21 May 2001

Physics Letters A 283 (2001) 323–326www.elsevier.nl/locate/pla

Effect of stochastic forcing on the Duffing oscillator

S. Datta∗, J.K. BhattacharjeeDepartment 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 2001Communicated by A.R. Bishop

Abstract

We consider the forced and damped anharmonic oscillator with an additional drive which is random in time. We find byanalysing the hysteresis that the random force effectively increases the damping. 2001 Elsevier Science B.V. All rightsreserved.

Hysteresis [1] is a fairly common occurrence andhas 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, wepoint out that a stochastic forcing is capable of re-moving hysteresis in the Duffing oscillator. In appli-cations where a jump phenomenon is undesirable, theintroduction 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 Landaand McClintock [3,4]. While the effect of noise (addi-tive and multiplicative) on different bifurcations, havebeen discussed, the effect on the hysteretic bifurca-tions has not been included. Another field of studywhich is currently of great interest is the control ofchaos. An easy way of controlling chaos in nonlin-ear mechanical vibration is through dissipation. In arecent work on dissipative control [5,6], it has beenshown that the increasing dissipation is a effective wayof 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 showthat having an added noise in a Duffing oscillator canmimic the effect of damping. We do this by studyingthe hysteresis in the Duffing oscillator. The usual Duff-ing oscillator has the equation of motion

(1)X + 2kX + X + λX3 = F cosΩt.

In the above,λ > 0 for the potential to be attractiveat largeX. It is possible to rescaleX andF suitablyand setλ = 1. However, we choose not to rescaleF

and hence keepλ as it is in the cubic term. To studythe hysteresis, one usually employs an equivalentlinearization which leads to

(2)X + 2kX + ω2X = F cosΩt,

where

(3)ω2 = 1+ 3

4λb2

with b the amplitude of oscillation. From Eqs. (2) and(3), the amplitude is self-consistently determined as

(4)b2 = F 2

(ω2 − Ω2)2 + 4k2Ω2

0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(01)00258-4

324 S. Datta, J.K. Bhattacharjee / Physics Letters A 283 (2001) 323–326

Fig. 1. The amplitude of oscillation is plotted against frequency.

which is a cubic inb2, the square of the amplitude.For the strongest response, we consider the forcingfrequenciesΩ close to the resonant frequency, i.e.,Ω = 1 + δ, whereδ is small. We now write Eq. (4)in the form

(5)b2[(ω2 − 1)4 − 4δ

(ω2 − 1

) + 4(δ2 + k2)] = F 2

which using Eq. (3) becomes

(6)

(3

4λ

)2

b6 − 3δλb4 + 4b2(δ2 + k2) − F 2 = 0.

Hysteresis occurs when there are three positive rootsof the above cubic. For a given dampingk, threepositive roots ofb2 occurs ifF exceeds a criticalFc ,given by [2]

(7)F 2c = 32

3

k3

λ.

For aF set aboveFc , a typicalb2 vs. δ plot is shownin Fig. 1. Over a range ofδ, there are three possiblevalues of the amplitude. Two of these are stable andthey are separated by an unstable branch. As thefrequency is increased an abrupt transition occurs fromthe topmost curve to the bottom curve at the turningpoint and this jump is the hysteretic jump. If we lowerthe frequency, an abrupt jump occurs to the topmostcurve but at a different point. This is the irreversibilityassociated with hysteresis.

We now switch off the periodic forcing and intro-duce a stochastic forcingf (t) which is a Gaussianwhite noise with the correlation

(8)⟨f (t)f (t ′)

⟩ = εδ(t − t ′).The equation of motion is

(9)X + 2kX + X + λX3 = f (t).

Once again, we carry out an equivalent linearizationwriting

(10)X + 2kX + ω2X = f (t),

whereω2 is given by Eq. (3) withb the amplitude ofmotion, i.e., the root mean square displacement in thiscase. In Fourier space,

(11)X(ω′) = f (ω′)ω2 − ω′2 + 2ikω′ .

From Eq. (8), we have

(12)⟨f (ω′)f (ω′′)

⟩ = εδ(ω′ − ω′′)

and, hence,⟨X(ω′)X(ω′′)

⟩

= 〈f (ω′)f (ω′′)〉(ω2 − ω′2 + 2ikω′)(ω2 − ω′′2 + 2ikω′′)

(13)= εδ(ω′ + ω′′)(ω2 − ω′2 + 2ikω′)(ω2 − ω′′2 + 2ikω′′)

.

The mean square amplitude

(14)b2 = ⟨X(t)2⟩

can be calculated as

b2 = ⟨X(t)2⟩

=∫

dω1

2π

dω2

2π

⟨X(ω1)X(ω2)

⟩ei(ω1+ω2)t

= ε

∫dω1

2π

dω2

2π

× δ(ω1 + ω2)ei(ω1+ω2)t

(ω2 − ω12 + 2ikω1)(ω2 − ω2

2 + 2ikω2)

= ε

∫dω1

2π

1

(ω2 − ω12)2 + 4k2ω1

2

(15)= ε

4kω2.

The amplitude is found from the positive roots of thequadratic equation

(16)3

4λb4 + b2 − ε

4k= 0

and it is clear that there can be no hysteresis. Thus ifthe periodic forcing were to be replaced by a stochasticforcing in the Duffing oscillator, the hysteresis woulddisappear. Consequently, we expect that if there is astochastic part in addition to the periodic forcing inthe Duffing oscillator, then for a critical value of thenoise correlation,ε the hysteresis present in Eq. (4)will disappear. To establish the above result, we begin

S. Datta, J.K. Bhattacharjee / Physics Letters A 283 (2001) 323–326 325

Fig. 2. The amplitude of oscillation is plotted against external force.

with the equation of motion

(17)X + 2kX + X + λX3 = F cosΩt + f (t)

whose equivalent linearization is

(18)X + 2kX + ω2X = F cosΩt + f (t),

where ω2 is given by Eq. (3). The correlation ofthe stochastic force prescribed in Eq. (8) and, since〈f (t)〉 = 0, it follows that

(19)b2 = F 2

(ω2 − Ω2)2 + 4k2Ω2+ ε

4kω2.

Since responses are strongest near resonance, i.e.,Ω 1, we work in terms of the variableδ = Ω − 1and write Eq. (19) as

(20)b2 = F 2/4

(λb2 − δ)2 + k2+ ε/4k

1+ λb2,

where λ = (3/4)λ. The amplitude is found from theroots of

λ3b8 + (λ2 − 2λ2δ

)b6

+(

λδ2 − 2λδ + λk2 − λ2ε

4k

)b4

+(

δ2 + k2 + λεδ

2k− λF 2

4

)b2

(21)= F 2

4+ ε(k2 + δ2)

4k.

Now we proceed to examine the situation whenε isturned on. The positive roots of Eq. (21), which isquartic inb2, are the roots that we are interested in.Since the product of the roots is negative, it is clear thatthere can be three positive roots or just one positiveroot. The existence of the three positive roots wouldindicate hysteresis. The curveb2 vs. F for a given

Fig. 3. The critical force, beyond which hysteresis sets in, in thepresence of noise is plotted against noise correlation. The solidline represents the linear scaling found analytically, while the dotsdenote the result of numerics.

value ofε would exhibit hysteresis if a critical valueof δ is exceeded. On the other hand,b2 vs.δ curve fora given value ofε would clearly show hysteresis if acritical value ofF is exceeded. The former variationfor a fixed value ofε = 0.001, λ = 1 is shown inFig. 2. In Fig. 3, which we obtained from a numericalstudy of Eq. (21), we show that when the value ofε

increases, the critical value of force also increases. Theturning points on theb2 vs.δ plot would correspond todδ/db2 = 0, a condition that can be put in the form

δ2 − 4λb2δ + 3λ2b4 + k2

(22)+ ελ

4k(1+ λb2)2

[δ2 + k2 + 2δ − λ2b4] = 0.

To the lowest order inε, the critical value of the exter-nal forcing for hysteresis as given in Eq. (7) is foundafter some straightforward but tedious algebra to begiven by

(23)Fcε2 = Fc

2 + 4εk = 32

3

k3

λ+ 4εk.

We can write this in the form

(24)Fcε2 = 32k

3λ

(k2 + 3ελ

8

)

which shows that the fluctuating force has the effect ofincreasing the damping constant. From Eq. (24), wecan conclude that 3ελ/8 is an effective contribution tok2, wherek is the damping constant of the unforcedoscillator.

326 S. Datta, J.K. Bhattacharjee / Physics Letters A 283 (2001) 323–326

Acknowledgements

S.D. would like to thank the CSIR, India, for pro-viding partial financial support.

References

[1] S.H. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, 1994;B.K. Chakraborty, M. Acharyya, Phys. Rep. 71 (1999) 847.

[2] L.D. Landau, E.M. Lifshitz, A Course in Theoretical Physics:Mechanics, Pergamon Press, 1959.

[3] P.S. Landa, P.V.E. McClintock, Phys. Rep. 323 (1999) 1.[4] S. Boccaletti, C. Grebogi, Y.C. Lai, H. Mancini, D. Maza, Phys.

Rep. 329 (2000) 103.[5] B. Ravindra, A.K. Mallik, J. Sound Vibration 211 (1998) 709.[6] W. Szemplinska Stupnica, J. Sound Vibration 178 (1994) 276.