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FITTING TRAPPING REGIONS FOR CHUA’S ATTRACTOR A NOVEL METHOD BASED ON ISOCHRONIC LINES



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International Journal of Bifurcation and Chaos Vol 10 No 1 2000 205 225 c World Scientific Publishing Company FITTING TRAPPING REGIONS FOR CHUA S ATTRACTOR A NOVEL METHOD BASED ON ISOCHRONIC LINES SORAYA BOUGHABA and RENE LOZI Laboratory of Mathematics Jean Alexandre Dieudonne CNRS UMR N o 6621 Nice Sophia Antipolis University France Received April 25 1997 Revised July 25 1999 We present in this paper a novel method for fitting trapping regions for a Spiral Chua s attractor For the values 0 0 465716 0 0 0932544 k 0 3279262 1 0 4152731 1 0 3446764 of the parameters the iterates of the attractor belong to two trapping regions P1 and P3 we construct with this method based uniquely on the isochronic lines Both P1 and P3 are bounded accurately with more than 450 segments of isochronic lines We show graphically that the inclusions P1 P3 P3 P1 hold The traps for the half Poincare map 0 have to be constructed 1 Introduction less form is 1 1 Chua s circuit dX Y h x dt We consider Chua s circuit governed by the equations dY X Y Z dt dVc1 G Vc2 Vc1 g Vc1 dt dVc2 1 C2 G Vc1 Vc2 iL dt diL L Vc2 dt where Vci i 1 2 and iL are respectively the voltages across the capacitors Ci i 1 2 and the current through the inductor L g is the voltage current characteristic of the nonlinear resistor See Fig 1 For more details on the circuit see Chua et al 1986 Wu 1987 Kennedy 1992 Dedieu 1993 Madan 1993 Equations 1 are transformed to a third order autonomous differential equation whose dimensionC1 2 dZ Y dt with h X X f X m1 X 1 m0 m1 X 1 X 1 2 via the rescaling X Vc1 Bp tG C2 Permanent address Institute of Mathematics Constantine University Algeria 205 Y Vc2 Bp C2 C1 Z iL Bp C2 LG2 206 S Boughaba R Lozi corresponding parameters to the 0 0 465716 normalized eigenvalue 0 0 0932544 k 0 3279262 1 0 4152731 1 0 3446764 a This Chua s system exhibits a strange attractor precisely a Spiral Chua s attractor the cross section of which with the plane U1 the Poincare section map presents four components Ai



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