PHYSICAL REVIEW E VOLUME 51, NUMBER 1 JANUARY 1995 Suppression of chaos by selective resonant parametric perturbations R. Chacon 13400 Almaden, Ciudad Real, Spain and Departamento de Fssica, I'acultad de Ciencias, Universidad de Extremadura, 06071 Badajoz, Spain* (Received 10 March 1994) It is shown that, depending on its amplitude, frequency, and initial phase, a time-dependent periodic parametric perturbation can suppress chaos in nonlinear oscillators. The example of the Du%ng-Holmes oscillator is used to demonstrate that all the numerically and experimentally observed phenomenology is theoretically explained by using the Melnikov-Holmes method and suppression of chaos is seen to be possible when certain resonant frequencies are involved. PACS number(s): 05. 45. + b, 05. 40. + j The problem of suppressing chaos has attracted great interest in recent years [1 — 4]. In particular, it has been observed both theoretically and experimentally [5 — 8] that resonant parametric perturbations [in the examples of the Josephson-junction model and the Duf6ng-Holmes (DH) equation] can suppress chaotic behavior arising from homoclinic bifurcations. In spite of this work, a complete comprehension of such inhibitory mechanisms is still far from being achieved. In fact, certain key ques- tions remain. (i) What exactly is the nature of the reso- nant condition imposed on the parametric perturbations in order to regularize the dynamics? From numerical and experimental results, regularization is only observed for a limited range of resonances between the frequencies of the parametric perturbation and the primary chaos- inducing forcing. (ii) What is the influence of the initial phase difFerence between such forces. Experimentally it is found that [7]: "Indefinitely long regularization is found at exact resonance, but this also requires an ap- propriate phase relation between the forcing and the parametric perturbation. " (iii) What is the nature of the route(s) from chaos to order underlying the inhibitory mechanism'? (iv) What type of agreement might one ex- pect between analytical and numerical results? In this article I attempt to answer these questions using the example of the DH oscillator with a parametric per- turbation of the cubic term [5, 7, 8] x — x+p[1+ricos(Qt+y)]x = — 5x+y cos(cot), where 0, g, and p are the frequency, amplitude, and ini- tial phase, respectively, of the parametric perturbation (g«1), which has a suppressory effect on the chaotic dynamics of the unperturbed system. By using the Melnikov-Holmes method (MHM) [9 — 13 ], analytical predictions are obtained for the threshold for chaos. In distinction to previous work [5, 7, 8], I obtain from these predictions a selective resonance condition involving co, Q, and y, and two threshold values (upper and lower) for g for the elimination of the chaotic dynamics present — B(g, Q)sin(Qto+y) — C(P, 5), 1/2 2 A (y, co) = gym sech B(r), Q)= (Q +4Q )csch 6 2 C(P, 5) = 45 (3) (The corresponding result for y=O from Lima and Pet- tini [Eq. (1), Ref. [8]] is incorrect. ) Suppose that for g=O we are in a chaotic situation for which the associated MF Mo(to) = A(y, to)sin(toto) — C(p, 5) changes sign at some to, 1. e. , at g=O. Numerical experiments are performed with Eq. (1) and the results are compared separately with the theoretically predicted values of 0, q and y that suppress chaos. Additionally, it is shown that type-II in- termittency appears as 0 approaches the resonance con- dition. As is well known, the MHM is only concerned with transient chaos [ll], i.e. , only necessary conditions for (steady) chaos are obtained from it, and therefore one al- ways has the possibility of finding sufhcient conditions for the elimination of (even transient) chaos. Observe that the validity of these statements is subject to two con- straints since the MHM is a perturbatiue (to first order) method: (a) its predictions are only valid for motions based at points su~ciently near the separatrix of the un- perturbed system; (b) the perturbative term's amplitude must be su+eiently small ( « 1). Let us now consider the concrete application of these ideas to system (1). From Cuadros and Chacon [8], the Melnikov function (MF) for this case (y WO) is written M(to)= A(y, co)sin(toto) A (y, co) — C(P, 5) = d~ 0, — (4) *Address for all correspondence. where the ~ equals sign corresponds to the case of tangency between the stable and unstable manifolds. 1063-651X/95/51(1)/761(4)/$06. 00 51 761 1995 The American Physical Society