21 May 2001 Physics Letters A 283 (2001) 323–326 www.elsevier.nl/locate/pla Effect of stochastic forcing on the Duffing oscillator S. Datta ∗ , J.K. Bhattacharjee Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Calcutta 700 032, India Received 5 September 2000; received in revised form 9 April 2001; accepted 17 April 2001 Communicated by A.R. Bishop Abstract We consider the forced and damped anharmonic oscillator with an additional drive which is random in time. We find by analysing the hysteresis that the random force effectively increases the damping.  2001 Elsevier Science B.V. All rights reserved. Hysteresis [1] is a fairly common occurrence and has been extremely studied in a wide variety of situa- tions. One of the commonest occurrences of hystere- sis is in nonlinear mechanical vibrations [2], as char- acterized by the Duffing oscillator. In this Letter, we point out that a stochastic forcing is capable of re- moving hysteresis in the Duffing oscillator. In appli- cations where a jump phenomenon is undesirable, the introduction of stochasticity could be a way of elim- inating the jump. The effect of noise on different dy- namical systems has recently been reviewed by Landa and McClintock [3,4]. While the effect of noise (addi- tive and multiplicative) on different bifurcations, have been discussed, the effect on the hysteretic bifurca- tions has not been included. Another field of study which is currently of great interest is the control of chaos. An easy way of controlling chaos in nonlin- ear mechanical vibration is through dissipation. In a recent work on dissipative control [5,6], it has been shown that the increasing dissipation is a effective way of controlling the chaos in nonlinear oscillators. In- creasing the dissipation may not always be practical, * Corresponding author. E-mail address: tpsd@mahendra.iacs.res.in (S. Datta). so it is worthwhile to study if it can be effectively in- creased by some other means. In this Letter we show that having an added noise in a Duffing oscillator can mimic the effect of damping. We do this by studying the hysteresis in the Duffing oscillator. The usual Duff- ing oscillator has the equation of motion (1) ¨ X + 2k ˙ X + X + λX 3 = F cos Ωt. In the above, λ> 0 for the potential to be attractive at large X. It is possible to rescale X and F suitably and set λ = 1. However, we choose not to rescale F and hence keep λ as it is in the cubic term. To study the hysteresis, one usually employs an equivalent linearization which leads to (2) ¨ X + 2k ˙ X + ω 2 X = F cos Ωt, where (3) ω 2 = 1 + 3 4 λb 2 with b the amplitude of oscillation. From Eqs. (2) and (3), the amplitude is self-consistently determined as (4) b 2 = F 2 (ω 2 - Ω 2 ) 2 + 4k 2 Ω 2 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0375-9601(01)00258-4