t− s) [A1A2 sin(ω1(s)s+ψ1) sin(ω2(s)s+ψ2)−ϕ(ω1(s)s−ω2(s)s+ψ1 −ψ2)] ds

= −A1A2

2

t∫

0

γ(t− s) [cos ((ω1(s) + ω2(s))s+ ψ1 + ψ2)] ds.

We consider the intervals [kδ, (k + 1)δ], where k = 0, . . . ,m, and the number m such that

t ∈ [mδ, (m + 1)δ].

It follows from conditions (3.1) that for any s ∈ [kδ, (k + 1)δ] the following relations are validover each interval [kδ, (k + 1)δ]:

γ(t− s) = γ(t− kδ) +O(δ), (3.2)ω1(s) + ω2(s) = ω1(kδ) + ω2(kδ) +O(δ). (3.3)

Then, for any s ∈ [kδ, (k + 1)δ] the estimate

cos ((ω1(s) + ω2(s))s + ψ1 + ψ2) = cos ((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2) +O(δ) (3.4)

follows from (3.3). And from (3.2) and (3.4) it follows that

t∫

0

γ(t− s) [cos ((ω1(s) + ω2(s))s + ψ1 + ψ2)] ds

=m∑

k=0

γ(t− kδ)

(k+1)δ∫

kδ

[cos ((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2)] ds +O(δ). (3.5)



The estimate(k+1)δ∫

kδ

[cos ((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2)] ds = O(δ2)

follows from the last inequality in (3.1) and the fact that R is sufficiently great as compared with δ.Whence,

t∫

0

γ(t− s) [cos ((ω1(s) + ω2(s))s+ ψ1 + ψ2)] ds = O(δ).

The theorem is proved.We consider the high-frequency pulse generators connected as shown in Fig. 8. Here,

fj(t) = Aj sgn sin(ωj(t)t+ ψj).

The oscillations of this kind are characteristic of the pulse sequence of the clock generators [37–47] centered relative to the zero voltage. Here, the oscillation multiplier can be built around the“Exclusive OR” gate [46, 47]. We assume here as before that conditions (3.1) are met.

Now, we consider the 2π-periodic function ϕ(θ) like

ϕ(θ) =

{A1A2(1 + 2θ/π) for θ ∈ [−π, 0]A1A2(1 − 2θ/π) for θ ∈ [0, π]

(3.6)

and the block diagrams from Figs. 9 and 10.

Theorem 6 ([42–44]). If conditions (3.1) are satisfied and the characteristic of the phase detectorϕ(θ) has the form (3.6), then the relation

|G(t) − g(t)| ≤ C4δ, ∀ t ∈ [0, T ]

is valid for the same initial states of the filter. Here, C4 is some number independent of δ.

Proof. Obviously, for t ∈ [0, T ]

g(t) −G(t) =t∫

0

γ(t− s) [A1A2 sgn [sin(ω1(s)s+ ψ1) sin(ω2(s)s+ ψ2)]

−ϕ(ω1(s)s− ω2(s)s + ψ1 − ψ2)] ds.

As in the proof of Theorem 5, we consider the intervals [kδ, (k + 1)δ] and make use of relations(3.2)–(3.4) in order to establish the estimate

t∫

0

γ(t− s)[A1A2 sgn

[sin(ω1(s)s+ ψ1) sin(ω2(s) + ψ2)

] − ϕ(ω1(s)s− ω2(s)s + ψ1 − ψ2)]ds

=m∑

k=0

γ(t− kδ)

⎡⎢⎣

(k+1)δ∫

kδ

A1A2 sgn[cos((ω1(kδ) − ω2(kδ)kδ + ψ1 − ψ2)

− cos((ω1(kδ) + ω2(kδ))s + ψ1 + ψ2)]ds− ϕ((ω1(kδ) − ω2(kδ) + ψ1 − ψ2)δ

⎤⎥⎦ +O(δ2)


1584 LEONOV

from which and the relation

A1A2

(k+1)δ∫

kδ

sgn [cosα− cos(Rs+ ψ0)] ds = ϕ(α)δ +O(δ2)

that is satisfied for any α ∈ [−π, π], any ψ0, and R sufficiently great as compared with δ, we getthe statement of Theorem 6.

Theorem 6 underlies design of the phase locked loops with pulse generators. For the pulseclock generators, it allows one to consider simultaneously two block diagrams: at the levels ofelectronic realization (Fig. 8) and phase relations (Fig. 11) where the general principles of the phasesynchronization theory can be applied. Therefore, it is possible to develop a phase synchronizationtheory of the distributed systems of clock generators in the multiprocessor clusters [42–44].

Now, we make a remark necessary to derive the PLL differential equations. We consider

θj(t) = ωj(t) + ωj(t)t.

For a correctly designed, that is, globally stable, PLL, ωj(t) decays exponentially:

|ωj(t)| ≤ Ce−αt.

Here, C and α are some positive numbers independent of t. Therefore, as a rule ωj(t)t is sufficientlysmall as compared with R (see condition (3.1)).

It may be concluded from the above relations that the following approximate equality exists:

θj(t) = ωj(t). (3.7)

The block diagram of Fig. 11 and relation (3.7) which is regarded as precise are used to derive thePLL differential equations.

We note that the law of control of the slave oscillator is assumed to be linear:

ω2(t) = ω2(0) + LG(t), (3.8)

where ω2(0) is the initial frequency of the slave oscillator, L is some number, and G(t) is the controlsignal which is the filter output (Fig. 11). Therefore, the PLL equation is given by

θ2(t) = ω2(0) + L

⎛⎝α0(t) +

t∫

0

γ(t− τ) ϕ(θ1(τ) − θ2(τ))dτ

⎞⎠ .

Hence, by assuming that the master oscillator is highly stable, that is, ω1(t) ≡ ω1(0), we obtainthe following PLL equations:

(θ1(t) − θ2(t))• + L

(α0(t) +

t∫

0

γ(t− τ)ϕ(θ1(τ) − θ2(τ))dτ

)= ω1(0) − ω2(0). (3.9)

This is the equation of a typical PLL [15, 21].We note, that if the filter is integrating with the transfer function

K(p) =1

p+ α

and ϕ(θ) = sin θ, then Eq. (3.9) is equivalent to Eq. (2.4) with θ = θ1−θ2, b = L, γ = ω1(0)−ω2(0).



By the same arguments as in the proofs of Theorems 5 and 6, we can obtain the relation

t∫

0

[f1(τ)f2(τ) − ϕ(θ1(τ) − θ2(τ))]dτ = O(δ)

which is a counterpart of Theorems 5 and 6 where K(p) = p−1. It follows from (3.4) and (3.5) thatthe slave oscillator can be regarded as an ideal integrator with the transfer function K(p) = Lp−1.Hence, filters with more general transfer functions must be used in PLL’s:

K(p) = a+W (p),

where a is a number and W (p) is a correct rational function. In this case, we obtain the equation

(θ1(t) − θ2(t))• + L

[a(ϕ(θ1(t) − θ2(t)) + α0(t) +

t∫

0

γ(t− τ)ϕ(θ1(τ) − θ2(τ))dτ

]

= ω1(0) − ω2(0) (3.10)

instead of Eq. (3.9).

4. AUTOSYNCHRONIZATION OF THE UNBALANCED ROTORS

The phenomenon of autosynchronization of unbalanced rotors resting on a common elastic foun-dation was discovered in the mid-Twentieth century [48–55]. Let us consider it by the example ofa two-vibrator system (Fig. 12) [55]. The electrical drive of the left vibrator develops the powermoment rotating the vibrator with the given angular speed ω. The drive of the right vibratordevelops a power moment M ≥ 0.

By disregarding the impact of dynamics of the right vibrator on the motions of platform andleft vibrator (recall a similar assumption for the stator windings when considering the models ofelectric motors), we get that the point of suspension of the physical pendulum represented by theright vibrator makes vertical oscillations of the form A cos ωt.

It is common knowledge [55] that in this case the dynamics of the right rotor obeys the differentialequation

Iϕ+ kϕ+meAω2 cosωt sinϕ = mge sinϕ+M, (4.1)

where I is the rotor moment of inertia, m is its mass, e is eccentricity, k is the viscous frictioncoefficient, and ϕ is the angle of rotor rotation.

Fig. 12. Platform with debalanced vibrators.


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We denote

θ = ϕ− ωt

and rearrange Eq. (4.1) in

Iθ + kθ +meAω2

2(sin θ + sin(θ + 2ωt)) −mge sin(θ + ωt) = M − kω. (4.2)

Now, we use the procedure developed at PLL analysis and design. We denote

α =k

I, β =

meAω2

2I, γ =

M − kω

I, ρ =

mge

I,

and assume that ω is large enough as compared with α,α−1, β, γ, and ρ. We also assume that

β > |γ|. (4.3)

In this case, in the phase space the equation

σ + ασ + β sinσ = γ (4.4)

has the attraction domains of the stable equilibrium positions (Fig. 7).We consider Eq. (4.4) as the comparison equation for Eq. (4.2). We demonstrate that over

any finite interval [0, T ] and for sufficiently great ω the difference of the solutions θ(t) and σ(t) ofEqs. (4.2) and (4.4) with the same initial data

θ(0) = σ(0), θ(0) = σ(0), |θ(0)| ≤ 2β + |γ| + ρ

α(4.5)

is sufficiently small. For that we put down Eq. (4.2) in the integral form

θ(t) = ν(t) +t∫

0

γ(t− τ)[sin θ(τ) + sin(θ(τ) + 2ωτ) − ρ sin(θ(t) + ωτ)

β− γ

β

]dτ.

A similar integral for Eq. (4.4) is given by

σ(t) = ν(t) +t∫

0

γ(t− τ)[sinσ(τ) − γ

β

]dτ,

where θ(0) = σ(0), θ(0) = σ(0)

ν(t) =1α

(1 − e−αt)θ(0) + θ(0), (4.6)

γ(t) =β

α(1 − e−αt). (4.7)

Like in the proof of Theorem 5, we obtain here the relation

t∫

0

γ(t− τ)[sin(θ(τ) + 2ωτ) − ρ

βsin(θ(τ) + ωτ)

]dτ = O(δ),



where it is taken into account that γ(t) and θ(t) are functions slowly varying over the intervals[kδ, (k + 1)δ]. Indeed, for the function γ(t) this follows from (4.7).

It is well known that the estimate

|θ(t)| ≤ 2β + |γ| + ρ

α, ∀ t ≥ 0

for Eq. (4.2) follows from conditions (4.5). Hence, θ(t) slowly varies over [kδ, (k + 1)δ], and thefunctions sin(θ(t) + 2ωt) and cos(θ(t) + ωt) rapidly oscillate over [kδ, (k + 1)δ]. Therefore, thefollowing result can be formulated.

Theorem 7. For any T > 0 and δ > 0, there exists a number ω > 0 which is sufficiently greatas compared with α,α−1, β, |γ| and ρ and such that the inequality

|θ(t) − σ(t)| ≤ δ, ∀ t ∈ [0, T ] (4.8)

is satisfied for the solutions θ(t) and σ(t) of Eqs. (4.1) and (4.4) with the initial data (4.5).

It follows from this theorem and inequality (4.3) that under fixed values of α, β, γ, ρ, T > 0,satisfying (4.3) and for a sufficiently great ω, rotation of the right rotor with frequency ω is observedfor Eq. (4.1) over the interval [0, T ] under the initial conditions ϕ(0) = 0, sinϕ(0) = γ/β, ϕ(0) ∈(0, π/2). The same effect is observed also for the initial conditions ϕ(0) = σ(0), ϕ(0) = σ(0)from the attraction domain of the stable equilibrium position σ(0) = 0, σ(0) ∈ (−π/2, π/2),sinσ(0) = γ/β. We recall that for M = 0 and some values of the parameters α, β, γ, and ρ andfor a sufficiently great ω stability of these equilibrium states is observed in some neighborhood ofthe initial conditions ϕ(0) = 0, ϕ(0) = 0 and ϕ(0) = 0, ϕ(0) = π [56–58].

5. EQUATIONS OF PHASE SYNCHRONIZATION SYSTEMS

We first note that consideration of synchronous machines, PLL’s, and unbalanced rotors leadsto analysis of the differential equation

θ + αθ + sin θ = γ. (5.1)

In a more complicated cases, however, the PLL equations and those of the synchronous machinesare structured differently. If the transfer function of the filter a +W (p) is nondegenerate, that is,its numerator and denominator have no common roots, Eq. (3.10) is equivalent to the system ofdifferential equations

z = Az + bψ(σ),σ = c∗z + ρψ(σ),

(5.2)

where A is a constant n × n matrix, b and c are constant n × n vectors, ρ is a number, ψ(σ) is a2π-periodic function satisfying the relations

ρ = −aL,W (p) = Lc∗(A− pI)−1b,

ψ(σ) = ϕ(σ) − ω1(0) − ω2(0)L(a+W (0))

.

We note that in (5.2) σ = θ1 − θ2.


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It follows from the nondegeneracy of a + W (p) [14, 19, 58] that the pair (A, b) is completelycontrollable and the pair (A, c) is completely observable. The last fact implies that the inequalities

det(b,Ab, . . . , An−1b) = 0,

det(c,A∗c, . . . , (A∗)n−1c) = 0

are satisfied. The discrete phase locked loops obey similar equations [34]

z(t+ 1) = Az(t) + bψ(σ(t)),σ(t+ 1) = σ(t) + c∗z(t) + ρψ(σ(t)),

(5.3)

where t ∈ Z, Z being the set of integers. Equations (5.2) and (5.3) describe the so-called standardPLL’s [15].

There exist many modifications of PLL’s whose equations can be found in [15–19, 25–36]. Wepresent here the equations of the floating PLL controlling the clock generators in the digital signalprocessors [42–44]. Its block diagram differs from that shown in Fig. 11 with the phase detectorcharacteristic (3.6) only in that a relay element with characteristic u(G) = sgnG is inserted afterthe filter. In this case, we get the following PLL equations:

z = Az + bϕ(σ),σ = g(c∗z),

(5.4)

where σ = θ1 − θ2, the matrix A and the vectors b and c are such that

W (p) = c∗(A− pI)−1b,

g(G) = −L(sgnG) + (ω1(0) − ω2(0)).

We recall that it is assumed here that the master oscillator θ1(t) ≡ ω1(t) ≡ ω1(0) is highly stable.A wide class of the synchronous electrical machines obeys the following system of differential

equations [14, 59, 60]:

dσ

dt= η,

dη

dt= −g(η, σ) + z∗Cf(σ) − ϕ(σ),

dz

dt= Az +Df(σ)η,

(5.5)

where A is the constant n × n matrix, D and C are the n × m matrices, f(σ) is a continuouslydifferentiable 2π-periodic m-dimensional vector function, and ϕ(σ) and g(η, σ) are continuouslydifferentiable functions satisfying the conditions

μ1 ≤ g(η, σ)η

≤ μ2, ∀ η = 0, ∀σ ∈ R1,

ϕ(σ + 2π) = ϕ(σ), g(η, σ + 2π) = g(η, σ), ∀ η ∈ R1, ∀σ ∈ R1,

where μ1 and μ2 are nonnegative numbers.



We note that system (2.1), (2.2) is representable as (5.5) with

σ = −θ, η = −θ, z =

(i1 − e/R

i2

),

g(η, σ) = 0, ϕ(σ) =βSBe

IRsinσ − M

I,

f(σ) =SB

L

(sinσcosσ

),

A =

⎛⎜⎝

−RL

0

0 −RL

⎞⎟⎠ , D =

(1 00 −1

),

C = −βLI

(−1 0

0 1

).

The synchronous motors with damping windings fed by a powerful network obey Eqs. (5.5) withthe three-dimensional vector z whose components are the currents in the excitation winding andtwo damping windings [12]. Various generalizations and modifications of the above Eqs. (5.5) canbe found in [17, 61].

6. SOME GENERAL NOTIONS OF THE PHASE SYNCHRONIZATION THEORY

The equations of phase synchronization systems were presented in the above sections. Theydiffer, but have a common trait, the phase coordinates. Stability is an important property of allsystems encountered in the applications.

Existence of the phase coordinate allows one to introduce the cylindrical phase space. For that,we consider the n-dimensional differential equation

dx

dt= F (t, x), x ∈ Rn, t ∈ R1, (6.1)

where F (t, x) is the vector function defined on R1 ×Rn.We assume that the identities

F (t, x+ dj) = F (t, x), ∀x ∈ Rn, ∀ t ∈ R1 (6.2)

are valid for the linearly independent vectors dj ∈ Rn (j = 1, . . . ,m).The scalar value d∗jx/|dj | is often called the angular coordinate. For example, for system (2.8),

d1 =

(2π0

), x =

(θη

),

d∗1x|d1| = θ.

We consider the discrete group

Γ =

⎧⎨⎩x =

m∑j=1

kjdj | kj ∈ Z, j = 1, . . . ,m

⎫⎬⎭ ,

where Z is the set of integers.We consider the factor group Rn/Γ whose elements are the classes of residues

[x] = {x+ u | ∀u ∈ Γ, x ∈ Rn}


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and introduce the so-called flat metric

ρ([x], [y]) = infz∈[x]v∈[y]

|z − v|, (6.3)

where z ∈ Rn, v ∈ Rn, | · | is the Euclidean norm in Rn. It may be easily seen that the followingproposition is true.

Proposition 1. If x(t) is a solution of Eq. (6.1) defined over the interval (t1, t2), then x(t)+kdj,k ∈ Z, also is a solution of Eq. (6.1) over the interval (t1, t2).

It follows from Proposition 1 that in the autonomous case, that is, F (t, x) ≡ F (x), the thus-introduced metric space Rn/Γ is the phase space of system (6.1), which means that this spacecan be decomposed into nonintersecting trajectories of system (6.1), provided that the trajectoriesin Rn exist and are unique. This space is often well-behaved also in the nonautonomous case witha time-dependent right-hand side of (6.1).

The space Rn/Γ is often called the cylindrical phase space because it is diffeomorphic to thecylinder surface

C × C × · · · × C︸︷︷︸m

×R1 × · · · ×R1︸︷︷︸

n−m

,

where C is a circumference. The space Rn/Γ with metric (6.3) is often more convenient than thespace Rn. First, for Rn/Γ the many-valuedness related with the angular coordinate disappears:only one state of the physical system corresponds to all values of the angular coordinate σ + 2kπ.Second, the notion of limitedness of the solutions of system (6.1) in the space Rn/Γ relative tometric (6.3) can be introduced here in a natural way. This limitedness implies limitedness of thesolutions in the original space Rn only with respect to the nonangular coordinates and, perhaps,also with respect to some angular coordinates if their number exceeds m. Third, a classificationof cycles is introduced in a natural way for the autonomous case, which will be explained in moredetail.

Let the right-hand side of Eq. (6.1) be time-independent, that is, F (t, x) ≡ F (x). The solutionx(t) of system (6.1) is as usual called the first-kind cycle, provided that x(t) is not the equilibriumstate and there exists a number T > 0 such that

x(t+ T ) = x(t), ∀ t ∈ R1. (6.4)

Definition 1. The solution x(t) will be called the second-kind cycle, provided that there arenumbers T > 0 and k ∈ Z, k = 0, such that the inequality

x(t+ T ) = x(t) + kdj , ∀ t ∈ R1, (6.5)

where dj is one of the previously introduced vectors from Rn, is satisfied.

It is clear that the second-kind cycles are the closed trajectories [x(t)] in the cylindrical phasespace which lose their closedness at the passage to the phase space Rn. Classification of the second-kind cycles can be continued by introducing homotopic classes of cycles whose number is definedby the connectivity order of the cylindrical phase space [9], although verification of property (6.5)often suffices for the theory of phase synchronization.



Definition 2. The solution x(t) of system (6.1) is called the circular solution if there are numbersε > 0 and T > 0 such that the inequality

d∗j x(t) ≥ ε, ∀ t ≥ T (6.6)

is satisfied.

We note that for system (2.8) this inequality assumes the form

θ(t) = η(t) ≥ ε/2π.

Such solutions exist for α < α(γ) and correspond to the circular motions of pendulum aboutthe point of suspension

Therefore, the introduced notion of circular solution is a sort of generalization of this notion forsystem (2.8) describing, in particular, the motion of pendulum.

Now we proceed to the definitions of global stability of system (6.1) which are substantial forthe phase synchronization systems. We assume that any solution of system (6.1) x(t, t0, x0) withthe initial data x(t0, t0, x0) exists and is defined for all t ≥ t0.

Definition 3 ([59]). System (6.1) will be referred to as dichotomic if any its solution limited on[t0,+∞) tends to the stationary set for t→ +∞.

Definition 4 ([59, 62]). System (6.1) will be referred to as gradient-like if any its solution tendsto the equilibrium state for t→ +∞.

If in the cylindrical phase space Rn/Γ the stationary set consists of one locally asymptoticallystable equilibrium state, the rest of the equilibrium states being Lyapunov-unstable, then such agradient-like system will be called the globally stable system.

From the point of view of the phase synchronization theory, the term “global stability” is morereasonable than “gradient-like system” because only one globally stable synchronism is physicallyobserved here. This terminology is most popular in practice.

A weaker type of stability, the so-called Bakaev-stability, is sometimes acceptable in the elec-tronic systems of phase synchronization [34]. To define this type of stability, we assume that m isthe maximum number of the linearly independent vectors dj for which identity (6.2) is valid, thesystem {dj} is orthogonal, that is, d∗kdj = 0, if k = j, and identities (6.2) are not satisfied for thevectors of the form λdj , where λ ∈ (0, 1).

Definition 5. We state that system (6.1) is Bakaev-stable if for any its solution x(t, t0, x0) thereexists a number T ≥ t0 such that the inequalities

|d∗j (x(t1, t0, x0) − x(t2, t0, x0))| ≤ |dj |2 (6.7)

are satisfied for all t1 ≥ T , t2 ≥ T , j = 1, . . . ,m.

We note that analysis of the Bakaev-stable PLL’s requires special study of the adequacy of thedifferential Eqs. (6.1) to the block diagrams describing the phase relations (see Fig. 11 and condition(3.7)). If Eq. (6.1) is globally stable, then the theory of first approximation in the neighborhoodof the stable equilibrium state usually allows one to make decision about exponential damping ofthe transients. This fact substantiates the passage to equality (3.7) and by that substantiates thepassage from the block diagram of Fig. 11 to the differential Eqs. (5.2). We note that references[63–68] present various approaches to defining the notion of “synchronization.”

It was possible to modify the classical direct Lyapunov method for studying the phase synchro-nization systems for global stability. This modification will be described in the following section.


1592 LEONOV

7. DIRECT LYAPUNOV METHOD FOR THE PHASE SYNCHRONIZATION SYSTEMS

Theorem 8. Let us assume that there exists a continuous function V (x) : Rn → R1 such thatthe following conditions are met:

(1) V (x+ q) = V (x), ∀x ∈ Rn, ∀ q ∈ Γ;

(2) V (x) +m∑

j=1(d∗jx)

2 → +∞ for x→ ∞;

(3) for any solution x(t) of system (6.1) the function V (x(t)) is nonincreasing.Then, any solution [x(t)] of system (6.1) is limited on [t0,+∞) in the cylindrical space Rn/Γ.

Proof. It follows from Condition (1) that the function V ([x]) : Rn/Γ → R1 can be determinedform the formula V ([x]) = V (x). At that, it follows from Condition (2) that

V ([x]) → +∞ for [x] → ∞. (7.1)

We assume now that the solution [x(t)] is not limited on [t0,+∞). In this case, there exists asequence tk → +∞ such that

[x(tk)] → ∞ for k → ∞.

It follows from this fact and condition (7.1) that

V ([x(tk)]) → +∞ for k → ∞,

which contradicts Condition (3) and proves limitedness of [x(t)] in Rn/Γ.Let us assume now that system (6.1) is autonomous (F (t, x) ≡ F (x)) and its stationary set

consists of isolated points.

Theorem 9. Let us assume that there exists a function V (x) : Rn → R1 such that Condi-tions (1)–(3) of Theorem 8 and condition

(4) if V (x(t)) ≡ V (x(0)), then x(t) ≡ constare satisfied.Then, (6.1) is a gradient-like system.

Proof. We recall that the point [p] is called the ω-limiting point for the trajectory [x(t)] if thereexists a sequence tk → ∞ such that

[p] = limk→∞

[x(tk)].

We also recall [14, 59] that the trajectory [y(t)], consisting wholly of the ω-limiting points of thetrajectory [x(t)] passes through the ω-limiting point [p]. Usually, this fact is called the invarianceof the ω-limiting set.

Limitedness of [x(t)] for t ≥ 0 follows from Theorem 8 from which fact and continuity of V ([x(t)])follows limitedness of the function [x(t)] for t ≥ 0. Therefore, existence of the limit

limt→+∞V ([x(t)]) = V0

follows from Condition (3). Then, V ([y(t)]) ≡ V0. In this case, we obtain from Condition (4) thaty(t) ≡ const. From this identity and isolatedness of the points of the stationary set of system (6.1)we get that it is a gradient-like system.



Theorem 9 is an extension of the well-known Barbashin–Krasovskii theorem [69] and the La Salleprinciple [70, 71] to the systems with cylindrical phase space. Its variants were described in [16,59, 72].

Now we consider the main features of applying Theorems 8 and 9 to systems like (5.2) for whichthe procedure for constructing the Lyapunov functions like the “quadratic form of the coordinate,that is, of the vector z, and, possibly, of the nonlinearity, that is, of ψ(σ), plus the integral ofnonlinearity”:

V =

(zψ

)∗H

(zψ

)+

σ∫

0

ψ(σ)dσ, (7.2)

where H is some symmetrical (n+ 1) × (n+ 1) matrix.For systems (5.2) with the angular coordinate σ, the dichotomization conditions can be estab-

lished using functions of this form. The gradient conditions can be obtained using the same methodonly under the condition

2π∫

0

ψ(σ)dσ = 0.

For the case where this condition is not satisfied, a procedure was developed for constructing theperiodic Lyapunov functions like [14, 59]

V =

(zψ

)∗H

(zψ

)+

σ∫

0

F (σ)dσ, (7.3)

where

F (σ) = ψ(σ) − ν|ψ(σ)|, (7.4)

where the number ν is chosen so that the function V is σ-periodic:

ν =2π∫

0

ψ(σ)dσ

⎛⎝

2π∫

0

|ψ(σ)|dσ⎞⎠

−1

. (7.5)

Let us formulate the frequency criterion obtained for system (5.2) using the Lyapunov functionslike (7.3)–(7.5) and the Yakubovich–Kalman frequency theorem [14, 34, 59].

We assume that the function ψ(σ) is differentiable and has isolated zeros, and that for somenumbers μ1 and μ2 the following conditions are satisfied:

μ1 ≤ dψ(σ)dσ

≤ μ2, ∀σ ∈ R1. (7.6)

We also consider the transfer function

K(p) = c∗(A− pI)−1b− ρ.

Theorem 10. Let the matrix A be stable and there exist numbers ε > 0, δ > 0, τ ≥ 0, and κ

such that satisfied are the inequalities

Re {κK(iω) − ε|K(iω)|2 − τ(K(iω) + μ−11 iω)∗(K(iω) + μ−1

2 iω)} > δ ∀ω ∈ R1, (7.7)

4εδ > (κν)2. (7.8)

Then, (5.2) is a gradient-like system.


1594 LEONOV

We recall that the matrix A is called stable if all its eigenvalues have negative real parts. Somegeneralizations of this theorem and its application to different PLL’s can be found in [14, 34, 59].

The σ-periodic Lyapunov function is readily constructed for system (5.4) in the case of n = 1:

V = −c∗z∫

0

g(y)dy + c∗bσ∫

0

ϕ(u)du.

It is easy to see that

V = −Ac∗zg(c∗z) < 0, ∀ c∗z = 0

for A < 0 and L > |ω1(0) − ω2(0)|. We recall that

2π∫

0

ϕ(u)du = 0.

This construction and some additional arguments allows us to prove the following theorem whichis important for designing the PLL’s in the digital signal processors.

Theorem 11 ([42–44]). If the inequalities A < 0 and

|L| > |ω1(0) − ω2(0)|are satisfied, then (5.4) with n = 1 is a gradient-like system.

If |L| < |ω1(0) − ω2(0)|, then all solutions of this system tend to infinity for t→ +∞.

It still is not clear how to extend this result to larger systems, that is, for n > 1. The topicalityof this problem is related with the possibility of using more complicated filters in such floatingPLL’s.

Now, we apply the method of constructing the periodic Lyapunov functions to the synchronousmachines and consider for system (5.5) the Lyapunov-like function

V = z∗Hz +12η2 +

σ∫

0

ϕ(σ)dσ. (7.9)

In the case of no load moment (M = 0), the equality

2π∫

0

ϕ(σ)dσ = 0 (7.10)

is true, and the function V is σ-periodic.The symmetrical positive definite matrix H is selected so as to satisfy the inequality

2z∗H(Az +Dξ) + z∗Cξ ≤ −ε|z|2, ∀ z ∈ Rn, ∀ ξ ∈ Rm (7.11)

for some number ε > 0. By the Yakubovich–Kalman theorem [59], such a matrix exists if A is astable matrix and the inequalities

K(iω) +K(iω)∗ > 0, ∀ω ∈ R1, (7.12)

limω→∞ω2(K(iω) +K(iω)∗) > 0, (7.13)



whereK(p) = C∗(A−pI)−1D, are valid. We recall that for the symmetrical matrixH the inequalityH > 0 means that the quadratic form z∗Hz is positive definite.

It is easy to see from (7.11) and the inequality μ1 ≥ 0 that the function V (z(t), η(t), σ(t)) isnonincreasing, where z(t), η(t), σ(t) is the solution of system (5.5).

Therefore, if conditions (7.10), (7.12), and (7.13) are satisfied, then Conditions (1)–(3) of The-orem 8 are satisfied for the function V . Let us verify that Condition (4) of Theorem 9 is satisfiedhere as well.

It follows immediately from V (z(t), η(t), σ(t)) ≡ const and inequality (7.11) that z(t) ≡ 0.Yet, then z(t) ≡ 0 and, therefore, Df(σ(t))η(t) ≡ 0. It is easy to demonstrate that the equalityrankD = m follows from (7.12). Therefore, f(σ(t))η(t) ≡ 0. We obtain from this fact and theequality η(t) = σ(t) that

σ(t)∫

σ(t0)

f(σ)dσ = 0, ∀ t ≥ t0. (7.14)

If at some point t1 > t0 the inequality σ(t1) = σ(t0) is valid, then we get from (7.14) the relation

σ∫

σ(t0)

f(σ)dσ = 0 (7.15)

for all σ such that (σ − σ(t0))(σ − σ(t1)) ≤ 0. By differentiating (7.15) with respect to σ, we getfor (σ − σ(t0))(σ − σ(t1)) ≤ 0 the equality f(σ) = 0, which is impossible if one assumes that

f(σ) ≡ 0, ∀σ ∈ (θ1, θ2), ∀ θ1 < θ2. (7.16)

Therefore, if condition (7.16) is met, then σ(t) ≡ σ(t0). Yet then also η(t) = σ(t) ≡ 0.So, if V (z(t), η(t), σ(t)) ≡ const, then the solution z(t), η(t), σ(t) is the equilibrium state. Simple

analysis of system (5.5) shows that the equilibrium states are isolated if and only if all zeros of thefunction ϕ(σ) are isolated, which proves the following theorem.

Theorem 12 ([59]). We assume that μ1 ≥ 0, A is a stable matrix, all zeros of the function ϕ(σ)are isolated, and conditions (7.10), (7.12), (7.13), and (7.16) are satisfied. Then (5.5) is a gradient-like system.

The following theorem is proved along the same lines.

Theorem 13 ([59]). If all conditions of Theorem 12, except for (7.10), are satisfied, (5.5) isdichotomic system.

Theorem 12 enables one to verify global stability of the unloaded synchronous machine, whichis important in two cases where the machine operates in the compensator mode [14] or is startedwithout load, is pulled in synchronism during the transient process, and only then loaded. Inthe last case, important is the problem of determining the limiting power surge under which themachine does not fall out step. This problem also yields to the Lyapunov function (7.9).

We assume here that the stationary solution of system (5.5) z(t) ≡ 0, η(t) ≡ 0, σ(t) ≡ 0corresponds to the synchronous no-operation, that is, no-load, mode. Then at some time instant τan instant power surge M occurs (see Eqs. (2.1) and (2.2) as a special case of system (5.5)). Afterthat, the right-hand side of (5.5) changes for t > τ .


1596 LEONOV

ϕ

γ

σ

0

σσ

1

Fig. 13. Method of areas.

Here, only the function ϕ(σ) is replaced by a constant:

ϕ(σ) → ϕ(σ) − γ,

and the point z = 0, η = 0, σ = 0 is no more the equilibrium state. The new locally stableequilibrium state has form z = 0, η = 0, σ = σ0, where ϕ(σ0) = γ, ϕ′(σ0) > 0 and σ0 ∈ [0, 2π](Fig. 13). We assume here that on [0, 2π] there are only two zeros σ0 and σ1 of the function ϕ(σ)−γfor which ϕ′(σ1) < 0.

The problem of the limiting load is formulated as follows: for what loads the synchronousmachine is pulled into a new synchronous mode z(t) ≡ 0, η(t) = σ(t) ≡ 0, σ(t) ≡ σ0 after thetransient ? In mathematical terms, this question is reformulated as follows. Needed is to determinethe conditions under which the solution z(t), η(t), σ(t) with the initial data z(τ) = 0, η(τ) = 0, andσ(τ) = 0 would be in the domain of attraction of the stationary solution z = 0, η = 0, σ = σ0:

limt→+∞ z(t) = 0, lim

t→+∞ η(t) = 0, limt→+∞σ(t) = σ0. (7.17)

The following result can be obtained by using a function like (7.9)

V = z∗Hz +12η2 +

σ∫

σ1

(ϕ(σ) − γ)dσ

and the arguments from the proof of Theorem 12. We recall that now for t > τ system (5.5) haschanged so that we have ϕ(σ) − γ instead of ϕ(σ)).

Theorem 14 ([59]). Let all conditions of Theorem 12 be satisfied. Then, if

σ1∫

0

(ϕ(σ) − γ)dσ ≥ 0,

relation (7.17) is valid for the solution of system (5.5) with ϕ(σ) − γ instead of ϕ(σ) and with theinitial data z(τ) = 0, η(τ) = 0, σ(τ) = 0.

Theorem 14 underlies the popular engineering method of areas which is formulated in simpleterms as follows: the maximum permissible instantaneous load of the synchronous machine γ isdefined so that the shaded areas in Fig. 13 are equal.



The equations of the synchronous machines often are considered in other coordinates for whichthe following general notation is more suitable:

dσ

dt= η,

dη

dt= −g(η, σ) + y∗C

df(σ)dσ

− ϕ(σ),

dy

dt= Ay + Df(σ) + q,

(7.18)

where f(σ) and ϕ(σ) are the 2π-periodic functions satisfying the same conditions as f(σ) and ϕ(σ),D is a constant n×m matrix, and q is the constant n-vector.

By means of the Lienard change

y = z −A−1Df(σ) −A−1q

this system can be reduced to the form (5.5) with D = A−1D, f(σ) =df(σ)dσ

,

ϕ(σ) = ϕ(σ) +

[df(σ)dσ

]∗C∗A−1

[Df(σ) + q

].

The introduced transfer matrices

K(p) = C∗(A− pI)−1D,

and

K(p) = C∗(A− pI)−1D

are related by

K(p) = p−1(K(p) − K(0)).

Therefore, the following result stems from Theorem 12.

Theorem 15. Let μ1 ≥ 0, all stationary points of system (7.18) be isolated, and the followingconditions be satisfied:

(1) K(0) is the diagonal matrix;

(2)K(iω)iω

+

(K(iω)iω

)∗> 0, ∀ω ∈ R1;

(3) limω→∞ω2

(K(iω)iω

+

(K(iω)iω

)∗)> 0;

(4)df(σ)dσ

≡ 0 on any (θ1, θ2);

(5)2π∫0ϕ(σ)dσ = 0.

Then, (7.18) is a gradient-like system.


1598 LEONOV

The problem of the maximum permissible load of the synchronous machine is considered simi-larly.

We consider the following equations of the synchronous machine with zero load as an exampleof using the above methods [73, 59]:

σ = η,η = (−α1y1 − α2y2) sinσ + (α3y3 cos σ − α4 sinσ cos σ),

y1 = α5 − α6y1 + α7y2 + α8 cos σ,y2 = α9y1 − α10y2 + α11 cos σ,y3 = −α12y3 + α13 sinσ,

(7.19)

where αj (j = 1, . . . , 13) are positive parameters such that

α6α10 − α7α9 > 0.

We show that all conditions of Theorem 15 are satisfied. Here,

g(η, σ) ≡ 0, ϕ(σ) = α4 sinσ cos σ,

f(σ) =

(cos σ− sinσ

), q =

⎛⎜⎝α5

00

⎞⎟⎠ , C =

⎛⎜⎝α1 0α2 00 −α3

⎞⎟⎠ ,

D =

⎛⎜⎝

α8 0α11 00 −α13

⎞⎟⎠ , A =

⎛⎜⎝α6 α7 0α9 −α10 00 0 −α12

⎞⎟⎠ ,

K(p) =

(K1(p) 0

0 K2(p)

),

K1(p) =−[(α1α8 +α2α11)p+α1α8α10 +α1α11α7 +α2α11α6 +α2α9α8][(p+α6)(p+α10)−α9α7]−1,

K2(p) = − α3α13

p+ α12.

Obviously, Conditions (1)–(3) of Theorem 15 are equivalent to the relations

Re

[Kj(iω)iω

]> 0, ∀ω ∈ R1, j = 1, 2,

limω→∞

{ω2Re

[Kj(iω)iω

]}> 0, j = 1, 2.

It is easy to check that these conditions are always satisfied and, consequently, (7.19) is the gradient-like system. In the case of power surge from the no-operation mode, the above method of areas(Fig. 13) can be used for system (7.19).

8. METHOD OF POSITIVE INVARIANT CONE GRIDS.AN ANALOG OF THE CIRCULAR CRITERION

This method, which was proposed independently in [74, 75], is sufficiently universal and “fine”in the sense that it makes use only of two properties of the system such as the availability of thepositive one-dimensional quadratic cone and the invariance of the field of system (6.1) under theshifts by the vector dj (see (6.2)).



Fig. 14. Positive invariant cone.

Fig. 15. Positive invariant cone grid.

We assume that such a cone of the form Ω = {x∗Hx ≤ 0}, where H is a symmetrical matrixwith one negative and the remaining positive eigenvalues, is positive invariant, which means thatthe relation

V (x(t)) < 0

is satisfied on the boundary of the cone ∂Ω = {xHx = 0} for all x(t) such that x(t) = 0, x(t) ∈ ∂Ω(Fig. 14).

Now we have recourse to the second property, invariance of the vector field to the shift by thevectors kdj , k ∈ Z, and multiply the cone so that

Ωk = {(x− kdj)H(x− kdj) ≤ 0}.Since positive invariance is, obviously, retained for the cones Ωk, we obtain a positive invariantcone grid shown in Fig. 15. As can be seen from this figure, all solutions x(t, t0, x0) of the systemwhich features the two aforementioned properties are bounded on [t0,+∞).

If the cone Ω has only one point of intersection with the hyperplane {d∗jx = 0} and all solutionsx(t) for which at the instant t the inequality

x(t)∗Hx(t) ≥ 0

is satisfied have property V (x(t)) ≤ −ε|x(t)|2 (here, ε is a positive number), then it is immediatelyclear from Fig. 15 that the system is Bakaev-stable. Thus, the proposed method is simple anduniversal. The Yakubovich–Kalman frequency theorem makes it practically efficient [14, 59].

Let us consider, for example, the system

dx

dt= Px+ qϕ(t, σ), σ = r∗x, (8.1)


1600 LEONOV

where P is the constant singular N ×N matrix, q and r are the constant N -dimensional vectors,and ϕ(t, σ) is the continuous 2π-periodic in σ function R1 ×R1 → R1 satisfying the relations

μ1 ≤ ϕ(t, σ)σ

≤ μ2, ∀ t ∈ R1, ∀σ = 0,

where μ1 and μ2 are some numbers which, by virtue of periodicity of ϕ(t, σ) in σ, without loss ofgenerality can be assumed to be, respectively, negative, μ1 < 0, and positive, μ2 > 0.

We introduce the transfer function of system (8.1)

χ(p) = r∗(P − pI)−1q

which is assumed to be nondegenerate and consider its quadratic forms V (x) = x∗Hx and

G(x, ξ) = 2x∗H[(P + λI)x+ qξ] + (μ−12 ξ − r∗x)(μ−1

1 ξ − r∗x),

where λ is a positive number.For existence of the symmetrical matrix H such that it has one negative and N − 1 positive

eigenvalues and the inequality G(x, ξ) < 0, ∀x ∈ RN , ξ ∈ R1, x = 0, is satisfied, it suffices by theYakubovich–Kalman theorem, that

(1) the matrix P + λI has N − 1 eigenvalues with the negative real part and(2) the frequency inequality

μ−11 μ−1

2 + (μ−11 + μ−1

2 )Reχ(iω − λ) + |χ(iω − λ)|2 < 0, ∀ω ∈ R1

is satisfied.It is easy to see that the condition G(x, ξ) < 0, ∀x = 0,∀ξ, gives rise to

dV (x(t))dt

+ 2λV (x(t)) < 0, ∀x(t) = 0.

This inequality guarantees the positive invariance of the aforementioned cone Ω. Therefore, wehave here the following counterpart of the well-known circular criterion.

Theorem 16 ([74, 14, 34, 59]). If there exists a positive number λ for which the above condi-tions (1) and (2) are satisfied, then any solution x(t, t0, x0) of system (8.1) is bounded over theinterval (t0,+∞).

The reader is referred to [14, 34, 59] for a more detailed proof of this fact. We also note thatthis theorem retains its validity if the nonstrict inequality in Condition (2) is satisfied also in thecases where μ1 = −∞ or μ2 = +∞ [14, 34, 59].

We demonstrate how the formulated (with regard for the above remark) counterpart of thecircular criterion can be applied to the simplest case of the second-order equation

θ + αθ + ϕ(t, θ) = 0, (8.2)

where α is a positive parameter. This equation is representable as system (8.1) with N = 2 andthe transfer function

χ(p) =1

p(p+ α).



Fig. 16. Sectors of stability and instability.

Obviously, Condition (1) of the theorem takes the form λ ∈ (0, α), and for μ1 = −∞ and μ2 = α2/4Condition (2) is equivalent to the inequality

−ω2 + λ2 − αλ+ α2/4 ≤ 0, ∀ω ∈ R1

which is satisfied for λ = α/2.Therefore, if in Eq. (8.2) the function ϕ(t, σ) is periodic in σ and satisfies the inequality

ϕ(t, σ)σ

≤ α2

4, (8.3)

then any its solution θ(t) is bounded over (t0,+∞).It is easy to demonstrate (see Theorem 1) that for ϕ(t, σ) ≡ ϕ(σ) Eq. (8.2) is dichotomic, that is,

t-independent ϕ(t, σ). Whence, if in the autonomous case (8.3) is satisfied, then for t → +∞ anysolution of (8.2) tends to some equilibrium state.

Here we encounter an interesting analog of the notion of absolute stability for the phase synchro-nization systems. If (8.1) is regarded as an absolutely stable system when any its solution tendsto some equilibrium state for any nonlinearity ϕ from the sector [μ1, μ2], then (−∞, α2/4] will bethis sector for Eq. (8.2) with ϕ(t, σ) ≡ ϕ(σ)

At the same time, in the classical theory of absolute stability (without the assumption of period-icity of ϕ), for ϕ(t, σ) ≡ ϕ(σ) we have two sectors (0,+∞) and (−∞, 0), respectively, of absolute sta-bility and absolute instability. Therefore, periodicity alone of ϕ enables one to encompass part of theabsolute stability sector and complete absolute instability sector: (−∞, α2/4] ⊃ (−∞, 0)∪ (0, α2/4](see Fig. 16). More complex examples of using the analogs of the circular criterion can be foundin [14, 34, 59].

9. METHOD OF NONLOCAL REDUCTION. EXTENSION OF THE TRICOMI RESULTSTO MULTIDIMENSIONAL SYSTEMS OF PHASE SYNCHRONIZATION

We describe the main stages of extending the theorems of Tricomi and his followers obtainedfor the equation

θ + αθ + ψ(θ) = 0, (9.1)

to higher-dimensionality systems and first consider the system

dz

dt= Az + bψ(σ),

dσ

dt= c∗z + ρψ(σ),

(9.2)


1602 LEONOV

describing the standard PLL. We assume as usual that ψ(σ) is 2π-periodic, A is the stable n × nmatrix, b and c are the constant n-vectors, and ρ is a number.

Let us consider the case where Eq. (9.1) or its equivalent system

η = −αη − ψ(θ),

θ = η(9.3)

is gradient-like. In this case, it is possible to demonstrate [9] (as was done at the beginning of thepresent paper for ψ(θ) = sin θ − γ) that for the equation

dη

dθ=

−αη − ψ(θ)η

(9.4)

equivalent to (9.3) there exists solution η(θ) such that η(θ0) = 0, η(θ) = 0, ∀θ = θ0,

limθ→+∞

η(θ) = −∞, limθ→−∞

η(θ) = +∞, (9.5)

where θ0 is a number such that ψ(θ0) = 0, ψ′(θ0) < 0.Now, we consider the function

V (z, σ) = z∗Hz − 12η(σ)2

inducing in the phase space {z, σ} the cone Ω = {V (z, σ) ≤ 0} which is a generalization of a kindof the quadratic cone shown in Fig. 14. We prove that under some conditions this cone will bepositive invariant. Indeed, we consider the expression

dV

dt+ 2λV = 2z∗H [(A+ λI)z + bψ(σ)] − λη(σ)2 − η(σ)

dη(σ)dσ

(c∗z + ρψ(σ))

= 2z∗H [(A+ λI)z + bψ(σ)] − λη(σ)2 + ψ(σ)(c∗z + ρψ(σ)) + αη(σ)(c∗z + ρψ(σ)),

where we made use of the fact that η(σ) satisfied Eq. (9.4).We note that if the frequency inequalities

Re K(iω − λ) − ε|K(iω − λ)|2 > 0,lim

ω→∞ω2(Re K(iω − λ) − ε|K(iω − λ)|2) > 0, (9.6)

where K(p) = c∗(A−pI)−1b−ρ, are satisfied, then by the Yakubovich–Kalman frequency theoremthere exists H such that for all z = 0 and ξ satisfied is the relation

2z∗H[(A+ λI)z + bξ] + ξ(c∗z + ρξ) + ε|(c∗z + ρξ)|2 < 0,

where ε is some positive number. If A+ λI is a stable matrix, then H > 0.Therefore, if A+ λI is stable and (9.6) and α2 ≤ 4λε are satisfied, then

dV

dt+ 2λV < 0, ∀z(t) = 0,

and consequently, Ω is a positive invariant cone.

The cones Ωk ={z∗Hz − 1

2ηk(σ)2 ≤ 0

}can be multiplied in the same way as in the last section,

and their cone grid (Fig. 15) where ηk(σ) is the solution η(σ) shifted along the axis σ by 2kπ can



be constructed. The cone grid proves boundedness of the solutions of system (9.2) over the interval(0,+∞).

Under the assumptions made, dichotomicity takes place as well, which is easily proved using theLyapunov function

z∗Hz +σ∫

0

ψ(σ)dσ.

Therefore, the following theorem was proved.

Theorem 17. If the matrix A + λI is stable for some λ > 0 and ε > 0, conditions (9.6) aresatisfied, and (9.3) with α = 2

√λε is a gradient-like system, then (9.2) is gradient-like system as

well.

The reader is referred to [76, 61, 60, 14, 17, 18, 59] for various generalizations of this theoremand numerous examples of applying the nonlocal reduction method, including also the synchronousmachines. Various criteria for existence of circular solutions and second-kind cycles were alsoobtained within the framework of this method [14, 17, 18, 59]. We demonstrate how it can be usedto estimate the limiting load of the induction electric motor obeying Eqs. (2.6).

We assume to this end that γ < 2c2 and denote

ψ(s) = −γcs2 + as− cγ,

Γ = 2 maxλ∈(0,c)

[λ

(c− λ− γ2

4c2(c− λ)

)]1/2

,

where the numbers a, c, and γ are the parameters of system (2.6). We assume that γ < a/2. Inthis case, the function ψ(s) has two zeros s0 ∈ (0, c) and s1 > c, and it follows from the inequalityγ < 2c2 that Γ > 0. We recall that the problem of the limiting power surge for the induction motorwhich was posed in Section 1 of the present review can be solved using the following result.

Theorem 18 ([77]). Let for the solution of the equation

θ + Γθ + ψ(θ) = 0 (9.7)

with the initial data θ(0) = θ(0) = 0 the condition

θ(t) ≤ s1, ∀ t ≥ 0 (9.8)

be satisfied. Then, the solution of system (2.6) with the initial data s(0) = y(0) = x(0) = 0 satisfiesthe relations

limt→+∞ s(t) = s0,

limt→+∞ y(t) = −γ

a,

limt→+∞x(t) = −γs0

ac.

Proof. By means of the change η = ay + γ, z = −x− γ

acs we reduce system (2.6) to

s = η,

η = −cη + azs− ψ(s),

z = −cz − 1asη − γ

acη.

(9.9)


1604 LEONOV

Condition (9.8) implies existence of the solution F (θ) of the equation

FdF

dθ= −ΓF − ψ(θ) (9.10)

defined over the interval [s2, s1] and such that F (s2) = F (s1) = 0, F (θ) > 0, ∀ θ ∈ (s2, s1), s2 < 0.For the function

V (s, η, z) .=a2

2z2 +

12η2 − 1

2F (s)2, s ∈ [s2, s1]

on the solutions of system (9.9) we have relations

V (s(t), η(t), z(t)) = −cη(t)2 − aγ

cη(t)z(t) − a2cz(t)2 − F ′(s(t))F (s(t))η − ψ(s(t))η

≤ −2λV (s(t), η(t), z(t)) +(F ′(s(t))F (s(t)) + ψ(s(t)))2

4ε− λF (s(t))2 = −2λV (s(t), η(t), z(t)),

where

ε = c− λ− γ2

4c2(c− λ)−1

and used was equality (9.10) with λ such that Γ = 2√λε.

It is easy to establish from

V (s(t), η(t), z(t)) + 2λV (s(t), η(t), z(t)) ≤ 0

that the set

Ω = {V (s, η, z) < 0, s ∈ [s2, s1]}is positive invariant and, by using the function

W (s, η, z) =a2

2z2 +

12η2 +

s∫

0

ψ(σ)dσ,

that system (9.9) is dichotomic. Whence, the theorem follows from boundedness and positiveinvariance of Ω and from the inclusions

⎛⎜⎝

000

⎞⎟⎠ ∈ Ω

⎛⎜⎝s000

⎞⎟⎠ ∈ Ω.

It is well known (see condition (8.19) on page 107 of [9]), that condition (9.8) is satisfied if Γ > 0and

2s1∫

0

ψ(σ)dσ + Γ2(s1 − s0)2 ≥ 0. (9.11)

The last inequality is satisfied if

2s1∫

0

ψ(σ)dσ = 2s1(− γ

3cs21 +

a

2s1 − cγ

)≥ 0.



This relation can be rearranged in

2γ3cs1 ≥ a

2. (9.12)

Since ϕ(s1) = 0 and s1 > c, the equality

s1 =c(a+

√a2 − 4γ2)2γ

is satisfied; whence, inequality (9.12) is satisfied if√

34a ≥ γ. (9.13)

Therefore, the inequalities γ < 2c2 and (9.13) estimate the permissible power surge. The condition8c2 >

√3a is satisfied for a large class of induction electric motors. Assuming that it is satisfied, we

use (9.13) to estimate the limiting permissible power surge. We recall that the maximum value ofthe static characteristic (2.7) is a/2. If we denote byMmax the maximum load for which system (2.6)is in the equilibrium state, that is, the induction motor has operating modes, then the permissiblesharp power surge of the idling motor is estimated simply as

M ≤√

32Mmax. (9.14)

In the case of massive rotor where I is a great number as compared to the parameters β, R, L,SB, and M , the parameters of system (2.6) can be represented as a = δa0 and γ = δγ0, where δ isa sufficiently small number relative to a0, γ0, and c. One can readily see that Γ = c + O(δ) andcondition (9.11) is satisfied for a0 > 2γ0, δ 1. In this case, the permissible sharp power surge isestimated as

M <Mmax, δ 1. (9.15)

This estimate coincides with the well-known result of asymptotic analysis where system (2.6) withthe small parameter δ is reduced to the first-order equation

s = − a0cs

c2 + s2+ γ0.

Estimates (9.14) and (9.15) illustrate effectiveness of applying the universal methods on the phasesynchronization theory to particular problems.

10. CONCLUSIONS

The theory of phase synchronization was created in the second half of the last century onthe basis of three applied theories of synchronous and induction electrical motors, phase lockedloops, and autosynchronization of the unbalanced rotors. Its basic principle lies in consideringthe problems of phase synchronization at the three levels of (i) mechanical, electromechanical,or electronic model, (ii) phase relations, and (iii) differential, difference, integral, and integro-differential equations. At that, the difference of oscillation phases is transformed in the controlaction realizing synchronization.

These general principles gave impetus to creation of universal methods of studying the phasesynchronization systems of which a modification of the direct Lyapunov method with constructionof the periodic Lyapunov-like functions, the method of positive invariant cone grids, and the methodof nonlocal reduction proved to be most effective. The last method, which combined the elements ofthe direct Lyapunov method and the bifurcation theory, enabled one to extend the classical resultsof F. Tricomi and his followers to the multidimensional dynamic system.


1606 LEONOV

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This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board


phase synchronization: theory and applications

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