On Global Dynamic Behavior of Weakly

On Global Dynamic Behavior of Weakly Connected Oscillatory Networks Marco Gilli, Michele Bonnin, and Fernando Corinto Department of Electronics, Politecnico di Torino, Italy. Abstract The global dynamics of weakly connected oscillatory networks is investigated: as a case study one-dimensional arrays of third order oscillators are considered. Through the joint application of the describing function technique and of Malkin’s Theorem a very accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase deviation due to the weak coupling) is derived. The total number of limit cycles and their stability properties are estimated via the analytical study of the phase deviation equation. The proposed technique significantly extends the results available in the literature and can be applied to almost all complex networks of oscillators. In particular two-dimensional, space variant and fully connected networks can be dealt with.

1

Introduction

Recent studies on thalamo-cortical systems have shown that weakly connected networks (WCNs) of oscillators represent a good architecture for a neurocomputer. In particular they have associative properties and can be exploited for dynamic pattern recognition [Hoppensteadt & Izhikevitch, 1997; Hoppensteadt & Izhikevitch, 1999]. Such structures can be adequately modelled as cellular neural/nonlinear networks (CNNs) (also called nonlinear dynamic arrays - NDAs), a new paradigm of analog dynamic processors, that were introduced some years ago in the electrical engineering community [Chua & Yang, 1988a; Chua & Yang, 1988b; Chua & Roska, 1993a; Chua, 1995; Chua et al., 1995; Chua & Roska, 2002]. CNNs are described as a 2 or n-dimensional arrays of mainly identical nonlinear dynamical systems (called cells). In most applications the connections are local and specified through space-invariant templates (that consist of small sets of parameters identical for all the cells). The local connectivity has allowed the realization of several high-speed VLSI chips [Vazquez et al., 2000]. The mathematical model of a CNN consists of a large system of locally coupled nonlinear ordinary differential equations (ODEs), that may exhibit a rich spatio-temporal dynamics, including several attractors and bifurcation phenomena [Chua, 1995]. A complete study of their dynamics would require to classify, for given sets of parameters, all the attractors and possibly to estimate their domains of attraction. Time domain numerical simulation has allowed one to discover several spatio-temporal dynamic phenomena in CNNs [Chua, 1995], but it is not suitable for discovering and classifying all the attractors of a high-dimensional network. In fact the global dynamic analysis, through the sole numerical simulation, would require to identify for each choice of network parameters all sets of initial conditions that converge to different attractors. This would be a formidable, and practically impossible task. Recently, some harmonic balance (HB) based techniques have been applied to the study of the dynamic behavior of space-invariant dynamic arrays (see [Civalleri & Gilli, 1996; Crounse & Chua, 1995; Gilli, 1995; Gilli, 1997; Gilli et al., 2004; Goras & Chua, 1995; Setti et al., 1998; Thiran 1

et al., 1998]). In [Crounse & Chua, 1995; Goras & Chua, 1995; Thiran et al., 1998] the authors have considered stable CNNs, composed by first-order cells; they have shown that a linear spectral technique, though approximate, can yield some insight on their asymptotic behavior. In [Civalleri & Gilli, 1996; Gilli, 1995] a spatio-temporal spectral approach, based on the describing function technique, was exploited for investigating generalized CNNs and NDAs with infinitely many cells. In [Setti et al., 1998] a HB based technique was applied to the study of one-dimensional CNNs with periodic boundary conditions. In [Gilli, 1997] a suitable extension of the describing function technique was applied to the study of one-dimensional CNNs and shown to be useful for predicting and classifying some significant spatio-temporal phenomena. In [Gilli et al., 2004] the authors presented a mixed time-frequency domain method, based on the joint application of two frequency domain techniques, i.e. describing function and harmonic balance, and of a numerical time-domain algorithm for computing limit cycle Floquet’s multipliers. The method is suitable for investigating periodic oscillations and their bifurcations in one dimensional space invariant CNNs. It was applied to the study of the global dynamic behavior of a one-dimensional array of third order oscillators (Chua’s circuit) in case of positive symmetric couplings. In this manuscript we focus on weakly connected networks of oscillators. As shown in [Hoppensteadt & Izhikevitch, 1997], such networks can be investigated through an equation (called phase deviation equation), that describes the evolution of the oscillator phase deviations with respect to the oscillator phases in absence of coupling. If the oscillating frequencies are commensurable, then Malkin’s Theorem yields a formal way for deriving the phase deviation equation, providing that the limit cycle trajectory of each uncoupled oscillator is known. Unfortunately the periodic trajectories of almost all non-trivial oscillators, can only be determined through numerical simulations. This implies that, in general, an explicit close form expression for the phase deviation equation is not available. It turns out that in most cases such an equation cannot be exploited for investigating the global dynamic behavior of the network and for classifying its periodic attractors. The main contribution of this manuscript is to show that an approximate analytic expression of the phase deviation equation can be derived, via the joint application of the describing function technique and of Malkin’s Theorem. The paper is organized as follows. In Section 2 the mathematical model of a weakly connected network is introduced, with particular emphasis on those networks that admits of a Lure-like model. Then a simplified version of Malkin’s Theorem is enunciated. In Section 3 we focus on a particular WCN composed by third order oscillators: a one-dimensional (1D) array of Chua’s circuits, that has been widely studied in several papers (see [Chua, 1995] and [Gilli et. al., 2004] for a review). In Subsections 3.1-3.3 we show that an approximate expression of the phase deviation equation can be analytically derived, through the application of the describing function technique. Then in Subsection 3.4 we show that a detailed analytical analysis of this equation allows one to estimate the total number of periodic orbits and their stability properties. All the results, although obtained through an approximate method, have been verified via extensive numerical simulations. The results presented in [Gilli et al., 2004], and partially based on a numerical approach, have been confirmed through our analytical approach and significantly extended. Furthermore we remark that, through the proposed technique, the phase deviation equation can be accurately approximated for general two-dimensional, space-variant, fully connected arrays, composed by oscillators with different and commensurable angular frequencies.

2

Weakly Connected Networks and Malkin’s Theorem

According to [Hoppensteadt & Izhikevitch, 1997] a weakly connected network (WCN), composed by n cells of dynamical order m, is described by the following system of nonlinear ordinary differential

2

equations (ODEs): X˙ i = Fi (Xi ) + ε Gi (X ),

X = [X1T , ... XnT ]T , Fi : Rm → Rm , Gi : Rm×n → Rm ,

(1 ≤ i ≤ n) (1) where T denotes transposition and ε represents a small parameter that guarantees a weak connection among the cells. In some significant cases each uncoupled cell admits of a Lur’e representation of the form [Genesio & Tesi, 1993]: L(D)xi (t) = fi [xi (t)] (2) where xi (t) represents one scalar component of Xi , fi (·) is a scalar Lipschitz nonlinear function and L(D) is a rational function of the time-differential operator D = d/dt. If the cells interact only through the state variables xi (t), then the WCN is described by the following simplified system of equations: L(D)xi = fi (xi ) + ε gi (x ), x = [x1 , ... xn ]T , gi : Rn → R

(3)

We assume that, in absence of coupling, each cell only exhibits the following invariant limit sets: a finite number of unstable equilibrium points, a finite number of either stable or unstable periodic limit cycles and at least one asymptotically stable limit cycle. According to these assumptions, we focus on a set of parameters and initial conditions such that the trajectory of each uncoupled cell is a periodic (either stable or unstable) limit cycle, described by a regular curve γi (t) ⊂ Rm . If we denote by ωi and θi ∈ S 1 = [0, 2π[ the angular frequency and the phase respectively of each limit cycle γi (t), then the WCN admits of the following description in term of phase variables: θi (t) = ωi t + φi (t) (4) where φi (t) ∈ S 1 represents the phase deviation from the natural oscillations, due to weak coupling. If the angular frequencies ωi are commensurable, then Malkin’s Theorem [Hoppensteadt & Izhikevitch, 1997] provides an explicit way for deriving the system of differential equations, that governs the phase deviation evolution. For the sake of the completeness and for introducing the proper notations, we report here a simplified version of Malkin’s Theorem, based on Theorem 9.2 of [Hoppensteadt & Izhikevitch, 1997]. Theorem 1 (Malkin’s Theorem for weakly coupled oscillator, with commensurable angular frequencies): Consider a WCN described by (1) and assume that each uncoupled cell X˙ i = Fi (Xi ), Xi ∈ Rm ,

(1 ≤ i ≤ N )

(5)

has a hyperbolic (either stable or unstable) periodic orbit γi (t) ⊂ Rm of period Ti and angular frequency ωi = 2π/Ti . Let τ = ε t be slow time and let φi (τ ) be the phase deviation from the natural oscillation γi (t), t ≥ 0. Then the vector of phase deviation φ = (φ1 , φ2 , ..., φn )T is a solution to: φ0i = Hi (φ − φi , ε) φ − φi = (φ1 − φi , φ2 − φi , ..., φn − φi )T ∈ [0, 2π[n = T n , where 0 =

d dτ

(1 ≤ i ≤ n) (6)

and Hi (φ − φi , 0) = µ

φ − φi γ t+ ω

¶

1 T ·

=

· µ

Z T

QTi (t)

0

µ

γ1T

φ − φi Gi γ t + ω ¶

µ

¶¸

dt

φ1 − φi φn − φi t+ , ..., γnT t + ω1 ωn

3

¶¸T

(7)

being T the minimum common multiple of T1 , T2 , ..., Tn . In the above expression (7) Qi (t) ∈ Rm is the unique nontrivial Ti -periodic solution to the linear time-variant system: Q˙ i (t) = −[DFi (γi (t))]T Qi (t)

(8)

satisfying the normalization condition QTi (0)Fi (γi (0)) = 1

(9)

Proof: The Theorem represents the extension of Theorem 9.2 of [Hoppensteadt & Izhikevitch, 1997] to the case of oscillators that exhibit hyperbolic limit cycles (either stable or unstable) with different angular frequencies ωi . The proof can be derived from [Hoppensteadt & Izhikevitch, 1997] in a straightforward way and therefore it is not reported here. It is only important to point out that Theorem 9.2 also holds if each limit cycle γi (t) is not asymptotically stable, but simply hyperbolic (i.e. all the Floquet’s multipliers, except one, does not lie on the unit circle).

3

Application of the Describing Function Technique

As pointed out in Section 1, we have assumed that the invariant limit sets of each uncoupled cell are a finite number of unstable equilibrium points and of periodic orbits (either stable or unstable) with the constraint that at least one of them is stable. In this Section we will show that the joint application of the describing function technique and of Malkin’s Theorem allows one to investigate the global dynamic behavior of weakly connected networks of oscillators; in particular it provides an approximate, but very accurate analytical method for estimating the total number of periodic cycles and their detailed stability properties. Hereafter we assume that, in absence of coupling, each periodic orbit can be detected by the describing function technique, i.e. that there exists a one-to-one correspondence between the number of periodic orbits and the solution provided by the application of the describing function technique. The rigorous conditions under which such an assumption holds are very difficult to check (see [Mees, 1981]); however for most oscillators the describing function solutions can be considered reliable if the normalized distortion index, defined in [Genesio & Tesi, 1993], is sufficiently small. In order to illustrate our method, we focus on a particular network of oscillators: a one-dimensional (1D) array of Chua’s circuit. However we remark that the proposed technique can be applied to any WCN, even if the connections are not local and space invariant and the cells are not identical. A 1D WCN composed by n Chua’s circuits is described by the following system of normalized equations (see also [Osipov & Shalfeev, 1995]): x˙ i = α [yi − xi − n(xi )] + ε (d1 xi−1 + d2 xi+1 − 2d xi ) y˙ i = xi − yi + zi z˙i

(1 ≤ i ≤ n)

(10)

= −β yi

The parameters α and β are defined in [Madan, 1993] and [Genesio & Tesi, 1993], whereas d, d1 and d2 represent the coupling coefficients; n(xi ) denotes the well known nonlinear memoryless resistance of the Chua’s diode (see [Madan, 1993] and [Genesio & Tesi, 1993]). For the sake of the simplicity, the boundary conditions are assumed to be fixed at x0 (t) = xN +1 (t) = 0. We also assume, according to [Genesio & Tesi, 1993], that the function n(xi ) admits of the following cubic approximation, that guarantees that each uncoupled cell possesses three equilibrium points, located at xi = 0, ±1.5 8 4 n(xi ) = − xi + x3i 7 63 4

(11)

According to [Khibnik et al., 1993], the parameters α and β are chosen in such a way that, in absence of coupling, the i-th cell exhibits the following invariant limit sets (depicted for α = 8 and β = 15 in Fig. 1): • Three unstable equilibrium points, corresponding to xi = ±1.5 and to xi = 0 respectively. − • Two asymmetric stable limit cycles (denoted by A+ i and Ai ) mainly lying in the regions xi > 1 and xi < −1 respectively.

• One stable symmetric limit cycle (denoted by Sis ). • One unstable symmetric limit cycle (denoted by Siu ). It is easily derived that equations (10) are a a particular case of (1) by assuming: 



xi   Xi =  yi  , zi





α [yi − xi − n(xi )]   Fi (Xi ) =  xi − yi + zi , −β yi





d1 xi−1 + d2 xi+1 − 2d xi   Gi (X ) =  0  (12) 0

Some algebraic manipulations (see [Gilli et al., 2004]) also allow one to recast equations (10) in the Lur’e form (3): L(D)xi = fi (xi ) + ε gi (x ) (13)

L(s) =

s3 + s2 (1 + α) + sβ + αβ , (s2 + s + β)

f (xi ) = − α n(xi ),

gi (x ) = d1 xi−1 + d2 xi+1 − 2d xi

(14)

Furthermore the two internal state variables of each cell, i.e. yi (t) and zi (t), are related to xi (t) via the following linear relations: yi (t) = Ly (D) xi (t),

Ly (D) =

zi (t) = Lz (D) xi (t),

D D2 + D + β

Lz (D) = −

β D2 + D + β

(15) (16)

In order to apply Malkin’s Theorem to the weakly connected array of Chua’s circuit (10) and to compute the phase deviation equation (6) the knowledge of γi (t) (i.e. the limit cycle trajectories in absence of coupling) is required. Our method is based on the idea that γi (t) can be approximately computed by exploiting the describing function technique. More precisely it consists of the following four fundamental steps (developed in detail in subsections 3.1 - 3.4): 1. A first harmonic approximation of each limit cycle γi (t) is determined, through the application of the describing function technique. Note that since in the case under study all the cells are identical, all the limit cycle trajectories γi (t) are also identical. 2. Once γi (t) is known, a first harmonic approximation of Qi (t) is computed, by exploiting (8) and the normalization condition (9). 3. The phase deviation equation (6) is derived by analytically computing the functions (7). 4. The phase equation is analyzed in order to determine the total number of stationary solutions (equilibrium points) and their stability properties: they correspond to the total number of limit cycles of the original weakly connected network, described by (10).

5

3.1

Describing function approximation of γi (t)

We briefly recapitulate how the describing function technique can be applied to a single uncoupled Chua’s circuit, described by the Lur’e model (13) (details are available in [Genesio & Tesi, 1993] and [Gilli et al., 2004]). The state xi (t) is represented through a bias term and a single harmonic, with suitable amplitude and angular frequency: xi (t) ≈ x ˆi (t) = Ai + Bi sin (ω t) (1 ≤ i ≤ n) (17) where Ai denotes the bias, Bi the amplitude of the first harmonic, and ω is the angular frequency. The output of the nonlinear function f (·), when the input is (17), admits of the following first harmonic representation (that in several cases can be expressed through a close analytical form): f (ˆ xi (t)) ≈ fˆ(ˆ xi (t)) = FiA (Ai , Bi ) + FiB (Ai , Bi ) sin (ω t) where: FiA (Ai , Bi ) =

1 2π

Z π −π

(18)

f [Ai + Bi sin(θ)] dθ

(19) Z π 1 f [Ai + Bi sin(θ)] sin(θ) dθ = π −π For the Chua’s circuit described by (13) and (14) the expressions above can be analytically computed. We have: ¶ µ Z π 4 2 2 2 1 8 FiA (Ai , Bi ) = Ai + Bi −α n [Ai + Bi sin(θ)] dθ = −α Ai − + 2π −π 7 63 21 µ ¶ (20) Z π 8 4 1 1 2 2 FiB (Ai , Bi ) = A + B −α n [Ai + Bi sin(θ)] sin(θ) dθ = −α Ai − + π −π 7 21 i 21 i FiB (Ai , Bi )

By substituting in (13) the approximate expressions (17) and (18) for xˆi (t) and fˆ(ˆ xi (t)) respectively, we derive a set of 3 nonlinear equations in the 3 unknowns Ai , Bi and ω. The set of 3 equations is reported below: L(0) Ai = FiA (Ai , Bi )

(21)

Re[L(jω)] Bi = FiB (Ai , Bi )

(22)

Im[L(jω)] = 0

(23)

The oscillation frequencies are determined by solving equation (23). By exploiting (14), the following admissible oscillation frequencies are computed: ω ˆ1 =

v u u t

1+α β− + 2

sµ

1+α 2

¶2

− β,

ω ˆ2 =

v u u t

1+α β− − 2

sµ

1+α 2

¶2

−β

(24)

The solution of the remaining two equations of the describing function system (21)-(22) provides an accurate characterization of each periodic orbit [Gilli et al., 2004]: − • The two stable asymmetric limit cycles A+ i and Ai are characterized by the following describing function parameters:

s

Ai = ±

63 20

µ

s

¶

15 2 − Re[L(j ω ˆ 2 )] , 7 α

Bi =

21 5

µ

¶

5 1 − + Re[L(j ω ˆ 2 )] , 7 α

ω ˆ=ω ˆ2 (25)

6

• The stable symmetric limit cycle Sis gives rise to the following describing function parameters: s

Ai = 0,

Bi =

µ

21

¶

1 8 − Re[L(j ω ˆ 1 )] , 7 α

ω ˆ=ω ˆ1

(26)

ω ˆ=ω ˆ2

(27)

• The unstable symmetric limit cycle Siu is described by parameters: s

Ai = 0,

Bi =

µ

21

¶

8 1 − Re[L(j ω ˆ 2 )] , 7 α

Once the approximate expression x ˆi (t) of xi (t) is known, the approximate expressions yˆi (t) and zˆi (t) of yi (t) and zi (t) respectively can be readily determined via the linear differential relations (15) and (16). We have: yˆi (t) = Ly (D) x ˆi (t) = Ly (0) Ai + Re[Ly (jω)] Bi sin(ωt) + Im[Ly (jω)] Bi cos(ωt)

(28)

zˆi (t) = Lz (D) x ˆi (t) = Lz (0) Ai + Re[Lz (jω)] Bi sin(ωt) + Im[Lz (jω)] Bi cos(ωt)

(29)

where ω equals either ω1 or ω2 (see 24), depending on the limit cycle under consideration. It turns out that each limit cycle trajectory γi (t) admits of the following first harmonic approximation:  



Ai + Bi sin(ωt)



  γi (t) ≈ γˆi (t) =  Ly (0) Ai + Re[Ly (jω)] Bi sin(ωt) + Im[Ly (jω)] Bi cos(ωt)    Lz (0) Ai + Re[Lz (jω)] Bi sin(ωt) + Im[Lz (jω)] Bi cos(ωt)

3.2

(30)

Describing function approximation of Qi (t)

In order to determine the vector function Qi (t) defined in (8) and (9), the Jacobian matrix DFi [γi (t)] has to be evaluated. The following expression is obtained by (12):    DFi [γi (t)] =   

−α {1 + n0 [xi (t)]}

α

1

−1

0

−β 0

0



  1   

(31)

If we denote with γ˜i (t) a small perturbation, with respect to the exact limit cycle solution γi (t), the variational equation is derived: d˜ γi (t) = DFi [γi (t)] γ˜i (t) dt

(32)

It is worth noting that only the explicit expression of xi (t) is needed for computing DFi [γi (t)]. By exploiting the scalar Lur’e model (13), (14), it is easily verified that an alternative scalar expression for the variational equation holds: L(D) x ˜i (t) = f 0 [xi (t)] x ˜i (t)

(33)

where x ˜(t) denotes a small perturbation with respect to xi (t). It is well known that γ˙ i (t) satisfies the vector variational equation (32) and hence x˙ i (t) satisfies the scalar one (33). 7

i (t) i (t) In the Proposition below we prove that the same property holds for dˆγdt and dˆxdt , if in (32) and (33), we substitute to γi (t) and xi (t) their first harmonic approximation γˆi (t) and x ˆi (t).

Proposition 1: The following two equations are satisfied, if all the harmonics of order higher than one are neglected L(D)

dˆ xi (t) dt

d2 γˆi (t) dt2

= f 0 [ˆ xi (t)]

dˆ xi (t) dt

= DFi [ˆ γi (t)]

(34)

dˆ γi (t) dt

(35)

Proof: It is readily verified that for any single-valued function p(·), the following equations hold: Z π

1 π 1 π

Z π −π

p[Ai + Bi sin(θ)] sin(2kθ) dθ = 0

(k ≥ 1)

p[Ai + Bi sin(θ)] cos[(2k + 1)θ] dθ = 0

(k ≥ 0)

−π

(36)

This implies that p[ˆ xi (t)] = f 0 [ˆ xi (t)] admits of the following Fourier expansion: p[ˆ xi (t)] = f 0 [Ai + Bi sin(ωt)] =

PiA (Ai , Bi ) + PiB (Ai , Bi ) sin (ωt) +

∞ X

C Pk,i (Ai , Bi ) sin [(2k + 1)ωt]

k=1

+

∞ X

D Pk,i (Ai , Bi ) cos (2kωt)

(37)

k=1

The time-derivative of x ˆi (t) can be expressed as: dx ˆi (t) = ω Bi cos(ωt) (38) dt Some algebraic manipulations allow one to write the first harmonic approximation of the right side of (34): · ¸ dˆ xi (t) 1 D 0 A f [ˆ xi (t)] ≈ Pi (Ai , Bi ) + P1,i (Ai , Bi ) ω Bi cos(ωt) (39) dt 2 On the other hand, due to (23), the left side of (34) yields: dˆ xi (t) = Re[L(jω)] ω Bi cos(ωt) dt We note that the following identity holds: L(D)

FiB (Ai , Bi )

=

1 π

=

1 π

=

Z π −π

Z π −π

Z π 1

π

−π

½

= Bi

(40)

f [Ai + Bi sin(θ)] sin(θ) dθ f 0 [Ai + Bi sin(θ)] Bi cos2 (θ) dθ f 0 [Ai + Bi sin(θ)] Bi

1 2π

Z π −π

1 + cos(2θ) dθ 2

f 0 [Ai + Bi sin(θ)] dθ +

·

= Bi PiA (Ai , Bi ) +

1 2π

Z π −π

¾

f 0 [Ai + Bi sin(θ)] cos(2θ) dθ

¸

1 D P (Ai , Bi ) 2 1,i 8

(41)

By using (41), the approximate relation (39) can be written as follows: f 0 [ˆ xi (t)]

dˆ xi (t) F B (Ai , Bi ) ≈ i ω Bi cos(ωt) dt Bi

(42)

Then by using (22), it is seen that second side of (42) exactly coincides with the second side of (40). This proves the first part of the Proposition, i.e. that (34) holds if all the harmonics of order higher than one are neglected. Under the same assumption, it is verified that the equation below holds:    d    dt   

dˆ xi (t) dt dˆ xi (t) Ly (D) dt dˆ xi (t) Lz (D) dt





    γi (t)]  = DFi [ˆ   

       

dˆ xi (t) dt dˆ xi (t) Ly (D) dt dˆ xi (t) Lz (D) dt

        

(43)

The second part of the Proposition is then proved, by noting that (15) and (16) imply: dˆ yi (t) dt

= Ly (D)

dˆ xi (t) dt

(44)

dˆ zi (t) dt

= Lz (D)

dˆ xi (t) dt

(45) Q.E.D.

We remark that the proof of the Proposition above is valid for any single-valued nonlinear function f (·). In the particular case of Chua’s circuit, f (·) is a third order polynomial (see (14) and (11)); hence f 0 (·) is a second order polynomial and all the coefficients of the Fourier expansion (37) are null, D. with the exception of PiA , PiB and P1,i According to Malkin’s Theorem, Qi (t) is the unique Ti -periodic solution to the linear time-variant system (see (8)):   

Q˙ i (t) = −{DFi [γi (t)]}T Qi (t) =   

α {1 + n0 [xi (t)]} −1 0 −α 0

1



  β   Qi (t) 

(46)

−1 0

If we focus on the first component of the vector Qi (t) = [qi (t), ri (t), si (t)]T a simplified scalar Lur’e like variational equation can be derived, through some straightforward algebraic manipulations: L(−D) qi (t) = f 0 [xi (t)] qi (t)

(47)

The second and the third component of Qi (t), i.e. ri (t) and si (t) are then related to qi (t) via a linear differential operator, that is readily derived from (46): ri (t) = α Ly (−D) qi (t) si (t) = −

α Lz (−D) qi (t) β

(48) (49)

The aim of this Section is to find a first harmonic approximation qˆi (t) of qi (t). Since L(s) is a hermitian function and Im[L(jω)] = 0, variational equation (47) is approximated by (34). Hence we expect that the for qˆi (t) an expression similar to x ˆi (t) holds. The result is enunciated in the following Proposition: 9

Proposition 2: If all the harmonics of order higher than one are neglected, then qˆi (t) = Ei cos(ωi t), (Ei ∈ R) satisfies the equation below (that is obtained from (47) by replacing xi (t) with its describing function approximation x ˆi (t)): L(−D) qˆi (t) = f 0 [ˆ xi (t)] qˆi (t) (50) Proof: Equation (50) is readily verified, by substituting qˆi (t) = Ei cos(ωi t) and by noting that Im[L(jω)] = 0 and Re[L(−jω)] = Re[L(jω)]. By exploiting (39) and (41), we obtain: L(−D) Ei cos(ωi t) = Re[L(jω)] Ei cos(ωt) · 0

f [ˆ xi (t)] qˆi (t) ≈

(51)

¸

PiA (Ai , Bi )

1 D F B (Ai , Bi ) + P1,i Ei cos(ωt) (Ai , Bi ) Ei cos(ωt) = 2 Bi

(52)

It is derived that (51) and (52) coincides for any Ei ∈ R, which proves the thesis of the Proposition. Q.E.D. ˆ i (t) of vector Qi (t) is then determined, by exploiting (48), The describing function approximation Q (49) and the fact that Re[Ly (−jω)] = Re[Ly (jω)], Im[Ly (−jω)] = −Im[L(jω)], Re[Lz (−jω)] = Re[Lz (jω)], Im[Lz (−jω)] = −Im[Lz (jω)]:  



cos(ωt)



ˆ i (t) = Ei  Qi (t) ≈ Q 

α {Re[Ly (jω)] cos(ωt) + Im[Ly (jω)] sin(ωt)}    α  − {Re[Lz (jω)] cos(ωt) + Im[Lz (jω)] sin(ωt)} β

(53)

The coefficient Ei is the only unknown in the above expression and should be determined by imposing the normalization condition (9). It is shown in [Hoppensteadt & Izhikevitch, 1997] that the exact solutions γi (t) and Q(t) exhibit the following property: QTi (t) · Fi [γi (t)] = QTi (t) ·

d γi (t) = const, dt

∀t

(54)

We will prove that the same property is valid if first order harmonic approximations are taken for γi (t) and Q(t) respectively. Proposition 3: The following relationship holds: ˆ Ti (t) · Fi [ˆ ˆ Ti (t) · d γˆi (t) = const, Q γi (t)] = Q dt

∀t

(55)

ˆ where γˆi (t) and Q(t) are given by (30) and (53). Proof: Firstly we compute the time derivative of γˆi (t), by exploiting (38), (44) and (45): 

d γˆi (t) = ω Bi dt



cos(ωt)

   Re[L (jω)] cos(ωt) − Im[L (jω)] sin(ωt)  y y    

(56)

Re[Lz (jω)] cos(ωt) − Im[Lz (jω)] sin(ωt)

The explicit computation of (55) yields: ˆ T (t) · d γˆi (t) Q i dt

·µ

= ω α Bi Ei

1 1 + {Re[Ly (jω)]}2 − {Re[Lz (jω)]}2 α β µ

+

1 {Im[Lz (jω)]}2 − {Im[Ly (jω)]}2 β 10

¶

cos2 (ωt)

¶

¸ 2

sin (ωt)

(57)

From (15) and (16) we derive real and imaginary parts of Ly (jω) and Lz (jω): Ly (jω) =

jω 2 (jω) + jω + β

=

ω2 (β − ω 2 )2 + ω 2

β β (β − ω 2 ) Lz (jω) = − = − 2 (jω) + jω + β (β − ω 2 )2 + ω 2 By substituting (58) in (57) we obtain:

+ j

ω (β − ω 2 ) (β − ω 2 )2 + ω 2

βω + j (β − ω 2 )2 + ω 2

ˆ T (t) · d γˆi (t) = ω α Bi Ei [Kc cos2 (ωt) + Ks sin2 (ωt)] Q dt Ã

Kc = Ã

Ks =

1 β (β − ω 2 )2 ω4 − + α [(β − ω 2 )2 + ω 2 ]2 [(β − ω 2 )2 + ω 2 ]2 ω2 β ω 2 (β − ω 2 )2 − 2 2 2 2 [(β − ω ) + ω ] [(β − ω 2 )2 + ω 2 ]2

(58)

(59)

!

(60)

!

(61)

Expression (59) is constant if and only if Kc = Ks . In order to prove this assertion, we firstly note that the following relation holds between L(jω) and Ly (jω) given by (14) and (15) respectively: L(jω) = jω + α − αLy (jω)

(62)

Since Im[L(jω)] = 0, this implies:

ω (63) α that, by exploiting (58), provides the following explicit constraint involving the variables ω, β, and α: Im[Ly (jω)] =

(β − ω 2 ) 1 = 2 2 2 (β − ω ) + ω α

(64)

We are now ready to show that Kc − Ks = 0 (i.e. Kc = Ks ). By using (60) and (61), we obtain: Kc − Ks =

1 ω 4 − β(β − ω 2 )2 − βω 2 + ω 2 (β − ω 2 )2 + α [(β − ω 2 )2 + ω 2 ]2

=

ω 2 (ω 2 − β) + (β − ω 2 )2 (ω 2 − β) 1 + α [(β − ω 2 )2 + ω 2 ]2

=

1 (ω 2 − β)[(β − ω 2 )2 + ω 2 ] + α [(β − ω 2 )2 + ω 2 ]2

=

1 (ω 2 − β) + α [(β − ω 2 )2 + ω 2 ]

(65)

By using (64) it is readily derived that Kc − Ks = 0. This implies that expression (59) is constant, i.e. the thesis of the Proposition. Q.E.D. Proposition 3 allows one to analytically determine the coefficient Ei and hence to complete the deˆ i (t), according to (53). By applying the normalization scribing function approximation of qˆi (t) and Q condition (9) to equation (59), by exploiting the fact that Kc = Ks and by using (61), we obtain: Ei (ω) =

1 [(β − ω 2 )2 + ω 2 ]2 ω 3 αBi β − (β − ω 2 )2 11

(66)

Since Bi and α are positive, the sign of Ei coincides with that of β − (β − ω 2 )2 , that depends on the oscillation frequency under consideration. By using (24) we derive the following: β − (β − ω12 )2

 



1+α = β − β − β − +  2

sµ

1+α 2

¶2

2  −β 

  sµ ¶2 µ ¶2 ¶2 µ 1 + α 1 + α 1 + α 1 + α + −β −2 −β = β −

2

1+α = 2 2 sµ

> 2

β − (β − ω22 )2

sµ

1+α 2

 

2

1+α 2

2

"µ

¶2

−β −2 sµ

¶2

−β

1+α 2



1+α − = β − β − β −  2

1+α 2

#

¶2

−β "µ

¶2

sµ

−β −2

1+α 2

¶2

2

1+α 2

#

¶2

−β

= 0

(67)

2  −β 

  sµ µ ¶2 µ ¶2 ¶2 1 + α 1 + α 1 + α 1 + α = β − + −β +2 −β

2

1+α = −2 2

sµ

2

1+α 2

2

"µ

¶2

−β −2

1+α 2

2

#

¶2

−β

0

(69)

Ei (ω2 ) < 0

(70)

Phase deviation equation

We will show in this subsection that an explicit and very accurate expression of the phase deviation ˆ i (t) equation can be derived by substituting in (7) the describing function expressions of γˆi (t) and Q provided in (30) and (53). By remembering that, according to (12) only the first component of Gi (X ) is different from zero, it is obtained: φ0i

1 ≈ Hi (φ − φi , 0) = T ≈

1 T

Z

Z 0

T

QTi (t) ·

T

[qi (t), ri (t), si (t)] · 0

· µ ¶¸ φ − φi Gi γ t + ωi

µ ¶ µ ¶ ¸T φi−1 − φi φi+1 − φi d1 x ˆi−1 t + + d2 x ˆi+1 t + − 2dx ˆi (t), 0, 0 dt ω ω

12

=

=

1 T 1 T

Z

T

0

Z

¶ µ ¶ ¸ · µ φi+1 − φi φi−1 − φi qˆi (t) d1 x ˆi−1 t + + d2 x ˆi+1 t + − 2dx ˆi (t) dt ω ω

T

Ei cos(ωt) {d1 [Ai−1 + Bi−1 sin(ωt + φi−1 − φi )] + d2 [Ai+1 + Bi+1 sin(ωt + φi+1 − φi )] 0

− 2 d [Ai + Bi sin(ωt)]} dt =

Ei [d1 Bi−1 sin(φi−1 − φi ) + d2 Bi+1 sin(φi+1 − φi )] 2

(71)

Since we have assumed that all the cells are identical (i.e. Bi+1 = Bi = Bi−1 ), the equation above admits of the following simplified form: Ei Bi [d1 sin(φi−1 − φi ) + d2 sin(φi+1 − φi )] (72) 2 In the case under study, i.e. a network composed by n cells, with zero boundary conditions, the following system of n phase deviation differential equations is derived: φ0i =

φ01 =

Ei Bi d2 sin(φ2 − φ1 ) 2

φ0i =

Ei Bi [d1 sin(φi−1 − φi ) + d2 sin(φi+1 − φi )] 2

(73) (2 ≤ i ≤ n − 1)

(74)

Ei Bi d1 sin(φn−1 − φn ) (75) 2 We will show in the next subsection that the above equation plays an important role for the analysis of the global dynamic behavior of networks of oscillators. In particular the following remarks hold: φ0n =

Remark 1: Equation (71) reduces a rather complex network of third order oscillators to a simple Kuramoto-like model [Kuramoto, 1984], that can be analytically dealt with. This opens the possibility of developing new applications, that exploit the rich dynamic behavior of nonlinear dynamic arrays, including dynamic pattern recognitions and dynamic associative memories [Hoppensteadt & Izhikevitch, 1999] Remark 2: The case under study is a one dimensional array of identical oscillators, with local spaceinvariant connections. However the proposed approach allows one to explicitly derive a very accurate analytical approximation of the phase deviation equation for more complex networks. In particular the general case of two-dimensional fully connected arrays, composed by oscillators with different and commensurable angular frequencies can be dealt with.

3.4

Analysis of the phase deviation equation

We focus on the phase deviation equations, derived in the previous subsection, for the 1D array of Chua’s circuits (73)-(75). We show that that the analytical form of the phase equations provides an accurate description of the global dynamic behavior of the WCN under study. As pointed out in subsection 3.1 (and depicted in Fig. 1) each uncoupled cell exhibits two stable − u asymmetric limit cycles (denoted by A+ i and Ai ) and one unstable symmetric limit cycle (Si ) with angular frequency ω2 , and one stable symmetric limit cycle (Sis ) with angular frequency ω1 . According to the assumptions of Malkin’s Theorem (that requires that the frequencies be commensurable), we consider the following three main cases: 1. Each uncoupled oscillator exhibits one of the two asymmetric stable limit cycles (denoted by A+ i and A− i ) with angular frequency ω2 . Hence the total number of cycles of the uncoupled network is 2n . 13

2. Each uncoupled oscillator exhibits the symmetric unstable limit cycle (denoted by Siu ) with angular frequency ω2 . This gives rise to a unique limit cycle for the uncoupled network. 3. Each uncoupled oscillator exhibits the symmetric stable limit cycle (denoted by Sis ) with angular frequency ω1 . This also yields a stable limit cycle for the uncoupled network. For each one of the 2n + 2 limit cycles of the uncoupled network, the phase equation (73)-(75) can be derived, by specifying the parameters ω, Bi and Ei , according to the theory developed in subsections 3.1-3.3. Then each equilibrium point of (73)-(75) corresponds to a periodic solution (either stable or unstable) of the weakly connected network. It is readily derived that the phase equation system (73)-(75) admits of an equilibrium point if and only if: ( 0 sin(φi+1 − φi ) = 0 → ηi = φi+1 − φi = (1 ≤ i ≤ n − 1) (76) π The total number of limit cycles of the WCN can accordingly be estimated. For each one of the admissible phase shifts {η1 , ..., ηn−1 } ∈ {0, π}, the WCN exhibits:

2n−1

± ± • 2n asymmetric limit cycles {A± 1 , A2 , ... AN }; they correspond to a angular frequency ω2 and to a solution of (21)-(22) with sign(A1 ) = ±1, sign(Ai ) = ±1, ... sign(An ) = ±1, respectively. s , S s }; it corresponds to a angular frequency ω and • One symmetric limit cycle {S1s , S2s , ... Sn−1 1 n to a solution of (21)- (22) with Ai = 0, ∀i. u , S u }; it corresponds to a angular frequency ω and • One symmetric limit cycle {S1u , S2u , ... Sn−1 2 n to a solution of (21)-(22) with Ai = 0, ∀i.

Note that the same results on the total number of limit cycles have already been derived in [Gilli et al., 2004] by using a completely different approach and they have been confirmed through extensive numerical simulations. However we point out that the results presented in [Gilli et al., 2004] are valid only if d1 /d2 > 0, whereas this assumption is not needed here. In addition the technique developed in [Gilli et al., 2004] shows several significant limitations. In particular it becomes untractable for two-dimensional networks and it cannot deal with space-variant and fully connected networks. The stability analysis of each limit cycle arises from the stability analysis of each stationary solution of the phase deviation equation, for given parameters d1 , d2 , Bi , Ei and ω. The jacobian matrix of system (73)-(75) has the following form:     E i Bi  J =  2   

−d2 cos(η1 )

d2 cos(η1 )

0

...

−[d1 cos(η1 ) + d2 cos(η2 )]

d2 cos(η2 )

...

.. .

..

..

..

0

...

d1 cos(ηn−2 )

−[d1 cos(ηn−2 ) + d2 cos(ηn−1 )]

0

...

0

d1 cos(ηn−1 )

d1 cos(η1 )

.

.

.



0

   ..   .  d2 cos(ηn−1 )   0

−d1 cos(ηn−1 ) (77)

We show that if d2 /d1 > 0 the sign of the eigenvalues of the Jacobian matrix (77) can be analytically determined. The results are summarized in the following Proposition. Proposition 4: Let us assume that d1 /d2 > 0, which implies that all the eigenvalues of the Jacobian matrix J are real. Let us denote with N the number of eigenvalues of J that have the same sign of Ei d2 , with M the number of eigenvalues of J that have sign opposite to Ei d2 , and with L the 14

number of null eigenvalues. Then L = 1, N equals the number of phase shifts ηi = π, and M equals the number of phase shifts ηi = 0. Proof: As a first step we introduce a non-singular diagonal matrix D (p = d2 /d1 ):             D =           

1

0

0 ...

0

0

0

1 2

0 ...

0

0

0

0

0

p ...

0

0

0

.. .

.. .

.. .

.. .

.. .

.. .

0

0

0 ... p

n−3 2

0

0

0

0

0 ...

0

n−2 2

0

0

0

0 ...

0

0 p

..

.

p

0

p

                      

(78)

n−1 2

Some straightforward algebraic computations allow one to verify that J can be transformed into a symmetric matrix P√through the similarity transform P = D J D −1 . The following expression holds for P (s = sign(d2 ) d1 d2 ):     E i Bi  P =  2   

−d2 cos(η1 )

s cos(η1 )

0

...

−[d1 cos(η1 ) + d2 cos(η2 )]

s cos(η2 )

...

.. .

..

..

..

0

...

s cos(ηn−2 )

−[d1 cos(ηn−2 ) + d2 cos(ηn−1 )]

0

...

0

s cos(ηn−1 )

s cos(η1 )

.

.



0

   ..   .  s cos(ηn−1 )   0

.

−d1 cos(ηn−1 ) (79)

As known, a similarity transform does not alter the eigenvalues: hence J and P have the same eigenvalues and they are all real. As a second step we note that matrix P can be put in diagonal form using the transformation T S PS = K where K is defined as follows:         Ei d2 Bi   K =−  2       



cos(η1 )

0

0

...

0

0

0

0

p cos(η2 )

0

...

0

0

0  

0 .. .

0 .. .

p2 cos(η3 ) . . . .. .. . .

0 .. .

0 .. .

0

0

0

. . . pn−3 cos(ηn−2 )

0

0

0

...

0

pn−1 cos(ηn−2 )

0

0

0

...

0

0

15

0



        0    0   

0 .. .

0

(80)

and S is the following upper triangular matrix            S =          

1

1 1

1 1

...

1 1

1 1

1 1

0 p2

p2

...

p2

p2

p2

0

0

p

...

p

p

p

.. .

.. .

.. .

.. .

.. .

.. .

.. .

0

0

0

... p

n−3 2

p

n−3 2

p

n−3 2

0

0

0

...

0

p

n−2 2

p

n−2 2

0

0

0

...

0

0

p

n−1 2

                     

(81)

The mapping P → S T P S = K is a congruence transformation of P. This transformation preserves symmetry but does not, in general, preserve eigenvalues. However, according to the Sylvester’s Inertia Theorem [Golub & Van Loan, 1996], [Demmel, 1997], the two matrices P and K have the same inertia, that is the same number of positive, negative and null eigenvalues. The following assertions hold: 1. P presents one null eigenvalue. 2. The number N of eigenvalues of P that have the same sign of Ei d2 equals the number of phase shifts ηi = π. 3. The number M of eigenvalues of P that have a sign opposite to Ei d2 equals the number of phase shifts ηi = 0. The thesis of the Proposition is then readily proved, by noting that matrices P and J possess the same eigenvalues. Q.E.D. Proposition 4 is useful for predicting limit cycle stability. As already pointed out, the coupled network exhibits (2n + 2) 2n−1 periodic limit cycles. Each uncoupled limit cycle is described by 3 Floquet’s multipliers (FMs), one of which is unitary; the other two ones are denoted by µ1 and µ2 . If the cycle s is stable (i.e. A± i and Si ) then the modulus of both the FMs is less than one; if the cycle is unstable (i.e. Siu ), then one FM has a modulus larger than one. A limit cycle of the coupled network exhibits 3n FMs. In case of weak connection, 2n of them have a numerical value very close to µ1 and µ2 respectively. The remaining n FMs (i.e. those that in absence of coupling equal 1) will be named Coupling FMs, because they describe the effect of the interactions among the cells. One of them is still unitary, whereas the other n − 1 are in general different from 1 and determine stability and/or instability, depending on their modulus. The absolute values of the Coupling FMs can be estimated by computing the sign of the eigenvalues of the Jacobian matrix J (77), according to the following rule: the unitary FMs corresponds to the zero eigenvalue of J ; the stable Coupling FMs (i.e. those with modulus less than one) correspond to the negative eigenvalues of J ; the unstable Coupling FMs correspond to the positive eigenvalues of J . In alternative the absolute values of the Coupling FMs can be evaluated as the eigenvalues of the matrix exp(K ), where K is given by (80). According to Proposition 4, and by taking into account that Ei (ω1 ) > 0 and Ei (ω2 ) < 0 (see (69), (70)), we derive the following:

16

1. If d2 > 0 (d2 < 0) then for limit cycles with angular frequency ω1 the number of stable (unstable) Coupling FMs equals the number of phase shifts ηi = 0; the number of unstable (stable) Coupling FMs equals the number of phase shifts ηi = π. 2. If d2 > 0 (d2 < 0) then for limit cycles with angular frequency ω2 the number of stable (unstable) Coupling FMs equals the number of phase shifts ηi = π; the number of unstable (stable) Coupling FMs equals the number of phase shifts ηi = 0. This allows us to classify the (2n + 2) 2n−1 periodic orbits of the coupled network, according to their stability properties. The following assertions hold. If d2 /d1 > 0 and d2 is positive then the weakly connected network exhibits: ± 1. 2n stable asymmetric limit cycles with angular frequency ω2 , i.e. those corresponding to {A± 1 , A2 , ... ± A± n−1 , An }, with all the phase shifts equal to π (ηi = π, i ≤ i ≤ n − 1).

2. 2n × (2n−1 − 1) unstable asymmetric limit cycles, with angular frequency ω2 ; they present as many Floquet’s multipliers |µ| > 1 as the number of phase shifts ηi = 0. 3. One stable symmetric limit cycle with angular frequency ω1 , i.e. that corresponding to {S1s , S2s , ... s , S s }, with all the phase shifts equal to zero (η = 0, i ≤ i ≤ n − 1). Sn−1 i n 4. 2n−1 − 1 unstable symmetric limit cycles with angular frequency ω1 , corresponding to {S1s , S2s , ... s , S s }, with at least one phase shift equal to π (∃ i: η = π). They exhibit as many Floquet’s Sn−1 i n multipliers |µ| > 1 as the number of phase shifts ηi = π. 5. 2n−1 unstable symmetric limit cycles, with angular frequency ω2 , corresponding to {S1u , S2u , ... u , S u }. The number of Floquet’s multipliers with |µ| > 1 can be computed as n plus the Sn−1 n number of phase shifts ηi = 0. If d2 < 0 the same classification holds, by simply exchanging ηi = 0 with ηi = π. In case of symmetric positive coupling, i.e. d1 = d2 > 0, the above results have been completely verified in [Gilli et al., 2004], where limit cycle Floquet’s multipliers have been numerically computed via a mixed time-frequency domain technique. The stability results presented here significantly improves those presented in [Gilli et al., 2004], for the following main reasons: 1. They are based on a simple analytical technique, whereas the method proposed in [Gilli et al., 2004] is partially numerical. 2. They address non-symmetric and non-positive coupling, that are not dealt with in [Gilli et al., 2004]. We also point out that the above classification cannot be considered rigorous because the proposed technique is based on the describing function approximation. Nevertheless all the results have been verified through extensive numerical simulations (based on the technique developed in [Gilli et al., 2004]. Some of them are reported in Figs 1 and 2, with reference to a one-dimensional array of Chua’s circuit, composed by 12 cells, and described by equations (10) with zero boundary conditions. Figs. 1 and 2 show the Coupling FM moduli, as a function of the coupling coefficient d2 , for the asymmetric + + + limit cycle {A+ 1 , A2 , ..., A11 , A12 }, with phase shifts [π, π, 0, 0, π, 0, π, π, π, π, π]. In Fig. 1, where the coupling parameters are assumed to be positive, the network exhibits, as expected, three unstable FMs, corresponding to the number of phase shifts ηi = 0. In Fig. 2 the coupling parameters are negative: the network presents as many unstable FMs as the number of phase shifts ηi = π, i.e. 8 unstable FMs. 17

As a final remark we note that the phase deviation equation can be explicitly written for any weakly connected network, including fully connected, space-variant and two-dimensional networks. In all cases (even those that cannot be dealt with analytically) the sign of the eigenvalues of the Jacobian matrix can be numerically evaluated in a straightforward way. Hence we are confident that the proposed technique is suitable for classifying periodic orbits in most weakly connected networks of oscillators.

4

Conclusions

We have investigated the global dynamic behavior of weakly connected networks of oscillators: as a case study we have considered a one dimensional array of third order oscillators (Chua’s circuits). As a main contribution we have shown that an accurate analytical expression of the phase deviation equation (i.e. the equation that describes the phase evolution, due to the weak coupling) can be derived via the joint application of the describing function technique and of Malkin’s Theorem. Then we have shown that a detailed analytical study of the phase deviation equation allows one to accurately estimate the total number of limit cycles and their stability characteristics (in terms of stable and unstable Floquet’s multipliers). In case of one dimensional arrays of Chua’s circuits all the results, though obtained through an approximate methods, have been verified via extensive numerical simulations. In addition the proposed technique has allowed us to significantly extends the results available in the literature. Finally we remark that an accurate analytical expression of the phase deviation equation can be derived for general two-dimensional, space variant, and fully connected networks of oscillators with commensurable frequencies.

References Chua, L. O. & Yang, L. [1988a] “Cellular neural networks: Theory,” IEEE, Trans. Circuits Syst. 35, 1257-1272. Chua, L. O. & Yang, L. [1988b] “Cellular neural networks: Applications,” IEEE, Trans. Circuits Syst. 35, 1273-1290. Chua, L. O. & Roska, T. [1993a] “The CNN paradigm,” IEEE, Trans. Circuits Syst.-I 40, 147-156. Chua, L. O. & Roska, T. [1993b] “The CNN universal machine: an analogic array computer,” IEEE, Trans. Circuits Syst.-I 40, 163-173. Chua, L. O. [1995] IEEE, Trans. Circuits Syst.-I: Special issue on nonlinear waves, patterns and spatio-temporal chaos in dynamic arrays 42, 559-823. Chua, L. O., Hasler, M., Moschytz, G. S. & Neirynck, J. [1995] “Autonomous cellular neural networks: a unified paradigm for pattern formation and active wave propagation,” IEEE, Trans. Circuits Syst.-I 42, 559-577. Chua, L. O. & Roska, T. [2002] Cellular Neural Networks and Visual Computing (Cambridge University Press, U.K.) Civalleri, P. P. & Gilli, M. [1996] “A spectral approach to the study of propagation phenomena in CNNs,” International Journal of Circuit Theory and Applications 24, 37-47. Crounse, K. L. & Chua, L. O. [1995] “Methods for image processing and pattern formation in cellular neural network: a tutorial,” IEEE, Trans. Circuits Syst.-I 42, 583-601.

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Demmel J. [1997] Applied Numerical Linear Algebra (SIAM, Philadelphia), pag. 114. Genesio, R. & Tesi, A. [1993] “Distortion control of chaotic systems: The Chua’s circuit,” Journal of Circuits Systems and Computers 3, 151-171. Gilli, M. [1995] “Investigation of chaos in large arrays of Chua’s circuits via a spectral technique,” IEEE, Trans. Circuits Syst.-I 42, 802-806. Gilli, M. [1997] “Analysis of periodic oscillations in finite-dimensional CNNs, through a spatiotemporal harmonic balance technique”, International Journal of Circuit Theory and Applications 25, 279-288. Gilli, M., Corinto, F. & Checco, P. [2004] “Periodic oscillations and bifurcations in cellular nonlinear networks,” IEEE, Trans. Circuits Syst.-I 51, 948-962. Golub G. & Van Loan C. [1996] Matrix Computations (The Johns Hopkins University Press, Baltimore, Third edition), pag. 198. Goras, L. & Chua, L. O. [1995] “Turing patterns in CNNs: Equations and behaviors,” IEEE, Trans. Circuits Syst.-I 42, 612-626. Hoppensteadt, F. C. & Izhikevitch, E. M. [1997] Weakly connected neural networks (Springer-Verlag, New York). Hoppensteadt, F. C. & Izhikevitch, E. M. [1999] “Oscillatory neurocomputers with dynamic connectivity,” Physical Review Letters 82, 2983-2986. Khibnik, A. I., Roose, D. & Chua, L. O [1993] “On periodic orbits and homoclinic bifurcations in Chua’s circuit with a smooth nonlinearity,” Journal of Circuits, Systems and Computers 2, 145-178. Kuramoto, Y. [1984] Chemical oscillations, waves and turbulence (Springer-Verlag, New York). Madan R. N. [1993] Chua’s circuit: A paradigm for chaos (World Scientific, Singapore). Mees, A. I. [1981] Dynamics of feedback systems (John Wiley, New York). Osipov, G. V. & Shalfeev, V. D. [1995] “Chaos and structures in a chain of mutually-coupled Chua’s circuits,”IEEE, Trans. Circuits Syst.-I 42, 693-699. Setti, G., Thiran, P. & Serpico, C. [1998] “An approach to information processing in 1-D cellular neural networks - Part II: Global propagation,” IEEE, Trans. Circuits Syst.-I 45, 790-811. Thiran, P., Setti, G. & Hasler, M. [1998] “An approach to information propagation in 1-D cellular neural networks - Part I: Local diffusion,” IEEE, Trans. Circuits Syst.-I 45, 777-789. V` azquez, A. R., Delgado-Restituto, M., Roca, E., Linan, G., Carmona, R., Espejo, S. & DominguezCastro, R. [2000] “CMOS Analogue Design Primitives,” in Towards the visual microprocessor - VLSI design and use of Cellular Network Universal Machines (Ed. by T.Roska and A. Rodriguez-Vazquez, J. Wiley, Chichester.), pp. 87-131.

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Stable symmetric LC (Ssi) Stable asymmetric LCs (A±i )

10

zi(t)

5

0

−5

−10 3 2

Unstable symmetric LC (Sui)

1

4 2

0

yi(t)

0

−1 −2

−2 −3

−4

xi(t)

Figure 1: Global dynamic behavior of a single uncoupled Chua’s circuit, described by equation (10) with α = 9 and β = 15. The circuit exhibits three unstable equilibrium points (depicted in red), two stable asymmetric limit cycles (depicted in blue and denoted by A± i ), one stable symmetric limit cycle s (depicted in green and denoted by Si ), and one unstable symmetric limit cycle (depicted in red and denoted by Siu ).

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1.4

Floquet’s multipliers moduli |µ|

1.2

1

0.8

0.6

0.4

0.2

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Coupling coefficient d2 Figure 2: One-dimensional array of Chua’s circuit, composed by 12 cells, and described by equations (10) with zero boundary conditions. The moduli of the Coupling FMs are represented as a function + + + of the coupling parameter d2 , for the asymmetric limit cycle {A+ 1 , A2 , ..., A11 , A12 } with phase shifts [π, π, 0, 0, π, 0, π, π, π, π, π] and d = d1 = 0.1. The system exhibits three Coupling FMs with a modulus greater than one (depicted in blue), one unitary FM (depicted in black) and 8 Coupling FMs with a modulus less than one. The latter are represented by four red curves, that denote a single FM, and by two green curves, each of which denote two almost coincident FMs. The moduli of the other FMs of the system are not represented, since they are all less than one.

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1.4

Floquet’s multipliers moduli |µ|

1.2

1

0.8

0.6

0.4

0.2

0 −0.1

−0.09

−0.08

−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

Coupling coefficient d2 Figure 3: One-dimensional array of Chua’s circuit, composed by 12 cells, and described by equations (10) with zero boundary conditions. The moduli of the Coupling FMs are represented as a function + + + of the coupling parameter d2 , for the asymmetric limit cycle {A+ 1 , A2 , ..., A11 , A12 } with phase shifts [π, π, 0, 0, π, 0, π, π, π, π, π] and d = d1 = −0.1. The system exhibits 3 Coupling FMs with a modulus less than one (depicted in red), one unitary FM (depicted in black) and 8 Coupling FMs with a modulus greater than one. The latter are represented by four blue curves, that denote a single FM, and by two green curves, each of which denote two almost coincident FMs. The moduli of the other FMs of the system are not represented, since they are all less than one.

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