International Journal of Bifurcation and Chaos, Vol. 9, No. 4 (1999) 723–733c© World Scientific Publishing Company

COMPOSITION METHODS FOR THE SIMULATIONOF ARRAYS OF CHUA’S CIRCUITS

YVES MOREAU and JOOS VANDEWALLEDepartment of Electrical Engineering,

Katholieke Universiteit Leuven,Kardinaal Mercierlaan 94, B-3001 Heverlee, Belgium

Received February 21, 1998; Revised September 9, 1998

Composition methods are methods for the integration of ordinary differential equations arisingfrom differential geometry, or more precisely, Lie algebra theory. We apply them here tothe simulation of arrays of Chua’s circuits. In these methods, we split the vector field ofthe array of Chua’s circuits into its linear part and its nonlinear part. We then solve theelementary differential equation for each part separately — which is easy since the equationsfor the nonlinear part are all decoupled — and recombine these contributions into a sequence ofcompositions. This splitting gives rise to simple integration rules for arrays of Chua’s circuits,which we compare to more classical approaches: fixed time-step explicit Euler and adaptivefourth-order Runge–Kutta.

1. Introduction

The standard methods for the integration of Or-dinary Differential Equations (ODEs) are schemessuch as the Euler method or the Runge–Kuttamethod. These schemes are straightforwardly ap-plied to the integration of continuous-time recur-rent neural networks, in particular Cellular NeuralNetworks (CNNs) consisting of arrays of Chua’s cir-cuits. However, physicists have recently introduceda completely different class of methods for the so-lution of ordinary differential equations: the com-position methods [Forest & Ruth, 1990; Yoshida,1990]. The spirit of these methods is that ifthe vector field of the differential equation is thesuperposition of some elementary vector fields, wecan approximate its solution by composing theflows of the elementary contributions. Composi-tion methods are particularly useful for numericallyintegrating ODEs when the equations have somespecial structure, of which we can take advantage[McLachlan, 1995]. The authors have recentlyintroduced them in the field of neural networks[Moreau & Vandewalle, 1997].

To demonstrate further the relevance ofcomposition methods to the field of neural net-works, we present their application to the integra-tion of the differential equations describing CNNsmade out of an array of Chua’s circuits. CNNs areeffective for parallel signal processing and real-timesimulation of phenomena described by partial dif-ferential equations. They have therefore been usedfor image processing and for the study of nonlinearspatio-temporal behaviors [Munuzuri et al., 1995].Their main interest from the applied viewpoint isthat a large array can be integrated into a mono-lithic VLSI chip and that CNNs can model a largenumber of practical systems — for example, in im-age processing.

We will first present the elements of dynam-ical system theory and Lie algebra theory neces-sary for the exposition of composition methods. Wewill then show how to use this theory for the inte-gration of ordinary differential equations using com-position methods. We will apply these methodsfirst to a single Chua’s circuit and then to an ar-ray of coupled Chua’s circuits. We will compare

723

724 Y. Moreau & J. Vandewalle

the performance of these methods with those ofstandard methods: fixed time-step explicit Eu-ler and adaptive fourth-order Runge–Kutta. Notonly do composition methods provide a new view-point on the dynamics of arrays of Chua’s circuits,but they provide enhanced performance for theirsimulation.

2. Lie Algebra Theory

The CNNs that we consider here are arrays ofChua’s circuits. These CNNs are described by or-dinary differential equations x(t) = A(x(t)) definedon the state-space Rn, where the vector field A de-scribes the dynamics of a number of nonlinear cir-cuits evolving with a linear coupling. We write theODE as x(t) = A(x(t)) to follow the conventions ofLie algebra theory. We can write its solution, as afunction of the initial condition x0, in two forms:as a flow x(t) = Φ(x0, t) (as is standard in the dy-namical system literature) or as an exponential so-lution x(t) = etAx0 (as in the Lie algebra literature[Arnold, 1989]). The latter notation should read:“x(t) is the image after time t of the initial condi-tion x0 under the flow of x = A(x)”. We stress that,in our notation, etA is a nonlinear transformation

of the state-space and not a matrix as in the case oflinear systems. The notation is also coherent withthe basic properties of the exponential. For exam-ple (using the dot “·” to denote the composition ofmaps), we have

e(t+s)Ax0 = esA · etAx0 ,

which is nothing but a restatement of the groupproperty of the flow Φ(x0, t + s) = Φ(Φ(x0, t), s)[Irwin, 1980]. This property states that, for anyinitial condition, evolving according to the differ-ential equation x = A(x) for t + s units of timeis equivalent to: First, evolving for units of time;then, starting from where we have arrived, evolvingfor another s units of time.

Lie algebra theory is an important tool inphysics [Arnold, 1989; Bluman & Kumei, 1989] andan essential part of nonlinear system theory [Isidori,1989]. It provides here the framework for the pre-sentation of composition methods for ODEs. TheLie algebra we consider here is the vector space ofall smooth vector fields, where we define a supple-mentary operation: the Lie bracket of two vectorfields, which is again a vector field. Bracketing is abilinear, anti-symmetric operation, which also sat-isfies the Jacobi identity [Arnold, 1989]:

[A, B] = −[B, A], antisymmetry ,

[aA, B] = a[A, B], bilinearity ,

[A, B + C] = [A, B] + [A, C], bilinearity ,

[[A, B], C] + [[B, C], A] + [[C, A], B] = 0, Jacobi identity .

(1)

Recalling that we take the product of exponen-tials to denote the composition of these maps, wecan define the vector field [A, B] as the Lie bracketof the vector fields A and B:

[A, B](x) =∂2

∂s∂t

∣∣∣∣∣t=s=0

e−sBe−tAesBetA(x) .

The bracket [A, B] is called the commutator ofthe vector fields, as it measures the degree of non-commutativity of the flows of the vector fields([A, B] = 0 ⇔ etAetB = etBetA). As we defineour vector fields on Rn, we can further express thebracket as

[A, B]i =n∑j=1

(Bj∂Aj∂xi−Aj

∂Bj∂xi

).

The last mathematical tool we need is the Baker–Campbell–Hausdorff (BCH) formula. This formulagives an expansion for the product of two exponen-tials of elements of the Lie algebra [Arnold, 1989]:

etAetB =et(A+B)+ t2

2[A,B]+ t3

12([A, [A,B]]+[B, [B,A]])+... .

(2)

It should be interpreted as follows. Letting theflow of x = B(x) act on the initial condition ofthe system for t, and then — from where we havearrived — letting the flow of x = A(x) act for an-other time-step t is equivalent to letting the flowof x = ((A + B) + t/2[A, B] + t2/12([A, [A, B]] +[B, [B, A]]) + . . .)(x) act on the initial conditionfor t.

Composition Methods and Arrays of Chua’s Circuits 725

2.1. Integration of ordinary differentialequations by compositions

Let us now try to solve ordinary differential equa-tions using compositions. Suppose we want to solvethe ODE x(t) = X(x(t)) for a time-step of ∆t.Since x(t) = e∆tXx0, the problem becomes thatof building an approximation to e∆tX . This prob-lem has recently been the focus of much atten-tion in the field of numerical analysis, especiallyfor the integration of Hamiltonian differential equa-tions [McLachlan, 1995]. The basic idea is that,if you can split the vector field X into elemen-tary parts for which you can solve the differentialequation directly, you can recombine these solutionsto approximate the solution of the more complexsystem.

Suppose, in a first step, that the vector field Xis the sum of two vector fields: X = A+ B, whereyou can integrate A and B either analytically ormuch more easily than X. Then we can use theBCH formula to produce a first-order approxima-tion to the exponential map:

BCH : etX = etAetB + o(t2) . (3)

You can check this relation by multiplyingthe left- and right-hand sides of Eq. 2 by etX(=et(A+B)), and then expanding using the BCH for-mula itself (2). The first-order approximation (3)between the solution of A and B, and the solutionof X is the essence of the method since it showsthat we can approximate an exponential map (thatis the mapping arising from the solution of an ODE)by composing simpler maps (Fig. 1).

By using the BCH formula to eliminate higher-order terms as we did for the first-order approxima-tion, but on the composition of three terms, we canshow that the following symmetric leapfrog schemeis second order:

Leapfrog : etX = et2AetBe

t2A + o(t3) ,

= S(t) + o(t3) .

Using this leapfrog scheme as a basis element, wecan build a fourth-order scheme:

Fourth order : etX = S(ct)S(dt)S(ct) + o(t5) ,

= SS(t) + o(t5) , (4)

with c = −21/3/(2 − 21/3) and d = 1/(2 − 21/3).Repeating the leapfrog strategy, Yoshida [1990]

����

��������

Flow of B

Flow of X

Flow of A

Fig. 1. First-order approximation by composition. Thevector field X of the system x = X(x) is the sum of twovector fields A and B. The Baker–Campbell–Hausdorff for-mula gives a first-order approximation with e∆tX(x0) ≈e∆tBe∆tA(x0). The bold solid line represents the evolution ofthe system between 0 and ∆t from the initial condition x0.The dashed lines represent the evolution from x0 to e∆tA(x0)and from e∆tA(x0) to e∆tBe∆tA(x0).

showed that it is possible to produce an approxima-tion to etX up to any order:

Arbitrary order : There exists k and

w1, v1, . . . , wk, vk such that

etX = ew1tAev1tB · · · ewktAevktB + o(tp+1) . (5)

This result is useful as long as t is sufficientlysmall to keep the higher-order terms negligible. If tbecomes too large, the previous result offers noguarantees. But Suzuki [1993] showed that higher-order repeated-leapfrog schemes converge every-where. Further, Forest and Ruth [1990] also showedthat approximations can be built for more than twovector fields: If X =

∑mi=1Ai then

there exists k and wij (i = 1, . . . , m;

j = 1, . . . , k) such that

etX =k∏j=1

m∏i=1

ewijtAi + o(tp+1) . (6)

3. Composition Integrators

We present a short-list of different composition in-tegrators (Table 1) for the case where the vectorfield can be split into two parts: X = A+B. These

726 Y. Moreau & J. Vandewalle

Table 1. Formulas for the composition methods.

Order Substeps Error Formula

2 3 0.070 S(∆t) = e(∆t/2)Ae∆tBe(∆t/2)A

2 5 0.026ea1∆tAeb1∆tBea2∆tAeb1∆tBea1∆tA

a1 = 0.1932, b1 = 0.5, a2 = 1− 2a1

4 7 0.098S(w1∆t)S(w2∆t)S(w1∆t)w1 = 1.3512, w2 = 1− 2w1

4 11 0.033S(w1∆t)S(w2∆t)S(w3∆t)S(w2∆t)S(w1∆t)w1 = 0.28, w2 = 0.6254,w3 = 1− 2w1 − 2w2

4 11 0.0046

ea1∆tAeb1∆tBea2∆tAeb2∆tBea3∆tAeb3∆tB

× ea3∆tAeb2∆tBea2∆tAeb1∆tBea1∆tA

a1 = 0.0893, b1 = 0.4,a2 = −0.0973, b2 = −0.1,a3 = 1− 2a1 − 2a2, b3 = 1− 2b1 − 2b2

methods can be recursively used for the case ofmore than two vector fields. For example, in thecase X = A + B + C, we can write X = A + D,where D = B + C. We then choose a compositionmethod for D, which we use as basis step for theintegration of X. The error constant associated toeach method is the effective error constant definedin [McLachlan, 1995]. This error measure accountsfor the fact that, for a given order, methods withmore substeps maybe more precise, but they alsorequire more computational work. This tradeoffwould allow methods composed of fewer substepsto use a smaller step size, therefore increasing theirprecision. For a given order, even if the effectiveerror constant decreases as the number of substepsincreases, it may not be advantageous to use themethod with the most substeps for finite time steps.This choice will depend on the system we are inte-grating, and on the error we require.

We present two types of methods: the sym-metric methods and the symmetric methods com-posed of symmetric steps. The symmetric methodsare presented directly as a product of exponentials,while the symmetric methods composed of symmet-ric steps are presented as a product of symmetricleapfrog steps S(∆t). The leapfrog is the simplestmethod to implement, while the fourth-order sym-metric method (error = 0.0046) is the most efficient.We therefore recommend the leapfrog for prelimi-nary experiments and the fourth-order symmetricmethod (error = 0.0046) for final implementations.

For the formulas of higher-order integrators, see[McLachlan, 1995]. A remark of fundamental im-portance is that these formulas are valid even with-out analytical solutions of e∆tA and e∆tB. Indeed,we can replace these solutions in the following sym-metric methods by any first-order integrator (e.g.,I + ∆tA = e∆tA + o(∆t2)) and still guarantee thesame order of approximation. Similarly, in the sym-metric methods composed of symmetric steps, wecan replace the leapfrog S(∆t) by any second-ordersymmetric integrator.

4. Application to CNNs

We will now apply this technique to arrays of Chua’scircuits. Chua’s circuit is a nonlinear electrical cir-cuit. We can write it as a state-space model orordinary differential equation of the form

x

y

z

=

−a a 0

b −b 1

0 −c 0

x

y

z

+

g(x)

0

0

, (7)

with g(x) a piecewise linear function: g(x) =−(a/2)[(s1 + s2)x+ (s0− s1)(|x−β1|− |β1|) + (s2−s0)(|x−β2|−|β2|)]. This system can exhibit chaoticbehavior for appropriate parameter values — for ex-ample, a = 6.3, b = 0.7, c = 7, s0 = −1.14286,s1 = s2 = −0.71429, β1 = −1 and β2 = 1. If wesplit the vector field of (7) into its linear part and

Composition Methods and Arrays of Chua’s Circuits 727

its nonlinear part, we see that we can solve bothparts explicitly. If we call A the matrix containingthe parameters a, b, c, the solution for the linearpart is x(∆t) = e∆tA1x0, where the exponential isthe classical matrix exponential. The solution forthe nonlinear part is x(∆t) = etg(x0). In the sim-ple case where β1 < 0 < β2, we can compute thepositive part of the solution as follows:

τ(x0) =

1

s2ln

(x0 − δβ2 − δ

), x > β2 ,

1

s0ln

(x0

β2

), 0 < x < β2 ;

x(∆t) =

es2(τ(x0)+∆t)(β2 − δ) , τ(x0) + ∆t > 0 ,

es0(τ(x0)+∆t)β2 , τ(x0) + ∆t ≤ 0 ;

(8)

where δ = β2(1 − s0/s2). As x = 0 is a fixedpoint, trajectories starting at positive initial con-ditions always remain positive. For negative initialconditions, we have the same formula except thatβ1 takes the place of β2, s1 takes the place of s2, andall signs are reversed. Therefore, we can integratethe equations of Chua’s circuit with a compositionmethod.

4.1. Arrays of Chua’s circuits

Moreover, this type of splitting applies to generalCNNs. For example, we can have a CNN composedby an array of Chua’s circuits (7) coupled togetherby some constant linear coupling. If we have n cir-cuits having each three state variables, the equa-tions describing the evolution of the system wouldbe

xi

yi

zi

=

−a a 0

b −b 1

0 −c 0

xi

yi

zi

+

g(xi)

0

0

+

ρΣn

j=1ξijxj

0

0

(9)

with i = 1, . . . , n and where the coupling term ξijis equal to one if circuit i is connected to circuitj, and is equal to zero else. Observe that, howeverlarge the number of circuits is, the system is madeof a large linear part (the first and third terms)

and of a nonlinear part that consists of n identicaldecoupled piecewise linear equations. This split-ting means that, once again, both the linear andthe nonlinear part can be solved explicitly (8), andcombined to find the solution of this large system ofODEs. If we call xtot the total state of the system:xtot = (x1, y1, z1, x2, y2, z2, . . . , xn, yn, zn)T ,A the matrix representing the linear part of thedynamics of the Chua’s circuits and the linear in-terconnections, and Γ the nonlinear part of the dy-namics; we have the following ODE:

xtot = Axtot + Γ(xtot) .

As a first toy problem, we present a simula-tion of the solution of such a CNN for the caseof a two-by-two array of Chua’s circuits. The cir-cuits are placed on the corners of a square andsymmetrically connected in a ring. We choose thevalue of the interconnection strength as ρ = 0.1.We realize the simulation in the Matlab environ-ment. We use a second-order integrator: xtot(∆t) =e(∆t/2)Ae∆tΓe(∆t/2)A(xtot). The time-step for the in-tegration is ∆t = 0.1. In Fig. 2, we see the evolutionof the state of the four circuits. The four circuitshave identical parameters but different random ini-tial conditions, hence the behavior of the system isnot symmetric.

An important point that comes under consider-ation is how to solve efficiently the one-dimensionalnonlinear ODEs arising through the splitting ofthe vector field of the array of Chua’s circuits.An analytical solution might not be difficult tocompute for this piecewise linear ODE, but its im-plementation might be rather inefficient. There-fore, we present a method for computing the so-lution of a one-dimensional ODE using a lookuptable.

4.2. Table lookups for the solutionof one-dimensional ODEs

The main observation is that the behavior of a one-dimensional ODE is monotone between two fixedpoints. Let us write the ODE as x = A(x) with x ∈R, and its solution as x(t) = Φ(x0, t). Suppose firstthat the ODE has no fixed points. Then, the trajec-tory passing through x0 = 0 will cover the whole ofR. Define this trajectory as the function ψ(t) =Φ(0, t). Because of the group property of theflow (Φ(Φ(x0, t), s) = Φ(x0, t + s)), we have thatΦ(x0, t) = Φ(Φ(0, ψ−1(x0)), t) = Φ(0, ψ−1(x0)+t).So that Φ(x0, t) = ψ(ψ−1(x0)+t). Thus, by storing

728 Y. Moreau & J. Vandewalle

−4 −2 0 2 4

−0.5

0

0.5

x1

x2

Chua circuit (1,1)

−10 −5 0 5 10−10

−5

0

5

10

x4

x5

Chua circuit (1,2)

−4 −2 0 2 4−1

−0.5

0

0.5

1

x7

x8

Chua circuit (2,1)

−4 −2 0 2 4

−0.5

0

0.5

x10

x1

1

Chua circuit (2,2)

Fig. 2. Evolution of the state of the CNN array. Simulation by a leapfrog composition method. The array consists offour Chua’s circuits in a ring. The circuits have identical parameters (a = 6.3, b = 0.7, c = 7, s0 = −1.14286, s1 = s2 =−0.71429, β1 = −1 and β2 = 1) and the coupling term is ρ = 0.1. The initial conditions of the circuits are different and chosenat random. We plot the evolution of the x and y variables of each of the four circuits.

the solution ψ of x = A(x), withx(0) = 0, and itsinverse in a lookup table, we are able to computethe solution of the one-dimensional ODE in just twotable lookups.

Suppose now that the ODE has k fixed points:z1, . . . , zk. Set also z0 = −∞ and zk+1 =∞. Definev0 = z1 − 1, v1 = (z1 + z2)/2, . . . , vk−1 = (zk−1 +zk)/2, vk = zk + 1. Define ψi(t) = Φ(vi, t), then weknow that this trajectory ψi(t) will cover the wholeof the interval ]zi, zi+1[. Thus, for all x0 ∈]zi, zi+1[:Φ(x0, t) = ψi(ψ

−1i (x0) + t) for all i = 0, . . . k. We

can thus store the function ψ−1i in a lookup table

and the function ψi in another lookup table con-sisting of k subtables (one per interval). This pro-cedure gives us a powerful way to compute repeatedintegrations of a one-dimensional ODE.

5. Simulations

To show that the composition method is an efficientintegration method, we ran a simulation of a spiralwave on an array of 50 × 50 Chua’s circuits (9) as

shown in [Pivka, 1995] and [Munuzuri et al., 1995].The appropriate set of parameters is

a = 10 , b = 1 , c = 0.3014987 ,

s1 = 0.078573 , s2 = 55 , s0 = −1.25719 ,

β1 = −1 , β2 = 0.023744 .

These parameters correspond to a bistable cellwith two stable equilibria: P− = [−1.238, 1.238],P+ = [0.02385, −0.02385]. The appropriate initialcondition is described as follows. All the x and yvariables are in the rest state except for a front ofexcited cells. The z variable has a circular gradientof excitation, which is smoothly decreasing. Thisinitial condition is described in details in [Pivka,1995].

We have compared the composition method toa Runge–Kutta adaptive time-step method and asimple Euler fixed-step method. We have chosena fourth-order composition method with differenttime-steps (∆t = 0.01, ∆t = 0.02, ∆t = 0.05), a

Composition Methods and Arrays of Chua’s Circuits 729

Fig. 3. Snapshot of the spiral wave for the Euler method (t0 = 0, t1 = 20, t2 = 40, t3 = 60, t4 = 80, t5 = 100). The picturerepresents the activity of the x variable on a 50 × 50 grid of Chua’s circuits. We do not present the activity of the y and z

variables.

730 Y. Moreau & J. Vandewalle

Fig. 4. Snapshot of the integration for the Euler method with ∆t = 0.005 at time t = 10. With this step size, the methodbecomes unstable and cannot give accurate results.

first-order Euler method with different time-steps(∆t = 0.001, ∆t = 0.002, ∆t = 0.005), and afourth-order Runge–Kutta method (the time-stepis automatically adaptively chosen). In a previousexperiment [Munuzuri et al., 1995], a fixed-step Eu-ler method with time-step ∆t = 0.0005 had beenused.

We found that it was possible to increase thetime-step of the Euler method to ∆t = 0.002 whilekeeping sufficient accuracy. We found that themethod was then about twice as fast as the adap-tive Runge–Kutta method. We show in Fig. 3 snap-shots of the activity of the x variable on a grid of50×50 Chua’s circuits at times t0 = 0, t1 = 20, t2 =40, t3 = 60, t4 = 80, t5 = 100 for the Euler method.We should not forget that there are two other vari-ables y and z for each Chua’s circuit that also evolveover time.

If we try to increase the time-step of the Eu-ler method, we see that instabilities appear for∆t = 0.005, which make the integration unreliable

as we can see in Fig. 4 where we show snapshots ofthe activity of the x variable at t = 10.

In contrast, we can use a much larger time-stepfor the composition method because of its good con-servation properties. This larger time-step comesat the price of a larger number of operations periteration since the method contains a number ofsubsteps. But as a whole, the fourth-order com-position method with a time-step of ∆t = 0.01 isalready as fast as the Euler method with a time-step of ∆t = 0.001 for the same accuracy. At∆t = 0.02 for the composition method, the integra-tion still proceeds normally and the method is twotimes as fast as the Euler method. This simulationis displayed in Fig. 5. In the case of a simulationfor a 50-by-50 array (N = 50) and a time span ofT = 100 seconds, the simulation for the Eu-ler method with ∆t = 0.002 takes 290 secondsfor a C code running on a Sun Ultra-1 worksta-tion. In contrast, the simulation for the fourth-order composition method with ∆t = 0.02 takes

Composition Methods and Arrays of Chua’s Circuits 731

Fig. 5. Snapshot of the spiral wave for the composition method (t0 = 0, t1 = 20, t2 = 40, t3 = 60, t4 = 80, t5 = 100). Thecomposition method uses a time step much larger than the Euler method.

732 Y. Moreau & J. Vandewalle

Fig. 6. Snapshot of the integration for the composition method with ∆t = 0.05 at time t = 10. With this time step, thecomposition method becomes unstable and does not give accurate results.

141 seconds. We took care to write the codes for thetwo methods with similar parts written in a similarway to make the comparison fair.

At ∆t = 0.05, instabilities appear and they aretoo severe to make the integration acceptable. Wecan see this behavior in Fig. 6.

Both methods are linear in 1/∆t so that tak-ing a time-step twice as small leads to twice aslong a computation; our experiments verified thisratio. In fact, we can estimate the complexity ofboth methods in terms of the number of multipli-cations and additions needed to perform the com-putation. The complexity is proportional to thetime span T of the simulation and to the num-ber of circuits (N2) of the array. Per circuit andper time step, the complexity of the Euler methodis 18 floating-point additions and 9 floating-pointmultiplications. In contrast, the fourth-order com-position method is composed of 11 substeps. Sixof these substeps account for the nonlinearity ofthe dynamics while five of the substeps account for

the linear part of the dynamics. The complexity ofthe nonlinear substep is 5 floating-point additionsand 1 floating-point multiplication. The complex-ity of the linear substep is 10 floating-point additionand 5 floating-point multiplications. Therefore, thecomplexity per circuit of a step of the compositionmethod is 6 × 5 + 5 × 10 = 80 floating-point ad-ditions and 6 × 3 + 5 × 5 = 43 floating-point mul-tiplications. This complexity is much higher thanthe complexity of the Euler method. However, wecan use a time step for the composition method(∆tcomp = 0.02) that is ten times as big as thetime step for the Euler method (∆tEuler = 0.002).When we take this difference into account, wehave that the total complexity of the compositionmethod (3N2T/∆tcomp(80 FPadd + 43 FPmult))is about twice as small as the total complexity ofthe Euler method (3N2T/(∆tcomp/10)(18 FPadd +9 FPmult)).

Although, the two integration methods give re-sults that are not numerically completely identical,

Composition Methods and Arrays of Chua’s Circuits 733

all the phenomena that have been described for thespiral wave in an array of Chua’s circuits are pre-served. The differences are mainly due to the trun-cation errors because we have tried to speed upboth methods to a maximum. For smaller time-steps both methods again coincide.

6. Conclusions

We have presented a new class of integration rulesfor arrays of Chua’s circuits: the composition meth-ods. We derived these methods from Lie algebratheory through the use of the Baker–Campbell–Hausdorff formula. Not only do these methods shednew light on the dynamics of arrays of Chua’s cir-cuits, but they provide enhanced performances forthe simulation of these arrays. The simplicity ofour integration rules makes them good candidatesfor efficient software implementation. They couldalso be used to design the templates of cellular neu-ral networks.

Acknowledgments

Joos Vandewalle is a full Professor at the K. U.Leuven. Yves Moreau is a research assistant of theF. W. O. Vlaanderen. This work is supported byseveral institutions:

1. The Flemish Government: Concerted ResearchAction GOA-MIPS (Model-based InformationProcessing Systems)

2. The Belgian State, Prime Minister’s Office —Federal Office for Scientific, Technical and Cul-tural Affairs — Interuniversity Poles of Attrac-tion Programme: (IUAP P4-02): Modeling,Identification, Simulation and Control of Com-plex Systems

3. The European Commission: ESPRIT projectDICTAM (Dynamic Image Coding Using Tera-Speed Analogic Visual Microprocessors)

4. The FWO Research Communities: ICCoS (Iden-tification and Control of Complex Systems) andAdvanced Methods for Mathematical Modeling.

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