Analysis of nonlinear oscillatory network dynamics via time-varying amplitude and phase variables

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt. J. Circ. Theor. Appl. 2007; 35:623–644Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cta.433

Analysis of nonlinear oscillatory network dynamics viatime-varying amplitude and phase variables‡

Valentina Lanza1, Fernando Corinto2, Marco Gilli2,∗,†and Pier Paolo Civalleri2

1Department of Mathematics, Politecnico di Torino, Turin, Italy2Department of Electronics, Politecnico di Torino, Turin, Italy

SUMMARY

The goal of this manuscript is to propose a method for investigating the global dynamics of nonlinearoscillatory networks, with arbitrary couplings. The procedure is mainly based on the assumption thatthe dynamics of each oscillator is accurately described by a couple of variables, that is, the oscillatorperiodic orbits are represented through time-varying amplitude and phase variables. The proposed methodallows one to derive a set of coupled nonlinear ordinary differential equations governing the time-varyingamplitude and phase variables. By exploiting these nonlinear ordinary differential equations, the predictionof the total number of periodic oscillations and their bifurcations is more accurate and simpler with respectto the one given by the latest available methodologies. Furthermore, it is proved that this technique alsoworks for weakly connected oscillatory networks. Finally, the method is applied to a chain of third-orderoscillators (Chua’s circuits) and the results are compared with those obtained via a numerical technique,based on the harmonic balance approach. Copyright q 2007 John Wiley & Sons, Ltd.

KEY WORDS: cellular nonlinear networks; nonlinear oscillatory networks; spectral techniques

1. INTRODUCTION

Oscillatory networks have been one of the most used paradigms in order to mimic repetitive dy-namical processes taking place in complex systems. Large collections of coupled oscillators (cells),having several periodic attractors, have been studied in many fields of science and technology,including, for example, Biology, Physics and Engineering [1–7]. The leading feature of oscillatorynetworks is the emergence of synchronized oscillations among the cells, i.e. it is observed that quite

∗Correspondence to: Marco Gilli, Department of Electronics, Politecnico di Torino, Turin, Italy.†E-mail: marco.gilli@polito.it‡Dedicated to Professor Sean Scanlan on the occasion of his 70th birthday.

Contract/grant sponsor: Ministero dell’Istruzione, dell’Universita e della Ricerca; contract/grant number:RBAU01LRKJContract/grant sponsor: CRT Foundation

Copyright q 2007 John Wiley & Sons, Ltd.

624 V. LANZA ET AL.

a lot of cells tune their rhythms so that numerous groups of oscillators exhibit highly correlatedbehavior.

From a mathematical point of view, the emergence of such coordinated behavior correspondsto the existence of global periodic oscillations, for networks described by a large set of couplednonlinear ordinary differential equations (ODEs) [8, 9].

It is evident that to understand how the interactions among the cells influence the appearanceand the properties of such global oscillations may be crucial to explain several phenomena of realcomplex systems.

The dynamics of oscillatory networks has been mainly analyzed by exploiting time-domainnumerical simulations [8] and spectral methods [10–12]. These techniques allow one to study theglobal dynamics for the given sets of coupling parameters, but some analytical results are neededin order to determine to what degree the global behavior of the network depends on the interactionsamong the cells.

By assuming that the coupling among the oscillators is weak, i.e. the dynamical beha-vior of each oscillator is not so perturbed by the interaction with other oscillators, approxi-mated analytic results can be obtained. Weakly connected oscillatory networks have been deeplyinvestigated, especially in neuroscience, because of their neuro-computational properties [13, 14].The global dynamic behavior of oscillatory networks, with weak couplings among the cells,can be studied through the phase deviation equation [13], i.e. the equation that describes theevolution of the phase deviations from the natural oscillations, due to the weak coupling. Ithas been shown that an accurate analytic expression of the phase deviation equation canbe derived, via the joint application of the describing function technique and of Malkin’sTheorem [15].

On the other hand, there are some significant cases where the hypothesis of weak coupling isa stringent requirement [16, 17]. Also in these cases, i.e. for arbitrarily coupled oscillatory net-works, one would like to obtain approximated analytical results. If the oscillators are not weaklycoupled, the representation only through the phase deviation is not so accurate. An interestingapproach, widely used in microwave theory, to cope with the analysis of general oscillatory arrayshas been proposed by York [16] and York et al. [17]. In these papers, the authors suggest a method,based on Kurokawa’s analysis [18], to derive a set of nonlinear ODEs, governing the amplitudeand phase variables associated with each oscillator, composing an oscillatory network defined inthe frequency domain.

The aim of this manuscript is to propose a method that permits us to unfold the global dynamicsof a large class of oscillatory networks made of cells described by a Lur’e model and connectedwith one another via arbitrary couplings. The procedure is mainly based on the representationof the oscillator dynamics as an oscillation with time-varying amplitude and phase (similar tothose described in [16, 17, 19]), that is in such a way a generalization of the describing functiontechnique. The outcome is a set of coupled nonlinear ODEs, governing the time-varying amplitudeand phase variables associated with each oscillator, that allow us to predict accurately the numberof global periodic oscillations and their bifurcations in a easier way with respect to the latestavailable methodologies.

In Section 2, we illustrate the method by making no assumption on the coupling parameters(in general, they might be frequency dependent) and by characterizing each oscillator in terms ofamplitude and phase variables that are assumed to be slowly varying function of time (generalizeddescribing function). It allows us to obtain a set of nonlinear ODEs, governing the amplitude andphase dynamics of each oscillator.

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Circ. Theor. Appl. 2007; 35:623–644DOI: 10.1002/cta

ANALYSIS OF NONLINEAR OSCILLATORY NETWORK DYNAMICS 625

In Section 3, the stability analysis for a one-dimensional network, composed of oscillatorscoupled through frequency-independent connections, is developed. It is derived that the proposedmethod can detect the whole set of periodic solutions also obtained via the application of mixedtime–frequency techniques [12]. In addition, we can easily study the stability properties and thebifurcations of the periodic solutions.

In Section 4, we prove that the phase deviation equations, obtained via the joint application ofthe describing function technique and of Malkin’s theorem to weakly connected networks, resultswith a particular case of the set of nonlinear ODEs, provided by the proposed method.

Finally, in Section 5 we consider, as case study, a network of four diffusively coupled third-orderoscillators (Chua’s circuits [20, 21]) and we predict the bifurcations of limit cycles depending onthe coupling parameters. The results are compared with those given in [12] based on the applicationof a mixed time–frequency technique.

2. OSCILLATORY NETWORKS

We consider a network of N identical coupled nonlinear oscillators of dynamical order m. Letus assume that the state equations of each uncoupled cell can be written in the Lur’e form byexploiting only a scalar variable xi (t):

L(D)xi (t) = n[xi (t)], i = 1, . . . , N (1)

where L(D) is a rational function of the time-differential operator D = d/dt and n[xi (t)] is a scalarLipschitz-continuous nonlinear function.

We also assume that the interaction between two oscillators is defined by an operator Yiq(D),that is, Yiq(D) is a rational function of D and describes the coupling between the i th and the qthoscillator, respectively.

The whole oscillatory network can be described by the following set of Lur’e-like equations(i = 1, . . . , N ):

L(D)xi (t) = n[xi (t)] +N∑

q=1Yiq(D)xq(t) (2)

Since we are interested in studying limit cycle bifurcations, we focus on a set of parameters andinitial conditions such that, in the absence of coupling, i.e. Yiq(D)= 0 for all (i, q), each cellhas at least a periodic (either stable or instable) limit cycle. We also assume that this limit cyclecan be estimated by exploiting the describing function technique. The rigorous conditions underwhich such an assumption holds are given in standard textbooks and are very difficult to check(see [22]); however, for most oscillators, describing function solutions can be considered reliableif the normalized distortion index, defined in [23], is sufficiently small.

2.1. Describing function technique

In order to apply the describing function technique to Equation (2), we have to represent eachstate variable xi (t) (i = 1, . . . , N ) through the following first harmonic expression:

xi (t) =Re(X0i + X1

i ej�t ) (3)

626 V. LANZA ET AL.

where X0i is the bias term, � and X1

i are the angular frequency and the amplitude of the firstharmonic, respectively.

By substituting the approximate expression (3) in Equation (2), the following set of N com-plex nonlinear equations in the 3N unknowns X0

1, . . . , X0N , Re(X

11), . . . ,Re(X

1N ), Im(X1

1), . . . ,

Im(X1N−1), � is easily derived:

Re(X0i L(0)+X1

i L( j�)ej�t ) = n[Re(X0i +X1

i ej�t )]+Re

q=1(Yiq(0)X

0q+Yiq( j�)X1

qej�t )

where the nonlinear term is approximated as

n[Re(X0i + X1

i ej�t )] = n[X0

i + Re(X1i ) cos�t − Im(X1

i ) sin�t]= N 0

i + N ci cos�t + N s

i sin�t + · · ·≈ N 0

i + Re((N ci − jN s

i )ej�t )

=Re(N 0i + N 1

i ej�t ) (5)

N 0i = N 0

i (X0i , X

N 1i = N 1

i (X0i , X

1i ) = N c

i − jN si

Hence, relations (4) can be written as (i = 1, . . . , N ):

X0i L(0) − N 0

i −N∑

q=1Yiq(0)X

0q = 0

X1i L( j�) − N 1

i −N∑

q=1Yiq( j�)X1

in agreement with the results shown in [12, 17].Equation (7) shows the application of the describing function technique to system (2) and allows

us to define the notation that is useful for the following sections. By solving Equation (7), thesteady-state periodic solutions can be accurately estimated.

It is worth noting that, in the absence of coupling, by denoting with � the angular frequencyof a single uncoupled oscillator, the application of the describing function technique yields (see[22, 23] for details):

Im(L( j�))= 0, Nis = 0 (8)

2.2. Time-varying amplitude and phase variables

We aim to extend the describing function technique by considering time-varying bias, amplitudeand angular frequency in order to obtain more information about the properties of the steady-stateperiodic solution.

Let us assume that the amplitudes X0i and X1

i are slowly varying functions of time, that is,

X0i (�) = A0

i (�) (9)

X1i (�) = A1

i (�)ej�i (�) (i = 1, . . . , N ) (10)

where A0i (�) and A1

i (�) are assumed to be real and �= �t is the slow time.According to these definitions, we have a first harmonic approximated solution of Equation (2)

having the following form (i = 1, . . . , N ):

xi (t, �) =Re(A0i (�) + A1

i (�)ej(�t+�i (�))) = x0i (�) + x1i (t, �) (11)

where we assume that � is the angular frequency of each uncoupled oscillator.It is worth observing that Equation (11) can be interpreted as a perturbation of the steady-state

solution obtained from Equation (7) when �≈ �.By analyzing the evolution of A0

i (�), A1i (�) and �i (�), we can investigate the stability properties

of the limit cycles provided by Equation (7). As first step, we consider how the linear term inEquation (2) can be approximated under these assumptions. We suppose to expand the operatorL(D) in a Taylor series around 0 when the operator is applied to the bias term of xi (t), and around� for the first harmonic term of Equation (11).

The approximation of L(D) to the first-order term yields

L(D)x0i (�) ≈ (a0 I + a1D)x0i (�)

L(D)x1i (t, �) ≈ [a0 I + a1(D − j�I )]x1i (t, �)(12)

By exploiting the method of multiple scales [24] and taking into account that we have the slowtimescale �, the time differential operator D can be written as

D = ��t

+ ��

�t��

= Dt + �D� (13)

Hence, we have

L(D)x0i (�) = L(Dt , D�)x0i (�)

= [a0 I + a1(Dt + �D�)]x0i (�)

= a0A0i (�) + a1�

= L(0)A0i (�) + �L ′(0)

d�(14)

628 V. LANZA ET AL.

L(D)x1i (t, �) = L(Dt , D�)x1i (t, �)

= [a0 I + a1(Dt + �D� − j�I )]x1i (t, �)

1i (�)e

j(�t+�i (�)) + a1A1i (�)

[j�ej(�t+�i (�))

+ j�

(d�i (�)

d�− j

A1i (�)

)ej(�t+�i (�)) − j�ej(�t+�i (�))

{L( j�)A1

i (�)ej(�t+�i (�))+j�L ′( j�)

(d�i

d�− j

)ej(�t+�i (�))

The second step is the approximation of the nonlinear term in Equation (2). By noting that thefunction n(·) is periodic in the variable t , expression (5) of the nonlinear term can be generalized as

ni [Re(A0i (�) + A1

i (�)ej(�t+�i (�)))] ≈Re(N 0

i (�) + N 1i (�)ej�t ) (16)

N 0i (�) = 1

∫ 2�

0ni [Re(A0

i (�) + A1i (�)e

j(�t+�i (�)))] d(�t)

∫ 2�+�i (�)

�i (�)ni [Re(A0

i (�) + A1i (�)e

js)] ds

∫ 2�

0ni [Re(A0

i (�) + A1i (�)e

js)] ds

= N 0i (A0

i (�), A1i (�)) (17)

N 1i (�) = 1

∫ 2�

0ni [Re(A0

i (�) + A1i (�)e

j(�t+�i (�)))]e−j�t d(�t)

∫ 2�+�i (�)

�i (�)ni [Re(A0

i (�) + A1i (�)e

js)]e−j(s−�i (�)) ds

2�ej�i (�)

∫ 2�

0ni [Re(A0

i (�) + A1i (�)e

js)]e−js ds

= N 1i (A0

i (�), A1i (�))e

j�i (�) = (N ci (�) − jN s

i (�))ej�i (�) (18)

It is extremely important to remark that the coefficients N 0i and N 1

i have formally the sameexpression of (6), but they are a function of slow time �.

Finally, the coupling term in (2) involves the operator Yiq(D) so we can apply the same procedureshown above for the linear operator L(D). We can easily conclude that

Yiq(D)x0q(�) = Yiq(0)A0q(�) + �Y ′

iq(0)dA0

d�(19)

Yiq(D)x1q(t, �) =Re

[Yiq( j�)A1

q(�)ej(�t+�q (�))

+ j�A1q(�)Y

′iq( j�)

(d�q

d�− j

)ej(�t+�q (�))

By substituting Equations (14)–(16), (19) and (20) in Equation (2) and taking into account theorthogonality of the functions e jk�t for different values of k, the following set of equations isderived (i = 1, . . . , N ):

F0(A0, A1) + P0dA0

d�= 0 (21)

F1(A0, A1,U) + P1

d�A1 − j

)= 0 (22)

where, without showing explicitly the dependency of the slow time �, A0 and A1 are vectorscontaining all bias and amplitudes, respectively, U is a diagonal matrix whose elements are theoscillators’ phases and

F0(A0, A1) = 1

�(L(0)A0 − N 0(A0, A1) − Y(0)A0)

F1(A0, A1,�) = 1

�(L( j�)A1 − N 1(A0, A1) − e−jUY( j�)ejUA1)

P0 = L ′(0)I − Y′(0)

P1 = j(L ′( j�)I − e−jUY′( j�)ejU)

(Y(0))iq = Yiq(0), (Y′(0))iq = Y ′iq(0)

(Y( j�))iq = Yiq( j�), (Y′( j�))iq = Y ′iq( j�)

630 V. LANZA ET AL.

By equating real and imaginary parts of (22), we obtain a set of 3N coupled nonlinear ODEsdescribing the evolution of bias terms A0, amplitudes A1 and phases U:

dA0(�)

d�= −P−1

0 F0(A0, A1)

dA1(�)

d�= Im[P−1

1 F1(A0, A1,U)]dU(�)

d�A1 = −Re[P−1

1 F1(A0, A1,U)]

3. FREQUENCY-INDEPENDENT CONNECTED OSCILLATORY NETWORKS

Let us consider oscillatory networks arranged as a one-dimensional array (chain) of oscillatorswhose interactions do not depend on the frequency, that is, Yiq( j�) = Yiq . As a direct consequence

dYiqd�

∣∣∣∣�

= 0 ∀(i, q)

and matrices P0 and P1 in Equation (23) reduce to diagonal ones. By denoting with

Hki ( j�) = L( j�)Ak

i (�) − Nki (�) −

N∑q=1

Yiq Akq(�)e

jk(�q ((�)−�i (�))) (k = 0, 1)

K ( j�) = jL ′( j�)

we can easily obtain that

(P−10 F0(A0, A1,U))i = j

H0i (0)

�K (0)(25)

(P−11 F1(A0, A1,U))i = H1

i ( j�)

�K ( j�)(26)

The real and imaginary parts of Equation (25) can be written as

Re((P−11 F1(A0, A1,U))i )

�|K ( j�)|2 [Re(K ( j�))Re(H1i ( j�)) + Im(K ( j�)) Im(H1

i ( j�))]

�|K ( j�)|2[Re(K ( j�))

(Re(L( j�))A1

i − N ci −

N∑q=1

Yiq A1q cos(�q − �i )

+ Im(K ( j�))

(Im(L( j�))A1

i − N si −

N∑q=1

Yiq A1q(�) sin(�q − �i )

)](27)

Im((P−11 F1(A0, A1,U))i )

�|K ( j�)|2 [Re(K ( j�)) Im(H1i ( j�)) − Im(K ( j�))Re(H1

i ( j�))]

�|K ( j�)|2[Re(K ( j�))

(Im(L( j�))A1

i − N si −

N∑q=1

Yiq A1q(�) sin(�q − �i )

− Im(K ( j�))

(Re(L( j�))A1

i − N ci −

N∑q=1

Yiq A1q cos(�q − �i )

)](28)

By substituting Equations (27), (28) and (25) in Equation (24), we obtain the following nonlinearsystem of differential equations describing an oscillatory network with resistive couplings:

d�= − A0

i (�)L(0) − N 0i (�) −∑N

q=1Yiq A0q(�)

�L ′(0)

d�= 1

�|K ( j�)|2[−Re(K ( j�))

N∑q=1

Yiq A1q sin(�q(�) − �i (�))

− Im(K ( j�))(Re(L( j�)A1i (�) − N c

i (�)))

+ Im(K ( j�))N∑

q=1Yiq A

1q(�) cos(�q(�) − �i (�))

d�= − 1

�|K ( j�)|2[Re(K ( j�))

(Re(L( j�)) − N c

i (�)

A1i (�)

−N∑

q=1Yiq

A1q(�)

A1i (�)

cos(�q(�) − �i (�))

− Im(K ( j�))N∑

q=1Yiq

A1q(�)

A1i (�)

sin(�q(�) − �i (�))

where, according to Equation (8), the conditions derived from the describing function technique,i.e. Im(L( j�))= 0 and N s

i = 0 ∀i = 1, . . . , N , have been exploited.For investigating the stability properties of the equilibrium points of system (29), let us focus

on oscillatory networks whose cells interact only with their adjacent ones. The resulting networkcan be described by the following equations (i = 1, . . . , N ):

L(D)xi (t) = n[xi (t)] ++1∑

k=−1Yi,i+k xi+k(t) (30)

where the parameter Yi,i−1, Yi,i and Yi,i+1 are identical for all cells, i.e. the coupling parametersare space invariant. Equation (30) has to be completed by specifying the initial and the boundaryconditions. In the following, we consider boundary conditions of the Dirichlet type.

632 V. LANZA ET AL.

These assumptions imply that Equations (29), governing the amplitudes and phases dynamics,become:

d�= − A0

i L(0) − N 0i −∑1

k=−1Yi,i+k A0i+k

�L ′(0)= Q0

i (A0, A1, �)

d�= 1

�|K ( j�)|2[−Re(K ( j�))

1∑k=−1

Yi,i+k A1i+k sin(�i+k − �i )

− Im(K ( j�))(Re(L( j�))A1i − N c

+ Im(K ( j�)1∑

k=−1Yi,i+k A

1i+k cos(�i+k − �i ))

i (A0, A1, �) (31)

d�= − 1

�|K ( j�)|2[Re(K ( j�))

(Re(L( j�)) − N c

−1∑

k=−1Yi,i+k

cos(�i+k − �i )

− Im(K ( j�))1∑

k=−1Yi,i+k

sin(�i+k − �i )

]= Ri (A

0, A1, �)

The equilibrium configurations can be derived by setting the right-hand side of Equation (31)equal to zero and solving the corresponding nonlinear system. From the second equation ofEquation (31), we have

Re(L( j�)) − N ci

−1∑

k=−1Yi,i+k

cos(�i+k − �i )

=−Re(K ( j�))

Im(K ( j�))

1∑k=−1

Yi,i+kA1i+k

sin(�i+k − �i ) (32)

that, substituted in the third equation of Equation (31), allows us to obtain

1∑k=−1

Yi,i+k A1i+k sin(�i+k − �i ) = 0 (33)

By denoting with �i = �i+1 − �i (1�i�N − 1), the phase shift between the (i + 1)th and i thoscillators, respectively, we can derive the following set of difference equations:

sin(�i ) = Yi,i−1A1i−1

Yi,i+1A1i+1

sin(�i−1), 2�i�N − 1

sin(�1) = 0

As a direct consequence, we deduce that only two values of phase shift �i are admissible:

�i = {0, �} (1�i�N − 1) (35)

It is worth pointing out that the result given above agrees with that shown in [12].In addition, by using Equations (32) and (35) in the first equation of Equation (31), we obtain

that the equilibrium amplitudes A0i and A1

i satisfy the following conditions:

A0i L(0) − N 0

i −1∑

k=−1Yi,i+k A

0i+k = 0 (36)

Re(L( j�))A1i − N c

i −1∑

k=−1Yi,i+k A

1i+k cos(�i+k − �i ) = 0 (37)

that are equivalent to those reported in [12].By exploiting Equations (35)–(37), the equilibrium configurations of the amplitudes A0

i , A1i and

phases �i , corresponding to the periodic oscillations of the whole network (30), can be derived.Furthermore, their stability properties can be easily studied by computing the Jacobian matrix.

The oscillatory network model (30) implies that there are at most seven nonzero elementsin every row of the Jacobian (except for those rows involving the oscillators on the boundary).Furthermore, the Jacobian can be seen as a 3× 3 block matrix in which every block is N × N .

If we denote the Jacobian matrix with J , from Equation (31) it can be easily derived that itselements for 1�i�N are

Ji,i−1 = �Q0i (A

0, A1, �)

�A0i−1

Ji,i = �Q0i (A

0, A1, �)

�A0i

Ji,i+1 = �Q0i (A

0, A1, �)

�A0i+1

Ji,i+N = �Q0i (A

0, A1, �)

�A1i

whereas for N + 1�i�2N , they are

Ji,i−N = �Q1i (A

0, A1,�)

�A0i

Ji,i−1 = �Q1i (A

0, A1, �)

�A1i−1

, Ji,i+N−1 = �Q1i (A

0, A1, �)

��i−1

Ji,i = �Q1i (A

0, A1, �)

�A1i

, Ji,i+N = �Q1i (A

0, A1, �)

��i

Ji,i+1 = �Q1i (A

0, A1, �)

�A1i+1

, Ji,i+N+1 = �Q1i (A

0, A1, �)

��i+1

634 V. LANZA ET AL.

and finally, for 2N + 1�i�3N

Ji,i−2N = �Ri (A0, A1,�)

�A0i

Ji,i−N−1 = �Ri (A0, A1, �)

�A1i−1

, Ji,i−1 = �Ri (A0, A1, �)

��i−1

Ji,i−N = �Ri (A0, A1, �)

�A1i

, Ji,i = �Ri (A0, A1, �)

��i

Ji,i−N+1 = �Ri (A0, A1, �)

�A1i+1

, Ji,i+1 = �Ri (A0, A1, �)

��i+1

Evaluating the Jacobian in the equilibrium points given by Equations (35)–(37) and computingits eigenvalues, we can analyze the stability properties of the periodic attractors of the oscillatorynetwork (30) (and consequently the related bifurcation processes). In particular, by exploitingcondition (35), the expressions for the nonzero elements of the Jacobian are the following: for1�i�N

Ji,i−1 = Yi,i−1

�L ′(0)

Ji,i = − L(0) − �N 0i /�A0

i − Yi,i�L ′(0)

Ji,i+1 = Yi,i+1

�L ′(0)

Ji,i+N = 1

�L ′(0)�N 0

�A1i

while for N + 1�i�2N

Ji,i−N = Im(K ( j�))

�|K ( j�)|2�N 1

�A0i

Ji,i−1 = Im(K ( j�))Yi,i−1 cos(�i−1 − �i )

�|K ( j�)|2

Ji,i = − Im(K ( j�))(Re(L( j�)) − �N 1i /�A1

i − Yi,i )

�|K ( j�)|2

Ji,i+1 = Im(K ( j�))Yi,i+1 cos(�i+1 − �i )

�|K ( j�)|2

Ji,i+N−1 = −Re(K ( j�))Yi,i−1A1i−1 cos(�i−1 − �i )

�|K ( j�)|2

Ji,i+N = −Re(K ( j�))∑1

k=−1,k �=0 Yi,i+k A1i+k cos(�i+k − �i )

�|K ( j�)|2

Ji,i+N+1 = −Re(K ( j�))Yi,i+1A1i+1 cos(�i+1 − �i )

�|K ( j�)|2

and for 2N + 1�i�3N

Ji,i−2N = Re(K ( j�))

�|K ( j�)|2�N 1

�A0i

Ji,i−N−1 = Re(K ( j�))

�|K ( j�)|2Yi,i−1 cos(�i−1 − �i )

Ji,i−N = −Re(K ( j�))

�|K ( j�)|2[− 1

�N 1i

�A1i

+ N 1i

(A1i )

1∑k=−1,k �=0

Yi,i+k A1i+k cos(�i+k − �i )

(A1i )

Ji,i−N+1 = Re(K ( j�))

�|K ( j�)|2Yi,i+1 cos(�i+1 − �i )

Ji,i−1 = Im(K ( j�))Yi,i−1 cos(�i−1 − �i )

�|K ( j�)|2A1i−1

Ji,i = − Im(K ( j�))

�|K ( j�)|2∑1

k=−1,k �=0 Yi,i+k A1i+k cos(�i+k − �i )

Ji,i+1 = Im(K ( j�))Yi,i+1 cos(�i+1 − �i )

�|K ( j�)|2A1i+1

In Section 5, the proposed method will be applied to a chain composed of four third-orderoscillators (Chua’s circuits) and the whole set of global periodic oscillations with their bifurcationswill be determined.

4. WEAKLY CONNECTED OSCILLATORY NETWORKS

In this section, we focus on a network of identical weakly coupled oscillators and we compare ourresults with those obtained in [13], through the joint application of Malkin’s theorem and of thedescribing function technique. We assume the coupling parameters, written as Yiq(D) = �Ciq ∀(i, q)

with ��1, are frequency independent and small. Under this assumption, it is reasonable to arguethat the amplitudes of the coupled oscillators will remain close to the amplitudes of the uncoupledoscillators, i.e. the amplitudes A0

i (�) ≈ X0i , A

1i (�) ≈ X1

i do not depend on time. It follows thatdA1

i /d�≈ 0 and from Equation (29) we can derive the following relationship:

Re(L( j�))−N ci

−�N∑

q=1Ciq

cos(�q(�)−�i (�))≈−Re(K ( j�))

Im(K ( j�))�

N∑q=1

CiqX1q

sin(�q(�)−�i (�))

636 V. LANZA ET AL.

that, used in the equation of the phase dynamics (the third of Equations (29)), gives

d�≈ 1

Im(K ( j�))

N∑q=1

Ciq sin(�q(�) − �i (�)) (44)

being X1i = X1

q due to the assumption that all oscillators are identical.Some nontrivial algebraic manipulations allow us to show that Equation (44) is analogous to

the phase deviation equation obtained in [25] by applying a method based on the joint applicationof Malkin’s Theorem and the describing function technique. Before proving this result, we brieflyoutline the approach used in [25] to make clear the notation.

In [25], each uncoupled oscillator is a dynamical mth-order system described by a state equationwritten in the following form:

xi = Ai11xi + Ai

12Zi + n[xi (t)]Zi =Ai

21xi + Ai22Zi

where the m-dimensional state vector is separated into a scalar state variable xi ∈ R and a (m−1)-dimensional state vector Zi ∈ Rm−1. By exploiting a linear relation between Zi and xi , one canrewrite Equations (45) as a Lur’e model involving the sole variable xi :

L(D)xi (t) = n[xi (t)] (46)

with L(D) having the following expression:

L(D) = D − Ai11 − Ai

12(D − Ai22)

−1Ai21 (47)

By applying the describing function technique and Malkin’s Theorem to a weakly oscillatorynetwork written as

L(D)xi (t) = n[xi (t)] + �N∑

q=1Ciq xq(t) (48)

with L(D) given by Equation (47), the authors of [25] obtained the following analytic expressionfor the phase deviation equation:

d�≈ Vi (�)

N∑q=1

Ciq sin(�q(�) − �i (�)) (49)

Vi (�) = 1

2 Im(NTi ( j�)) · Im(Mi ( j�))

Mi (D) = (D − Ai22)

−1Ai21 (51)

Ni (D) = −(D + (Ai22)

T)−1(Ai12)

T (52)

Hence, the goal is to show that Equation (44) is equivalent to Equation (49), that is, we have toprove that

Vi (�) = 1

Im(K ( j�))(53)

The proof of Equation (53) requires some preliminary results that are collected in the followinglemmas. The proof of the first lemma is obtained through straightforward algebraic computationsand therefore it is omitted.

Lemma 1Let M=M1 + jM2 be a Cn×n matrix. Then, its inverse M−1 has the following form:

M−1 =A + jB

A=M−12 M1(M1M

−12 M1 + M2)

B= −(M1M−12 M1 + M2)

Lemma 2Let A=A(t) be a nonsingular n × n matrix. Then, it holds

dA−1

dt= −A−1 dA

dtA−1

ProofBy definition, we have that AA−1 = I, where I is the n × n identity matrix. Deriving once thisexpression, it easily follows that

d(AA−1)

dt= dA

dtA−1 + A

dA−1

Hence, the lemma is proved. �

We are now able to demonstrate the following result.

Proposition 1Given a weakly connected oscillatory network as in Equation (48), the following condition holds:

Vi (�) = 1

Im(K ( j�))(54)

namely

2 Im(NTi ( j�)) · Im(Mi ( j�))= Im( jL ′( j�))=Re(L ′( j�)) (55)

638 V. LANZA ET AL.

ProofBy recollecting that

NTi (D) = −[(D + (Ai

22)T)−1(Ai

12)T]T

= −Ai12[(D + (Ai

22)T)−1]T (56)

we have

Mi ( j�) = ( j� − Ai22)

−1Ai21 (57)

NTi ( j�) = −Ai

12[( j� + (Ai22)

T)−1]T

= −Ai12[(( j� + Ai

22)T)−1]T

= −Ai12( j� + Ai

22)−1 (58)

In order to compute the imaginary parts of Mi ( j�) and NTi ( j�), we apply Lemma 1 for the

inverse of a complex matrix. It follows that the expression for Im(Mi ( j�)) and Im(NTi ( j�)) are

Im(Mi ( j�)) = Im(( j� − Ai22)

−1Ai21)

= −(1

�(Ai

22)2 + �I

= −�((Ai22)

2 + �2 I )−1Ai21 (59)

Im(NTi ( j�)) = −Ai

12 Im[( j� + Ai22)

−1]=Ai

12�((Ai22)

2 + �2 I )−1 (60)

Im(NTi ( j�)) Im(Mi ( j�))= −�2Ai

12((Ai22)

2 + �2 I )−2Ai21 (61)

From Equation (47), we derive the right-hand side of Equation (54):

L ′i (D) = 1 − Ai

dD((D − Ai

22)−1)Ai

= 1 + Ai12(D − Ai

22)−2Ai

21 (62)

having used Lemma 2 for the derivative of the inverse matrix.Hence, it is readily derived that

L ′i ( j�) = 1 + Ai

12( j�I − Ai22)

−2Ai21 (63)

and it follows

Re(L ′i ( j�)) = 1 + Ai

12 Re(( j�I − Ai22)

−2)Ai21

= 1 + Ai12(A

2((Ai22)

2 + �2 I )−2Ai21 − �2Ai

12((Ai22)

2 + �2 I )−2Ai21

= 1 + Ai12((A

2 + �2 I )−1Ai21 − 2�2Ai

12((Ai22)

2 + �2 I )−2Ai21 (64)

The expression of Re(L ′i ( j�)) can be simplified by noting that the first term is zero because,

according to Equation (8), the describing function technique provides the condition Im(L( j�))= 0(being � the angular frequency of each uncoupled oscillator):

Im(L( j�)) = � + �Ai12((A

2 + �2 I )−1Ai21

= �(1 + Ai12((A

2 + �2 I )−1Ai21) = 0 (65)

that is,

1 + Ai12((A

2 + �2 I )−1Ai21 = 0 (66)

By using Equations (61), (64) and (66), we can finally conclude that

Re(L ′i ( j�))=−2�2Ai

12((Ai22)

2 + �2 I )−2Ai21 = 2 Im(NT

i ( j�)) · Im(Mi ( j�)) (67)�

RemarkA direct consequence of the previous proposition is that for a weakly connected oscillatory network,the phase deviation equation (49) (obtained via the joint application of Malkin’s Theorem and thedescribing function technique) is identical to that shown in Equation (44) (provided by the method,based on the time-varying amplitude and phase variables, proposed in this manuscript).

5. CASE STUDY: 1-D ARRAY OF CHUA’S CIRCUITS

We consider a chain of N Chua’s circuits characterized by the following normalized state equations:

xi = �[yi − xi − n(xi )] + dxi−1 + dxi+1 − 2 dxi

yi = xi − yi + zi

zi = −�yi

where the space-invariant coupling parameters Yi,i−1 = Yi,i+1 = d , Yi,i =−2d realize a diffusiveinteraction among the oscillators and the scalar nonlinear function is n(xi ) =− 8

7 xi + 463 x

The circuit parameters � and � are chosen in such a way that every uncoupled cell exhibitsthe following invariant limit sets: three unstable equilibrium points, two asymmetric stable limitcycles A±

i , one stable symmetric limit cycle Ssi and one unstable symmetric limit cycle Sui (see[12] for more details). In our simulations, we have considered � = 8 and � = 15.

640 V. LANZA ET AL.

Equation (68) can be recast in the following Lur’e form:

L(D)xi (t) =−� fi (xi ) + dxi−1 − 2 dxi + dxi+1 (69)

where L(D) is given by

L(s) = s3 + s2(1 + �) + s� + ��

s2 + s + �(70)

It can be easily derived that there are two admissible angular frequencies �1 and �2 for theuncoupled oscillator (see [15] for more details).

By applying the proposed method (see the second part of Section 3), we obtain the set ofequations given by Equation (31). It follows that the amplitude and phase equilibrium configurationssatisfy Equations (35)–(37), respectively. Hence, as shown also in [12], for each one of the 2N−1

admissible phase shifts {�1, . . . , �N−1} ∈ {0, �}, system (68) has:

• one symmetric limit cycle {Ss1, Ss2, . . . , SsN } that corresponds to the frequency �1 and to thesolution of Equations (36) and (37) with A0

i = 0 ∀i ;• one symmetric limit cycle {Su1 , Su2 , . . . , SuN } that corresponds to the frequency �2 and to thesolution of Equations (36) and (37) with A0

i = 0 ∀i ;• 2N asymmetric limit cycles {A±

1 , A±2 , . . . , A±

N } that correspond to the frequency �2 and to thesolution of Equations (36) and (37) with sign(A0

1) = ±1, sign(A02) = ±1, . . . , sign(A0

N ) = ±1,respectively.

Under the assumption of symmetric couplings (Yi,i−1 = Yi,i+1 = d), the study of the stabilityproperties can be carried out by considering only few amplitude and phase equilibrium configu-rations, i.e. only a restricted number of limit cycles (see also [12]). By exploiting d as a bifurcation

Table I. Bifurcation values for the limit cycles Ss�� and Su��.

Cycle Proposed method Bifurcation Method in [12] Bifurcation

Ss000 d∗ = 18.2 H ∗ ∗Ss00� d∗ = 1.426 T d∗ = 1.2163 T

Ss0�0 d∗ = 5.02 H d∗ = 5.0170 H

Ss0�� d∗ = 0.9696 T d∗ = 0.8678 T

Ss�0� d∗ = 1.445 T d∗ = 1.3166 T

Ss�� d∗ = 1.124 T d∗ = 1.1206 T

Su000 d∗ = 6.2 H d∗ = 6.1560 H

Su00� d∗ = 0.484 T d∗ = 0.5381 T

Su0�0 d∗ = 1.706 H d∗ = 1.7010 H

Su0�� d∗ = 0.3290 T d∗ = 0.3627 T

Su�0� d∗ = 0.491 T d∗ = 0.5171 T

Su�� d∗ = 0.382 T d∗ = 0.4203 T

Table II. Bifurcation values for the limit cycles Aabcd�� ({a, b, c, d} ∈ {+,−}, {�, �, } ∈ {0, �}).

Cycle Proposed method Method in [12] Cycle Proposed method Method in [12]Apppp000 d∗ = 0.2377 d∗ = 0.2065 Apmmp

000 d∗ = 0.1076 d∗ = 0.0999

Apppp00� d∗ = 0.2610 d∗ = 0.2227 Apmmp

00� d∗ = 0.1090 d∗ = 0.0997

Apppp0�0 d∗ = 0.1943 d∗ = 0.1665 Apmmp

0�0 d∗ = 0.1053 d∗ = 0.0995

Apppp0�� d∗ = 0.1959 d∗ = 0.1673 Apmmp

0�� d∗ = 0.1055 d∗ = 0.0995

Apppp�0� d∗ = 0.5740 d∗ = 0.6250 Apmmp

�0� d∗ = 0.2448 d∗ = 0.2440

Apppp�� d∗ = 0.4601 d∗ = 0.4466 Apmmp

�� d∗ = 0.2650 d∗ = 0.2615

Apmpm000 d∗ = 0.0741 d∗ = 0.0690 Appmm

000 d∗ = 0.1164 d∗ = 0.1050

Apmpm00� d∗ = 0.0712 d∗ = 0.0650 Appmm

00� d∗ = 0.1109 d∗ = 0.0955

Apmpm0�0 d∗ = 0.0896 d∗ = 0.0845 Appmm

0�0 d∗ = 0.1681 d∗ = 0.1520

Apmpm0�� d∗ = 0.1014 d∗ = 0.0955 Appmm

0�� d∗ = 0.2329 d∗ = 0.2077

Apmpm�00 d∗ = 0.0712 d∗ = 0.0650 Appmm

�00 d∗ = 0.1107 d∗ = 0.0970

Apmpm�0� d∗ = 0.1738 d∗ = 0.1755 Appmm

�0� d∗ = 0.7470 d∗ = 0.7140

Apmpm��0 d∗ = 0.1014 d∗ = 0.0955 Appmm

��0 d∗ = 0.2330 d∗ = 0.2080

Apmpm�� d∗ = 0.1852 d∗ = 0.1795 Appmm

�� d∗ = 0.5674 d∗ = 0.5570

Apppm000 d∗ = 0.1102 d∗ = 0.0955 Appmp

000 d∗ = 0.1645 d∗ = 0.1383

Apppm00� d∗ = 0.1570 d∗ = 0.1340 Appmp

00� d∗ = 0.1008 d∗ = 0.0929

Apppm0�0 d∗ = 0.1052 d∗ = 0.0896 Appmp

0�0 d∗ = 0.1545 d∗ = 0.1383

Apppm0�� d∗ = 0.2018 d∗ = 0.1780 Appmp

0�� d∗ = 0.1842 d∗ = 0.1440

Apppm�00 d∗ = 0.1131 d∗ = 0.0955 Appmp

�00 d∗ = 0.0677 d∗ = 0.0860

Apppm�0� d∗ = 0.1471 d∗ = 0.1242 Appmp

�0� d∗ = 0.0970 d∗ = 0.0860

Apppm��0 d∗ = 0.1053 d∗ = 0.0887 Appmp

��0 d∗ = 0.1356 d∗ = 0.1210

Apppm�� d∗ = 0.2769 d∗ = 0.2680 Appmp

�� d∗ = 0.1746 d∗ = 0.1709

parameter, the eigenvalues i of the Jacobian, evaluated in the amplitude and phase equilibriumconfigurations, permit to investigate the limit cycle stability and bifurcations.

By focusing on a chain composed of N = 4 oscillators, Table I compares the results obtainedby the proposed method with those given in [12] for the stable Ss�� and unstable Su�� limitcycles (where {�, �, } ∈ {0, �} denote phase shifts), while Table II shows the bifurcation valuesfor the asymmetric limit cycles Aabcd

�� (with {a, b, c, d} ∈ {+, −} and {�, �, } ∈ {0, �}). The valueof the parameter d for which the bifurcation occurs is denoted as d∗. Tangent bifurcation andHopf bifurcations are labeled as T and H, respectively. In Figure 1, the absolute values of theFloquet multipliers depending on the coupling d are shown in case of a symmetric limit cycle. InFigures 2 and 3, the absolute values of Floquet multipliers, evaluated with the proposed technique,are compared with those obtained in [12].

642 V. LANZA ET AL.

0 0.05 0.1 0.15 0.2 0.25 0.30

2π/ω

Figure 1. Absolute values of the Floquet multipliers versus coupling parameterd for the symmetric limit cycle Su0��.

0 0.05 0.1 0.15 0.20

2π/ω

0 0.05 0.1 0.15 0.20

Figure 2. Absolute values of the Floquet multipliers versus coupling parameter d for the asymmetric limitcycle Apmpm

�� . Comparison between (a) the proposed method and (b) that given in [12].

It is worth noting that the proposed method may provide Floquet multipliers, evaluated for agiven coupling strength d , that are slightly different with respect to those obtained by applyingthe time–frequency domain technique shown in [12] (see Figures 2 and 3). Nevertheless, the twomethods yield quantitatively the same conclusions about the bifurcation occurrences.

In addition, it is remarkable that the proposed procedure consists only in solving a low-dimensional nonlinear algebraic system and in evaluating the eigenvalues of the Jacobian 3N × 3Nmatrix; on the other hand, the time–frequency domain techniques are computationally more

2π/ω

0 0.02 0.04 0.06 0.08 0.10

d(b)0 0.02 0.04 0.06 0.08 0.1

2π/ω

Figure 3. Absolute values of the Floquet multipliers versus coupling parameter d for the asymmetric limitcycle Apmmp

0�� . Comparison between (a) the proposed method and (b) that given in [12].

onerous, because it requires the application of HB-based technique for estimating periodic os-cillations and time-domain methods to compute the Floquet multipliers as eigenvalues of thefundamental matrix. It follows that the proposed method is able to predict accurately the occur-rence of limit cycle bifurcations without numerical overload.

6. CONCLUSION

We have considered nonlinear oscillatory networks, wherein each cell admits of a Lur’e modeland is arbitrarily coupled to the others. We have proposed a method that permits to describe eachperiodic oscillation of the cells in terms of a couple of variables: the time-varying amplitudeand phase variables. It allows us to derive a set of nonlinear ordinary differential equations whoseequilibrium configurations represent the global periodic solutions of the whole network. As a directconsequence, the limit cycle stability and their bifurcations can be easily analyzed for differentcoupling conditions.

We have also proved that for weakly connected oscillatory networks, the proposed techniquegives analogous results to those reported in the literature, i.e. each oscillator can be describedjust through its phase dynamics and that the phase deviation equation has the same analyticalexpression as that in [15].

Finally, as case study, we have considered a chain of four diffusively coupled Chua’s oscillators.By exploiting the proposed technique, we have detected the periodic oscillations of the wholenetwork and the related limit cycle bifurcations. It is shown that the proposed method providesresults that agree with those obtained in [12] via a numerical time–frequency approach.

ACKNOWLEDGEMENTS

This work was partially supported by the Ministero dell’Istruzione, dell’Universita e della Ricerca, underthe FIRB project no. RBAU01LRKJ and by the CRT Foundation under the Lagrange Fellow project.

644 V. LANZA ET AL.

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Analysis of nonlinear oscillatory network dynamics via time-varying amplitude and phase variables

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