Emergence of stability in a stochastically driven pendulum: beyond the Kapitsa effect
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Emergence of stability in a stochastically driven pendulum: beyond the Kapitsa effect
Yuval B. Simons and Baruch Meerson
Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel
We consider a prototypical nonlinear system which can be stabilized by multiplicative noise: anunderdamped non-linear pendulum with a stochastically vibrating pivot. A numerical solution ofthe pertinent Fokker-Planck equation shows that the upper equilibrium point of the pendulum canbecome stable even when the noise is white, and the “Kapitsa pendulum” effect is not at work. Thestabilization occurs in a strong-noise regime where WKB approximation does not hold.
PACS numbers: 05.40.-a, 05.10.Gg
It has been known for a long time that multiplica-tive noise can enhance stability of nonlinear systems.Examples are numerous indeed and culminate at noise-induced phase transitions far from equilibrium [1]. Thispaper deals with a noise-induced stabilization of oscillat-ing systems. As a prototypical example we consider anunderdampled nonlinear pendulum with a stochasticallyvibrating pivot. The stochastic driving introduces bothmultiplicative and additive noise, see Fig. 1. Our numeri-cal simulations clearly show that the multiplicative noisecan stabilize the otherwise unstable upper equilibriumpoint of the pendulum. The mechanism for this stabi-lization is markedly different from, and more subtle than,the “Kapitsa pendulum” mechanism. The Kapitsa pen-dulum involves a (deterministic) monochromatic para-metric driving of the pendulum at a frequency that ismuch higher than the natural frequency of the pendulum[2]. Here the upper equilibrium point becomes stable ifthe driving acceleration is higher than a critical valuedepending on the pendulum length and the gravity ac-celeration. In the Kapitsa pendulum problem the changeof stability of the upper equilibrium point comes from achange in the effective potential of the pendulum [2]. ξ g θξ l FIG. 1: (color online). Schematic of the stochastically drivensimple gravity pendulum.
Extensions of the Kapitsa pendulum effect to mul-tiplicative stochastic driving have also been considered[3, 4], see Ref. [5] for a review. In these extensions thenoise spectrum is strongly peaked at a single frequencywhich is much higher than the natural frequency of thependulum. The presence of a high-frequency noise of a sufficient strength modifies the effective potential whichcan stabilize the upper equilibrium point. Theory-wise,this setting introduces a time-scale separation which per-mits a perturbative treatment [5]. In this work we con-sider a model stochastic driving with a flat spectrum: awhite noise. Here all frequencies from 0 to ∞ are equallypresent, and there is no time-scale separation. We willshow that, for such a noise, the upper position of the pen-dulum can also become stable. However, the stabilizationcannot be traced to a change in the effective potential ofthe pendulum.A stochastically driven simple gravity pendulum canbe described by a Langevin equation:˙ θ = Ω , (1)˙Ω = − ω sin θ − γ Ω + 1 l sin θ √ µ ξ ( t ) + √ α ξ ( t ) , (2)where θ is the deviation angle of the pendulum, see Fig.1, Ω is the angular velocity, ω = p g/l is the harmonicfrequency of the pendulum, γ is the damping factor, l isthe the pendulum length, g is the gravity acceleration,and µ and α are the magnitudes of the multiplicativeand additive noises ξ and ξ , respectively. The noisesare assumed to be guassian, white with zero mean andmutually uncorrelated: h ξ i ( t ) i = 0 , h ξ i ( t ) ξ j ( t ′ ) i = 2 δ ( t − t ′ ) δ i,j , i, j = 1 , . (3)The Langevin equations (1) and (2) are equivalent (see,e.g. Ref. [6]) to the following Fokker-Planck equation forthe probability distribution W ( θ, Ω , t ): W t = − Ω W θ + ω sin θ W Ω + 2 γ ∂∂ Ω (Ω W )+ (cid:16) α + µl sin θ (cid:17) W ΩΩ , (4)where the indices θ , Ω and t denote the correspondingpartial derivatives of W ( θ, Ω , t ). As the noises are Ω-independent, there is no difference between the Ito andStratonovich interpretations. Introducing the dimension-less variables ˜ t = ω t , ˜Ω = Ω /ω and ˜ W ( t, θ, Ω) = ω W ( t, θ, Ω), we can rewrite the Fokker-Planck equationin a dimensionless form: W t = − Ω W θ + sin θW Ω + 2Γ ∂∂ Ω (Ω W )+ (cid:0) ε + δ sin θ (cid:1) W ΩΩ , (5)where Γ = γ/ω , ε = α/ω and δ = µ/ ( l ω ) are therescaled parameters of the system, and the tildes areomitted.We assume that, after a transient, the stochasticsystem approaches a smooth steady state for which W ( θ, Ω , t ) is independent of time: W ( θ, Ω , t → ∞ ) =¯ W ( θ, Ω). The steady-state probability distribution¯ W ( θ, Ω) obeys the equation − Ω ¯ W θ + sin θ ¯ W Ω + 2Γ ∂∂ Ω (Ω ¯ W )+ (cid:0) ε + δ sin θ (cid:1) ¯ W ΩΩ = 0 . (6)We classify a point ( θ, Ω) as a stable point of the systemif it is a local maximum of the stationary probabilitydistribution ¯ W ( θ, Ω). The necessary and sufficient con-ditions for a function of two variables f ( x, y ) to have alocal maximum at ( x , y ) are (see, e.g. Ref. [7]): f x ( x , y ) = f y ( x , y ) = 0 , (7) f xx ( x , y ) < f yy ( x , y ) < , (8) f xx ( x , y ) f yy ( x , y ) − f xy ( x , y ) > , (9)where the indices x and y denote partial derivatives. Letus examine the stability properties of the upper equi-librium point ( θ = π, Ω = 0) of the driven pendu-lum. Equation (6) is invariant under the transformation θ → π − θ, Ω → − Ω, that is under reflection of the axes θ and Ω around the point ( π, W ( θ, Ω) mustobey the same symmetry. Therefore, the first derivatives¯ W θ and ¯ W Ω must vanish at ( π, W ΩΩ ( π,
0) = − ε ¯ W ( π, < , so Eq. (8) is also satisfied at ( π ,0). As a result, the nec-essary and sufficient condition for ( π,
0) to be a stablepoint is given by Eq. (9):∆ ≡ ¯ W ΩΩ ( π,
0) ¯ W θθ ( π, − ¯ W θ Ω ( π, > . (10)For δ = 0 (only additive noise), the steady-state equation(6) is soluble analytically, see Ref. [8]:¯ W ( θ, Ω) = Γ / π / ε / I (2Γ /ε ) exp (cid:20) − Γ ε (cid:0) Ω − θ (cid:1)(cid:21) , (11)where I ( . . . ) is the modified Bessel function. In this casethe point ( π,
0) is unstable, as the stability parameter ∆,defined in Eq. (10), is negative:∆ = − Γ π ε I (2Γ /ε ) exp (cid:18) − ε (cid:19) < . (12) FIG. 2: (color online). A steady-state probability distributionwith no multiplicative noise. The parameters are ε = 0 . . δ = 0. This solution evolved from the initialcondition W = (2 π √ π ) − exp( − Ω ) after 60 normalized timeunits. It coincides with the analytical solution (11). The probability distribution (11) is depicted in Fig. 2.What happens when δ >
0? First, we show that thereis no stabilization in the weak-noise limit, δ ∼ ε ≪ , Γ.In this limit one can use a dissipative variant of WKBapproximation, see e.g.
Ref. [6], and make the ansatz¯ W ( θ, Ω) = A ( θ, Ω) exp (cid:20) − S ( θ, Ω) ε (cid:21) , assuming that the pre-factor A varies on scales muchlarger than 1 /ε . In the leading order in 1 /ε one obtainsΩ S θ − sin θS Ω − S Ω + (cid:18) δε sin θ (cid:19) S = 0 . (13)This first-order PDE for the action S ( θ, Ω) has the formof a stationary Hamilton-Jacobi equation (see e.g.
Ref.[9]) with the Hamiltonian H ( θ, Ω , p , p ) of the form H = Ω p − sin θ p − p + (cid:18) δε sin θ (cid:19) p . (14)Here p = S θ and p = S Ω are the canonical momentaconjugated to the coordinates θ and Ω, respectively. Asthe Hamilton-Jacobi equation (13) is stationary, we areinterested in the zero-energy dynamics H = 0. TheHamilton’s equations˙ θ = Ω , ˙Ω = − sin θ − (cid:18) δε sin θ (cid:19) p , (15)˙ p = cos θ p − δε sin 2 θ p , ˙ p = − p + 2Γ p , (16)have two zero-energy fixed points: a = (0 , , ,
0) and a = ( π, , , a is de-termined by the quadratic approximation to the Hamil-tonian around a , H ( θ, Ω , p , p ) ≃ Ω p + θp − p + p . (17)This quadratic approximation is noiseless, as the noiseterm in Eq. (14), ( δ/ε ) sin θ p , is bi-quadratic in smalldeviations from the fixed point. Therefore, a weak mul-tiplicative noise does not generate any correction to thepotential of the pendulum, in agreement with an earlyobservation [3], and cannot change the stability proper-ties of the system.However, for a strong multiplicative noise numericalsolutions of the time-dependent Fokker-Planck equation(5) do show emergence of stability of the upper equilib-rium point. We obtained these numerical results with a“Mathematica” PDE solver. The numerical domain was0 < θ < π and − Ω max < Ω < Ω max with Ω max chosen,separately for each set of parameters, sufficiently large.Periodic boundary conditions in θ were used. As to theboundary conditions in Ω, we checked that, at sufficientlylarge Ω max , the steady-state solution remained the sameup to 1 per cent, whether we imposed periodic or zero- W conditions at the Ω-boundary. Larger Ω max neededto be taken when Γ became smaller than ε and when δ became large compared to Γ and ε . The values of Ω max that we used ranged from Ω max = 2 for Γ = 0 . ε = 0 . δ ≃ .
15 to Ω max = 15 for Γ = 0 . ε = 0 . δ ≃ .
43. After verifying that Ω max is sufficiently large,we used the zero- W boundary conditions in Ω, as thischoice reduced the computational time. For each set ofparameters δ, Γ and ε we used the analytical solution (11)for δ = 0 as the initial condition.We ran the solution until t = t max such that the valueof ∆, evaluated at t = t max , was within 1 per cent fromits value at t = t max /
2. The larger the parameter Γ, thesmaller t max was needed. The values of t max that weused ranged from 40 for Γ = 1, ε = 0 . δ = 0 . . ε = 0 .
05 and δ = 0 . /ε or large δ the steady-state probabilitydistribution broadens which demands a larger Ω max , alonger computation time and more computer memory.At large Γ /ε or small δ the distribution becomes too nar-row to numerically resolve it with confidence. We ver-ified that, starting from an arbitrary initial condition,the probability function always converges to the samesteady-state solution. We also checked that for δ = 0the steady-state solution coincides, with a high accuracy,with the analytical solution (11).Figure 3 gives a typical numerical example of a steady-state probability distribution having a distinct local max-imum at ( π, , FIG. 3: (color online). Multiplicative noise stabilizes the up-per equilibrium point of the pendulum. Shown is the steady-state probability distribution ¯ W ( θ, Ω) which exhibits a localmaximum at the upper equilibrium point ( θ = π, Ω = 0). Theparameters are ε = 0 .
1, Γ = 0 . δ = 2. The steady-stateprobability distribution was found by solving numerically thetime-dependent Fokker-Planck equation (5). It evolved fromthe initial condition (11) after 60 dimensionless time units. To determine the stability properties of the upper equi-librium point ( π, δ , for different values of Γ and ε .Several examples of such plots are shown in Fig. 4. Usinginterpolation, we found the critical values δ c = δ c (Γ , ε )of δ , at which ∆ = 0. FIG. 4: (color online). The stability parameter ∆, normalizedto | ∆ | from Eq. (12), is plotted versus the rescaled magnitudeof the multiplicative noise δ for different values of ε at Γ = 0 . δ c . Figure 5 shows the plot of δ c versus Γ at different ε .One can see that δ c slowly decreases with Γ and ap- = 0.5 = 0.3 = 0.1 = 0.05 c FIG. 5: (color online). The critical rescaled magnitude of themultiplicative noise δ c vs. the rescaled damping factor Γ fordifferent values of ε . The lines are only given for guiding theeye. = 0.5 = 0.3 = 0.1 c FIG. 6: (color online). The critical rescaled magnitude ofthe multiplicative noise δ c vs. the rescaled magnitude of theadditive noise ε for different values of Γ. The lines are onlygiven for guiding the eye. proaches a finite value at Γ →
0. The plot of δ c versus ε atdifferent Γ, presented in in Fig. 6, shows a much stronger,square-root-like dependence at small ε . Our data for dif-ferent Γ do not contradict a power-law behavior δ c ∼ ε α with α ≃ .
44, as shown in Fig. 7 for Γ = 0 .
1. When ε approaches 0, δ c also goes to zero, and one would expectto always see stability in this case. As ε →
0, however,the probability distribution develops singularities bothat (0 , π ,0), but the peak at (0 ,
0) becomesmuch higher than that at ( π, π,
0) is physically insignificant in this case. Only with a sufficiently large additive noise the probability distri-bution becomes sufficiently broad to allow a reasonableprobability for the pendulum to be at and around ( π ,0).In summary, we have investigated the stabilization of a -4 -3 -2 -1-2.4-2.0-1.6-1.2-0.8 l n ( c ) ln( ) FIG. 7: The log-log plot of δ c vs. ε for Γ = 0 .
1. A linear fitgives a slope of 0 . ± .
01 with the coefficient of determination R = 0 . prototypical nonlinear oscillating system by a multiplica-tive white noise. The stabilization is clearly observedin the numerical solution of the Fokker-Planck equationand requires a super-critical noise magnitude. The sta-bilization cannot be traced to a change in the effectivepotential of the system and is not predicted by a WKBanalysis which assumes a weak noise.We are grateful to Pavel V. Sasorov for useful discus-sions. This work was supported by the Israel ScienceFoundation (Grant No. 408/08). [1] W. Horsthemke and R. Lefever, Noise-Induced Transi-tions. (Springer-Verlag, Berlin, 1984).[2] L.D. Landau and E.M. Lifshitz,
Mechanics (Pergamon,Oxford, 1960).[3] V.E. Shapiro and V.M. Loginov, Phys. Lett. A, 287(1979).[4] P.S. Landa and A.A. Zaikin, Phys. Rev. E , 3535 (1996).[5] R.A. Ibrahim, J. Vibration and Control , 1093 (2006).[6] C.W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 2004).[7] G.M. Fikhtengol’ts,