International Journal of Bifurcation and Chaos, Vol. 10, No. 1 (2000) 205–225c World Scientific Publishing CompanyFITTING TRAPPING REGIONS FORCHUA’S ATTRACTOR ANOVELMETHODBASED ON ISOCHRONIC LINESSORAYA BOUGHABA∗and RENE LOZILaboratory of Mathematics Jean Alexandre Dieudonn´e,CNRS – UMR No6621,Nice – Sophia Antipolis University, FranceReceived April 25, 1997; Revised July 25, 1999We present in this paper a novel method for fitting trapping regions for a Spiral Chua’s attractor.For the values: σ0= −0.465716 ..., γ0=0.0932544 ..., k =0.3279262 ..., σ1=0.4152731 ...,γ1=−0.3446764 ... of the parameters, the iterates of the attractor belong to two trappingregions P1and P3we construct with this method based uniquely on the isochronic lines. BothP1and P3are bounded accurately with more than 450 segments of isochronic lines. We showgraphically that the inclusions π(P1) ⊂ P3, π(P3) ⊂ P1hold. The traps for the half-Poincar´emap π0have to be constructed.1. Introduction1.1. Chua’s circuitWe consider Chua’s circuit governed by theequations:C1dVc1dt= G(Vc2− Vc1) − g(Vc1)C2dVc2dt= G(Vc1− Vc2)+iLLdiLdt= −Vc2(1)where Vci,(i=1,2), and iLare respectively thevoltages across the capacitors Ci,(i=1,2), andthe current through the inductor L.g(·) is the voltage current characteristic of thenonlinear resistor. See Fig. 1. For more detailson the circuit see [Chua et al., 1986; Wu, 1987;Kennedy, 1992; Dedieu, 1993; Madan, 1993].Equations (1) are transformed to a third-orderautonomous differential equation whose dimension-less form is:dXdt= α(Y − h(x))dYdt= X − Y + Z (2)dZdt= −βYwithh(X)=X+f(X)=m1X+12(m0−m1)(|X +1|−|X−1|)via the rescalingX =Vc1Bp,Y=Vc2Bp,Z=iLBp,τ=tGC2,α=C2C1,β=C2LG2.∗Permanent address: Institute of Mathematics, Constantine University, Algeria.205206 S. Boughaba & R. Lozi(a)(b)Fig. 1. (a) Chua’s circuit model. (b) The three-segmentpiecewise-linear characteristic of the nonlinear voltage con-trolled resistor (Chua’s diode) [Lozi & Ushiki, 1991].For the following choice of the parameters whichgovern the system (2):α = −6.800 ,β=−1.520 ,m0=−17,m1=27corresponding to the normalized eigenvalueparameters:σ0= −0.465716 ... , γ0=0.0932544 ... ,k =0.3279262 ...σ1=0.4152731 ... , γ1=−0.3446764 ...This Chua’s system exhibits a strange attractor,precisely a Spiral Chua’s attractor, the cross-sectionof which with the plane U∗1(the Poincar´e sectionmap) presents four components Ai,(i=1,2,3,4).(Figures 2 and 3.)1.2. Fitted trapping regionsOur aim in this paper is to build a sufficiently sharptrap of the solutions which is useful and appropriateto deepen the application of the theory of Confinorsto the Double Scroll system. See Appendix B [Lozi& Ushiki, 1991, 1993].The main result of this paper is the construc-tion in T of both the components P1and P3of thetrap satisfying the inclusions:A1⊂ P1and A3⊂ P3π(P1) ⊂ P3and π(P3) ⊂ P1See Fig. 4.Fig. 2. Spiral Chua’s attractor (projection in the (X, Y ) plane). Numerical integration with the initial value: (X0,Y0,Z0)=(−0.001, 0.001, 0.001) for: σ0= −0.465716 ..., γ0=0.0932544 ..., σ1=0.4152731 ..., γ1=−3446764 ..., k =0.3279262 ....Fitting Trapping Regions for Chua’s Attractor 207Fig. 3. Cross-section view (Poincar´e section map) of thespiral Chua’s attractor.Fig. 4. Schematic representation of the traps in the planeU∗1.The boundaries of P1and P3are composedof approximatively 450 segments of isochronic lines(see Appendix A). The particularity of the methodis that the tools used are exclusively the isochroniclines. This fact is precisely the reason for our inter-est on confinors, which are based on the symbolicdynamic of intervals. In the simplest case, the quo-tient of the time spent to go from one point to an-other point by the necessary time spent to do anexact “turn” around the axis of rotational symme-try of the linear part of the vector field, may beinterpreted as an estimate of the number of “turns”around the axis.The time needed to go from a point on someset to another set is bounded by:tmin≤ t ≤ tmaxThe symbolic dynamics can be described as:Iτ IθwhereIτ=[τmin,τmax]andIθ=[θmin,θmax] .2. The Fitting Method2.1. BackgroundIn order to analyze the qualitative behavior in theChua’s system (1) and provide a rigorous proof thatthis system is chaotic, Chua [Chua et al., 1986] usedPoincar´emapswithU1and U−1as cross-sectionplanes and showed that the triangular region AEB(called T ) of the plane U1(respectively the symmet-ric triangular region A−E−B−of the plane U−1)bounded by the three lines L0, L1and L2,isatrap-ping region for the flow inD0and D1(respectivelyinD0and D−1). See Fig. 5.A complete description of the Poincar´emapsrequires the definition in each regionD0, D1, D−1,of the half-maps π0and π1. Since the dynamics inthe regionD−1is a symmetry through O of what isobserved inD1, our study is focused on the regionsD0and D1.2.1.1. Half-return maps π0and π1Consider the region D0representing the image ofD0under the affine transformation Ψ0(see [Lozi &Ushiki, 1991] for all the equations and formulae ofthe linear change of coordinates Ψ0and Ψ1). Anytrajectory originating inside the region ∆A0E0B0must move down initially and toward V0. Hence itdefines the map:π+0:∆A0E0B0→V0,208 S. Boughaba & R. Lozi(a) (b)Fig. 5. Geometrical structure and typical trajectories of the original piecewise-linear system and their images in the D0unitand D1unit of the transformed system: (a) Original system. (b) D0and D1units and half-return maps [Chua et al., 1986].via the obvious image:π+0(X0)=ϕτ0(X0), (X0≡(x0,y0,z0)⊥).ϕτ0(X0) denotes the flow (in D0)fromX0to thefirst return point where the trajectory first inter-sects V0at some time τ ≥ 0, with:τ = τ(X0)=inf{t>0|ϕt0(X0)∈V0},On the other hand, any trajectory originatingfrom a point in the infinite wedge∠A0E0B0to theright of (A0E0) must move downward and eventu-ally intersect V−0. It defines the map:π−0: ∠A0E0B0\∆A0E0B0→ V−0,via the obvious image:π−0(X0)=ϕτ0(X0),whereτ = τ(X0)=inf{t>0|ϕt0(X0)∈V−0}is the time for which the trajectory first enters V−0.Definition 1 (Definition of π0and π1). Given bothcomponent half-maps π+0and π−0,denoteπ0as themap defined such that:π0(X0)=(π+0(X0):X0∈∆A0E0B0π−0(X0):X0∈∠A0E0B0\∆A0E0B0.One can define the half-map π1in D1π1: ∠A1E1B1→ V1,via the obvious