CHIN.PHYS.LETT.
Vol. 25, No. 4 (2008) 1209
Numerical Investigation of Noise Enhanced Stability Phenomenon in a Time-Delayed Metastable System JIA Zheng-Lin(_)∗ Department of Physics, Yuxi Normal University, Yuxi 653100
(Received 13 January 2008) The transient properties of a time-delayed metastable system subjected to the additive white noise are investigated by means of the stochastic simulation method. The noise enhanced stability phenomenon (NES) can be observed in this system and the effect of the delay time on the NES shows a critical behaviour, i.e., there is a critical value of the delay time τc ≈ 1, above which the time delay enhances the NES effect with the delay time increasing and below which the time delay weakens the NES effect as the delay time increases.
PACS: 05. 40. Ca, 02. 50. −r, 02. 60. Cb In recent years, the investigation of nonlinear dynamics and instabilities in systems away from equilibrium has led to the discovery of some counterintuitive and resonance-like phenomena, among which the noise enhanced stability (NES)[1,2] and the resonant activation (RA)[3] are two typical noise-induced effects. The NES effect can be characterized by the dependence on the noise intensity of the mean first-passage time (MFPT) for metastable and unstable systems, has a maximum at some noise intensity.[1,2,4−16] The signature of the RA phenomenon is a minimum of the lifetime of the metastable state as a function of a driving frequency.[2,3] Therefore, the NES effect increases the average lifetime of the metastable state, while the RA phenomenon minimizes this lifetime. In particular, the NES effect implies that the stability of metastable or unstable states can be enhanced by the noise and has been observed in different physical systems.[1−16] The investigation of the NES effect has revealed that the role of the NES on the physical systems may be positive or negative.[2,5] However, the previous studies on the NES effect are just performed in the nondelayed systems. Indeed, due to the finite transmission times of the signals or other key quantities (for example, information, matter, energy etc.), the time delay is ubiquitous in many physical systems,[17,18] and the effects of noise on dynamical systems with time delay recently have gained considerable attention.[19−26] These investigations have shown that the interplay between the noise and the time-delay can fundamentally change the statistical properties of the system.[21,22] Thus, the NES effect in the time-delayed metastable system need to be investigated. The effects of the time delay on the NES effect deserve further discussion. In this Letter, we introduce a time-delay into a stochastic cubic potential system and numerically investigate the effects of the delay time on the transient properties of this system. The primary aim is to test ∗ Email:
[email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd °
whether the NES effect can be observed in the timedelayed metastable system or not and to investigate how the delay time influence the MFPT of the system, especially for the case of large delay time. Because any metastable state can be described through a local cubic potential even if the real potential has other local or global stable states,[4] the cubic potential is usually used as an archetypal model in study of the NES phenomenon.[4,16] The Langevin equation (LE) of the cubic potential system driven by the additive white noise reads[4] dU (x) √ x˙ = − (1) + Dξ(t), dx where U (x) = 0.3x2 − 0.2x3 is a cubic potential and ξ(t) is zero mean (hξ(t)i = 0) and δ-correlated in time (hξ(t)ξ(t0 )i = δ(t − t0 )), and D is the noise intensity. We introduce a time delay into the system described by Eq. (1), and assume that Eq. (1) with time delay should be rewritten as √ x(t) ˙ = 0.6x2 (t) − 0.6x(t − τ ) + Dξ(t), (2) where τ is the delay time. Equation (2) describes a noise-driven metastable system with time-delayed feedback. We simply refer to the system described by Eq. (2) as the time-delayed metastable system in the present study. Such an extension is important since it provides a possible way to control the NES effect. Under the condition of the small delay time and the firstorder approximation, Eq. (2) can be approximated by the following LE:[17,18] √ (1 − 0.6τ )x˙ = 0.6x2 (t) − 0.6x(t) + Dξ(t). (3) Using the following scaling relations 1 t, 1 − 0.6τ
(4)
1 D, (1 − 0.6τ )2
(5)
e t= and e= D
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Eq. (3) can be rewritten as x( ˙ e t) = 0.6x2 (e t) − 0.6x(e t) +
p
e e Dξ( t),
(6)
which has the same form as that of Eq. (1). Accordingly, for the case of small delay time, after fixing a given target position xF , the MFPT (the average time for a Brownian particle starting from an initial position x0 to reach xF ) can be obtained as[4,27] Z Z 2 xF 2u(x) x −2u(y) e e dydx, (7) T (x0 , xF ) = e x0 D −∞ e Equation (7) can be evaluwhere u(x) = U (x)/D. ated via the method proposed by Fiasconaro et al. in Ref. [4]. Obviously, based on Eq. (7) and the results obtained in Ref. [4], one can conclude that the NES effect can be observed in this time-delayed metastable system for the case of small delay time.
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to obtain the MFPT of a stochastic delayed bistable system in Ref. [23]. An absorbing boundary condition at x = xF and a reflecting boundary condition at x = −∞ are used in the numerical simulations of Eq. (2). The evaluation of each MFPT has been performed through 5 × 104 simulation runs, which is calculated as the ensemble average of the first passage times through xF for different noise realizations. In the present study, x0 = 1.6 is used as an initial unstable position and xF = 2.2 as a target position, which follows from Ref. [4]. To check the validity of the numerical simulation method, for the case of small delay time, we compare the numerical estimations of the MFPT with the corresponding estimations of the MFPT from Eq. (7) in Fig. 1, from which one can see that the agreement between the result of the numerical simulation of the Eq. (2) and that of the theoretical evaluation from Eq. (7) is very good when the noise intensity is not very large.
Fig. 1. Plot of the MFPT as a function of the noise intensity D with x0 = 1.6, xF = 2.2, and ∆t = 0.001 for the case of small delay time: the numerical estimations (Eq. (2)) vs the theoretical evaluations (Eq. (7)).
However, for the case of large delay time, so far there is no analytical method to obtain the MFPT. Thus, to obtain the MFPT for various values of the delay time, we perform a series of direct numerical simulations of the stochastic delayed differential equation (Eq. (2)), using the Euler forward algorithm with a small time step ∆t,[26,28] x(t + ∆t) = x(t) + [0.6x2 (t) − 0.6x(t − τ )]∆t + Γ + O(∆t3/2 ),
(8)
where Γ = [−2D∆t ln(γ1 )]1/2 cos(2πγ2 ) and γ1 and γ2 are two independent random numbers uniformly distributed in the interval (0, 1). We use ∆t = 0.01 for the cases of large delay times and ∆t = 0.001 for the cases of small delay times. When t ≤ τ , x = x0 is assumed, i.e., we assume that the Brownian particle is initially located at position x0 in the time interval [−τ, 0]. This kind of initial condition has been used
Fig. 2. Plot of the MFPT as a function of the noise intensity D with x0 = 1.6 and xF = 2.2: (a) for small values of the delay time(τ < τc ) with time step ∆t = 0.001 and (b) for large values of the delay time (τ > τc ) with time step ∆t = 0.01.
The simulative results of the MFPT as a function of the noise intensity D are shown in Fig. 2 for different delay times. It is clearly seen that a nonmonotonic behaviour of the MFPT as a function of the noise intensity occurs, i.e., the MFPT has a maximum at some noise intensity and the NES effect can be observed for all values of the delay time. More interestingly, Figs. 2(a) and 2(b) indicate that the effect of the de-
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lay time on the NES shows a critical behaviour. That is, there is a critical value of the delay time τc , below which the height of the maximum of the MFPT decreases with the delay time increasing and above which it increases with the delay time increasing. This fact implies that the time delay weakens the NES effect for the case of τ < τc , while the time delay enhances the NES effect for the case of τ > τc . One can conclude that for the case of τ > τc , the time delay can further enhance the stability of the metastable state in this time-delayed metastable system due to the combination of the interplay between the noise and the time-delay with the NES effect. In addition, Fig. 2 also shows that the influences of the delay time on the MFPT depend on the different values of the noise intensity. In the regime of noise intensity where the NES effect takes place, the effect of the delay time on the MFPT is more pronounced than that of other noise intensities due to the NES effect. At the same time, one should note that as the delay time changes from the case of τ < τc to the case of τ > τc , the maximum of the MFPT shifts towards a higher value of noise intensity, i.e., the maximum of the MFPT is located at about D = 0.5 when τ < τc and the maximum of the MFPT is located at about D = 1.0 when τ > τc .
Fig. 3. Plot of the MFPT as a function of the delay time τ for various values of the noise intensity (a) D = 0.15 and (b) D = 1.0. Other parameter values are the same as those in Fig. 2.
Figure 3 is the plot of the MFPT as a function of the delay time τ for different values of the noise inten-
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sity D. Figure 3 further confirms the dependence on the D of the effect of the τ on the transient properties of the system. From Fig. 3(b), one can determine the critical value of the delay time τc is about 1. Moreover, Fig. 3 shows that the curve of MFPT vs τ presents a minimum at some noise intensity, which has an analogy to the resonant activation effect. A possible explanation of this effect is that the time-delayed feedback leads to the fluctuating of the force acting on the Brownian particle and the fluctuation rate depends on the delay time, which plays the similar role as that of the fluctuation of the potential barrier in non-delayed metastable system, namely the MFPT exhibits a minimum at a resonant fluctuation rate. This mechanism explains why the effect of the delay time on the NES shows a critical behaviour. In conclusion, we have numerically investigated the effects of the time delay on the transient properties of a time-delayed stochastic metastable system. It is found that the NES effect can be observed in this system for an arbitrary value of the delay time and the influence of the time delay on the NES effect shows a critical behaviour. That is to say, there is a critical value of the delay time τc . When the delay time τ < τc , as τ increases, the NES effect becomes weaker and weaker. However, for the case of τ > τc , an enhancement of the NES effect can be observed with the delay time increasing, i.e., the time delay can further enhance the stability of the system. In general, the time delay usually induces the instability of the system in the absence of noise, while the present study shows that the large delay time can enhance the stability of the time-delayed system due to the combination of the interplay between the noise and the time delay with the NES effect. Additionally, it is found that the effect of the delay time on the MFPT depends on the different noise intensities, and the effect of the delay time on the MFPT becomes more pronounced in the regime of noise intensity where the NES effect occurs. Finally, it must be pointed out that the effect of the delay time on the NES also depends on how the timedelayed feedback is introduced into the system. For example, if the time delay is added to the first term of the right-hand side of Eq. (2), the time delay always weakens the NES effect. The results of this study suggest that the appropriate introduction of the timedelayed feedback can provide a feasible control of the NES effect in some physical systems.
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