analytical study of vibrational resonance in an ... - World Scientific
July 9, 2008
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International Journal of Bifurcation and Chaos, Vol. 18, No. 6 (2008) 1767–1773 c World Scientific Publishing Company
ANALYTICAL STUDY OF VIBRATIONAL RESONANCE IN AN OVERDAMPED BISTABLE OSCILLATOR V. N. CHIZHEVSKY B. I. Stepanov Institute of Physics, NASB, 220072 Minsk, Belarus Received February 1, 2007; Revised June 16, 2007 The results of analytical study of vibrational resonance (VR) occurring in overdamped bistable system driven by two periodic signals with very different frequencies are presented. Approximate solutions for responses at the low-frequency as a function of the amplitude, and the frequency of the additional high frequency modulation which describe well the main features of vibrational resonance are obtained. Scaling laws for the gain factor and the switching threshold in VR are also found. Analytical results are compared with results of the numerical simulation, showing a good agreement. Keywords: Bistability; vibrational resonance; biperiodic excitation.
1. Introduction An overdamped bistable oscillator is one of the basic models in studying different phenomena in real physical systems. Recently, much attention has been focused on investigations of dynamics of such a system excited by two periodic signals with highly different frequencies [Landa & McClintock, 2000; Gittermann, 2001; Zaikin et al., 2002; Ullner et al., 2003; Baltanas et al., 2003; Chizhevsky et al., 2003; Blekhman & Landa, 2004; Casado-Pascual & Baltanas, 2004; Chizhevsky & Giacomelli, 2004, 2005]. In these conditions the response to the lowfrequency (LF) periodic modulation passes through a maximum depending on the amplitude of the high-frequency (HF) modulation [Landa & McClintock, 2000]. Such a phenomenon has been named vibrational resonance (VR). An evidence of VR has been demonstrated in analog electric circuits utilized to model noise-induced structures [Zaikin et al., 2002], excitable systems [Ullner et al., 2003], an overdamped bistable oscillator [Baltanas et al., 2003], and in a vertical cavity surface emitting laser within the regime of polarization bistability for both
symmetric and strongly asymmetric quasipotentials [Chizhevsky et al., 2003]. Besides, the possibility to make use of VR for detection of weak signals has been also demonstrated [Chizhevsky et al., 2003]. Recently, the effect of noise on VR was studied both theoretically [Casado-Pascual & Baltanas, 2004; Chizhevsky & Giacomelli, 2004, 2005] and experimentally [Chizhevsky & Giacomelli, 2004, 2005]. In fact, it was shown that vibrational resonance can be used for the control of stochastic resonance. A theoretical explanation of the phenomenon was given in [Baltanas et al., 2003] on the basis of the analytical treatment of the problem. It has been shown that VR takes place in the vicinity of the critical point corresponding to transition from bistability to monostability controlled by the HF modulation. Because of the difference of time scales associated with LF and HF modulations, a rapidly oscillating double-well potential can be transformed into an effective potential with a parametric dependence on the amplitude and frequency of the HF modulation. In such a formulation, one can note that VR has a clear analogy with a parametric
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amplification of near-resonant perturbations near the first period-doubling bifurcation point in nonautonomous systems [Bryant & Wiesenfeld, 1986]. In spite of a number of theoretical papers [Landa & McClintock, 2000; Gittermann, 2001; Zaikin et al., 2002; Ullner et al., 2003; Baltanas et al., 2003; Blekhman & Landa, 2004] devoted to VR, there is no adequate analytical description of VR in overdamped bistable oscillator until now. Here we present the approximate analytical solutions for VR occurring in overdamped oscillator with symmetrical double-well potential that well describe the main features of VR. These solutions are in a good agreement with the numerical results. Besides, scaling laws relating the switching threshold and the maximum gain factor in VR with the amplitude of the LF signal have been also found.
2. Theoretical Model and Basic Relationships We consider the phenomenon of VR using the model of overdamped bistable oscillator driven by two forces. One of them is a slowly time-varying input signal f (t) with a characteristic time τL and the amplitude AL , where f (t) can be periodic or aperiodic functions. The second signal is a high frequency periodic modulation with an amplitude AH and a frequency ΩH so that ΩH τL 1. The dynamics of the system in this case can be described by the following equation ∂x = −V (x) + AL f (t) + AH sin ΩH t ∂t
(1)
where V (x) is the derivative with respect to x of a bistable potential function αx2 βx4 + (2) 2 4 with the local minima x± 0 = ± α/β, the barrier height ∆V0 = α2 /4β, the statical threshold of crossing the potential barrier µ0 = 4α3 /27β, and the relaxation time τr0 = 1/2α, where α and β are positive numbers. The parameter µ0 corresponds to a transition from the symmetric double-well potential to an asymmetrical one-well configuration. Taking into account the expression for the potential, the equation takes the form V (x) = −
dx = αx − βx3 + AL f (t) + AH sin ΩH t. dt
(3)
Generally, we have in the problem two timescales: the slow one, which is associated with the LF signal, and the fast one relating to the high frequency modulation. For the sake of clarity, we recall some of the results presented in [Baltanas et al., 2003]. We look for the solution as follows: x(t) = y(t) + η(AH , ΩH ) sin ΩH t,
(4)
where y(t) denotes a slow part of the solution, η(AH , ΩH ) is the coefficient. Substituting (4) into (3) and averaging over the period TH = 2π/ΩH we obtain the following equation governing the slow dynamics of the bistable system [Chizhevsky & Giacomelli, 2004] dY = α(1 − ξ 2 )Y − βY 3 + AL F (t), (5) dt where Y = y(t)TH , F (t) = f (t)TH . The ξ is a normalized amplitude of the HF modulation ξ = AH /Ac and Ac is the switching threshold Ac = (2α/3β)(Ω2H + (2α2 /9)) [Chizhevsky & Giacomelli, 2004]. Correspondingly, the effective potential Vξ (y) takes the form
α(1 − ξ 2 )y 2 βy 4 + . (6) 2 4 Obviously, after averaging operation, all parameters ±, of the potential such as the location of minima ym the barrier height ∆V , the crossing threshold µ and the relaxation time τr depend now on the parameter ξ in the following way: α(1 − ξ 2 ) ± , (7) ym = ± β Vξ (y) = −
α2 (1 − ξ 2 )2 , 4β 4α3 (1 − ξ 2 )3 , µ= 27β
∆V =
τr =
1 . 2α(1 − ξ 2 )
(8)
(9) (10)
For the approximate solution in the response at low frequency, some conclusions can be made from expressions (7)–(10). Figure 1 shows the view of potential Vξ (y) for different values of the normalized amplitude of the HF modulation ξ. The corresponding locations of the minima of Vξ (y) are depicted in Fig. 2. As ξ increases one can see the transition from the doublewell configuration of Vξ (y) to the one-well potential. This transition occurs for the critical value ξ = 1.
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Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
3. Adiabatic Approximation
2.5
First we consider the effect of the low-frequency periodic modulation F (t) = sin(ΩL t) when ΩL 2α. In this case we can neglect the time derivative in (5) and obtain the following equation: 4α3 sin(ΩL t) = 0. (11) α(1 − ξ 2 )Y − βY 3 + ε 27β
2 1.5 1 V(y)
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0.5 3
0
4
2
−0.5 1
−1 −1.5 −1.5
−1
−0.5
0 y
0.5
1
1.5
Fig. 1. View of the potential Vξ (y) for different values of ξ = 0 (1), 0.7 (2), 1 (3), 1.5 (4).
We here use the normalized LF amplitude ε = AL / 4α3 /27β. In this case, locations of extremums are determined by the following solutions: 1 1 − ξ2 cos arccos γ , (12) y1 = 2y0 3 3 y2 = −2y0
1 π 1 − ξ2 cos arccos γ − , 3 3 3
(13)
+ y0
y
0.5
yu 0
0 −0.5
−
y0
−1 0 Fig. 2.
0.5
ξ
1
1.5
The location of extrema y0+ , y0− and y0u versus ξ.
This means that driving the bistable system by the HF modulation, we place the system near the bifurcation point corresponding to the transition from bistability to monostability. At such conditions one can expect a high sensitivity of bistable systems to external perturbations, since nonlinear systems operating near simple critical points are known to be very sensitive to the effect of periodic perturbations and noise. It should be noted also that the relaxation time τr tends to infinity as ξ approaches the bifurcation point as it follows from (10). This can be considered as a manifestation of the general phenomenon of critical slowing down which is well known in the theory of bifurcations. On the other hand, one can also note that the periodic modulation with the amplitude larger than Ac leads to the elimination of bistability in the same sense as was found for the periodically modulated nonlinear systems possessing generalized bistability [Pisarchik & Goswami, 2000].
1 π 1 − ξ2 cos arccos γ + , (14) yu = 2y0 3 3 3 where y0 = a/b and γ = ε(1 − ξ 2 )−3/2 . Figure 3 shows a set of all real solutions corresponding to locations of two stable (y1 and y2 ) and one unstable (yu ) fixed points as a function of ξ. From the condition ε(1 − ξ 2 )−3/2 = 1 it follows that the critical amplitude ξc needed for the switching between two states depends on the normalized amplitude ε of the LF signal as follows: (15) ξc = 1 − ε2/3 . Analysis of the solutions (12)–(14) shows that the response of bistable system to the LF signal for ξ < ξc can be associated with the solution yu . As 1
y1
0.5
y
1
yu
0 y2 −0.5 −1 0
0.5
ξ
1
1.5
Fig. 3. Set of all real solutions obtained from (11) showing locations of extrema as a function of ξ (AL = 0.25).
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Now one can evaluate the gain factor GVR . We define the gain factor GVR as a ratio of responses of the bistable system to the LF periodic signal in the presence [RL (ΩL )] and the absence [R0 (ΩL )] of the HF modulation: RL (ΩL ) . (17) GVR = R0 (ΩL ) In the linear approximation AL , R0 = Ω2L + 4α2
(18)
and in the case of ΩL 2α the response R0 can be reduced to AL . (19) R0 ∼ = 2α In this case, one can find from (12) that 1/2 1/3 4α , (20) RL = ε 3β and the maximum gain Gmax VR is determined by the expression −2/3 . Gmax VR = 6ε
1
50 2
40 30
3 4
20 6
10 0
0.6
5
0.8
1.2
Fig. 4. Analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for different values of the LF amplitude AL = 0.05 (1), 0.1 (2), 0.2 (3), 0.33 (4), 0.5 (5), 0.75 (6).
α = β = 4. Depending on the context in the paper, along with the quantities AL and AH we use the dimensionless amplitudes ε and ξ defined previously as ε = AL /µL and ξ = AH /µH where µL and µH are the switching thresholds at the frequencies ΩL and ΩH , respectively, in the absence of any additional modulation. For a quantitative characterization of VR in the simulation we used the gain factor GVR (17), which was evaluated from the Fourier spectra of temporal responses of the system. Figure 4 shows the analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for different values of the amplitude AL of the LF signal for the case of low frequency (ΩL /2π = 10−3 in the numerical simulation) with respect to Ωr = 2α(ΩL ≪ Ωr ) using the expression 1.9
1
(21)
It should be noted such a scaling law has been experimentally observed in a nonautonomous system (a loss-modulated CO2 laser) for the gain factor in the amplification of near-resonant perturbations near the first period-doubling bifurcation [Corbalan et al., 1995]. In parallel, in order to check the analytical results we numerically integrated Eq. (3) with
ξ
1
log10Gmax VR
For ξ ≥ 1 the real solution y1 is determined by the expression 1 1 − ξ2 sinh arcsinh γ . y1 = 2y0 3 3
60
GVR
ξ increases from zero to ξc the response follows the solution yu until ξ reaches its critical value ξc where purely real solutions yu and y2 become the imaginary ones. This condition corresponds to the disappearance of bistability. At the point ξ = ξc the system response jumps across into solution y1 which results in the appearance of a discontinuity in the response. For ξ > ξc the response of the system follows the solution y1 which is purely real. Based on these speculations one can build the response of a bistable system to the LF modulation as a function of ξ by the following rule: yu for ξ < ξc (16) RL = y1 for ξ ≥ ξc
ξc
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0
0
0.5 ε (a)
1
0.9 −1.5
−1 −0.5 log10ε (b)
Fig. 5. (a) Switching threshold ξc and (b) the maximum gain Gmax VR as a function of ε. Solid lines — analytics, open circles — numerics.
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Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
(16) for RL . One can note that analytical results well agree with the numerical ones and well describe the main regularities of VR — the strong dependence of the response on the LF amplitude AL , the shift of the optimal value of ξ and the broadening of curves as AL increases. In Fig. 5 we compare our analytical results for the switching threshold (15) and the maximum gain Gmax VR (21) with the numerical results as a function of ε. One can note a good agreement between them.
4. General Case We now consider the general case for F (t) = sin(ΩL t) when one cannot neglect the time derivative in (5). In this case, we look for the approximate solution in the form y(t) = σ sin(ΩL t + ϕ). Substituting it into Eq. (5) and neglecting the trig function of argument 3ΩL , we obtain two equations for a determination of σ and ϕ as a function of AL and ξ. 3βσ 2 cos ϕ = 0 −ω sin ϕ + 4 − α(1 − ξ 2 ) 3βσ 3 sin ϕ − AL = 0 σω cos ϕ + 4 − ασ(1 − ξ 2 ) Resolving the system with respect to σ we get the following equation: 6
2
δ + pδ + q = 0 where δ=
(22)
σ2 = −
κ2 u + κ1 v + 8α(1 − ξ 2 ) 9β
− 8α(1 9β
− ξ2)
p=
16 3Ω2L − α2 (1 − ξ 2 )2 27 β2
q=
16 8α3 (1 − ξ 2 )3 + 72α(1 − ξ 2 )Ω2L − 81βA2L 729 β3
Equation (22) can be resolved with respect to δ giving the location of stable and unstable fixed points. The complete set of solutions σ of one sign can be represented as follows: u + v + 8α(1 − ξ 2 ) (23) σ1 = 9β κ1 u + κ2 v + 8α(1 − ξ 2 ) (24) σu = 9β
(25)
where
√ √ q q + D, v = − − D, u= − 2 2 √
p 3 q 2 (−1 ± i 3) D= . + , κ1,2 = 3 2 2 Figure 6 shows an evolution of solutions σ1 , σ2 and σu versus ξ as ΩL increases (from (1) to (4) in the figure). In fact, these solutions allow us to study the influence of the frequency of the LF signal on the response of a bistable system. Similar to the previous case (11) the response to the LF signal for ξ < ξc can be associated with the solution σu . But in this case ξc depends on the frequency ΩL as well. This critical value ξc can be found from the condition that the discriminant D = 0. As ξ increases, the response follows the solution σu until ξ reaches its critical value ξc . At the point ξ = ξc the response of the system jumps across to the solution σ1 leading to the appearance of a discontinuity in the response for ΩL Ωr . As ΩL approaches Ωr , σ1
1 0.5
0.5
σu
0
σ1
1
σu
0
−0.5
σ2
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−0.5 σ2
−1 0
0.5
ξ
σ2
−1 1
1.5
0
(a)
0.5
1
1.5
1
1.5
σ1
1 0.5
σu
0
ξ
(b)
σ1
1
0.5
σu
0
−0.5
−0.5 σ2
−1 0
0.5 (c)
ξ
σ2
−1 1
1.5
0
0.5
ξ
(d)
Fig. 6. Solutions σ1 , σ2 and σu as a function of ξ for different values of ΩL (ΩL /2π = 0.01 (a), 0.0825 (b), 0.1 (c), 0.15 (d), AL = 0.25).
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the discontinuity is replaced by a smooth transition in the response (Fig. 6). For ξ > ξc the response of the system follows the solution σ1 which is purely real. Thus, in this case the response is determined by same rule as (16): for ξ < ξc for ξ ≥ ξc
(26)
In Figs. 7 and 8 we compare analytical and numerical results for the gain factor GVR in the general case using RL in the expression (26). Figure 7 shows GVR for ΩL /2π = 0.075. In this case, the smooth shape of GVR is noted, also the weak dependence of GVR on AL as well as the small shift of Gmax VR on AL for the weak LF amplitudes in contrast with the previous case shown in Fig. 4. For the given amplitude of the LF signal, an increase of ΩL leads to a significant lowering of GVR , smoothing of its shape and to the location of Gmax VR close to the value ξ = 1 (Fig. 8). Also seen is a good agreement between analytical and numerical results in Figs. 7 and 8. It should be noted that there are some systematic quantitative discrepancies in the solutions for the values of parameter ξ less than the critical value corresponding to the switching threshold as well as some difference in the analytical shapes of GVR compared to numerical ones (Figs. 4, 7 and 8). Such discrepancies are conditioned by the approaches used for obtaining approximate solutions. Nevertheless, in spite of these discrepancies, approximate
1 30 GVR
Re σu RL = Re σ1
40
2
20
3 4
10
0
0.8
0.9
1 ξ
1.1
1.2
Fig. 8. Analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for different values of ΩL /2π = 0.025 (1), 0.05 (2), 0.075 (3), 0.1 (4). (AL = 0.1.)
solutions well describe all main regularities of vibrational resonance in overdamped bistable oscillator. Since ξ explicitly depends on AH and ΩH , the solutions (16) and (26) also contain the dependences of RL on ΩH . But in order to observe a resonancelike behavior depending on the frequency ΩH , the initial amplitude AH should be taken larger than µ0 . In fact, these solutions parametrically depend on the amplitude and the frequency of the additional HF modulation. From this point of view, VR can be considered as a parametric amplification near the onset of bistability controlled by the HF modulation.
30 2
GVR
5. Conclusions
4
20 15
1
3
25
5
10 5 0 0.6
0.8
ξ
1
1.2
Fig. 7. Analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for ΩL /2π = 0.075 and different values of the LF amplitude AL = 0.05 (1), 0.1 (2), 0.2(3), 0.33 (4), 0.5 (5).
To conclude, the analytical study of vibrational resonance showing a good agreement with the numerical results has been presented. These results are very general and can be applicable to bistable systems in different fields as well as to a broad class of nonautonomous systems displaying the perioddoubling bifurcation being subjected to the effect of near-resonant perturbations, since the normal form equation [Bryant & Wiesenfeld, 1986] coincides with the equation for overdamped bistable oscillator with a parametrically dependent potential.
Acknowledgment Author acknowledges the partial support from BRFFI (project F06-265).
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Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
References Baltanas, J. P., Lopez, L., Blekhman, I. I., Landa, P. S., Zaikin, A., Kurths, J. & Sanjuan, M. A. F. [2003] “Experimental evidence, numerics, and theory of vibrational resonance in bistable systems,” Phys. Rev. E. 67, 066119-7. Blekhman, I. I. & Landa, P. S. [2004] “Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation,” Int. J. Non-Lin. Mech. 39, 421–426. Bryant, P. & Wiesenfeld, K. [1986] “Suppression of period-doubling and nonlinear parametric effects in periodically perturbed systems,” Phys. Rev. E 33, 2525–2543. Casado-Pascual, J. & Baltanas, J. P. [2004] “Effects of additive noise on vibrational resonance in a bistable system,” Phys. Rev. E 69, 046108-7. Chizhevsky, V. N., Smeu, E. & Giacomelli, G. [2003] “Experimental evidence of vibrational resonance in an optical system,” Phys. Rev. Lett. 91, 220602-4. Chizhevsky, V. N. & Giacomelli, G. [2004] “Experimental and theoretical study of the noise-induced gain degradation in vibrational resonance,” Phys. Rev. E. 70, 062101-4.
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Chizhevsky, V. N. & Giacomelli, G. [2005] “Improvement of signal-to-noise ratio in a bistable optical system: Comparison between vibrational and stochastic resonance,” Phys. Rev. A 71, 011801(R)-4. Corbalan, R., Cortit, J., Pisarchik, A. N., Chizhevsky, V. N. & Vilaseca, R. [1995] “Investigation of a CO2 laser response to loss perturbation near perioddoubling,” Phys. Rev. A 51, 663–668. Gittermann, M. [2001] “Bistable oscillator driven by two periodic fields,” J. Phys. A: Math. Gen. 34, L355– L357. Landa P. S. & McClintock, P. V. E. [2000] “Vibrational resonance,” J. Phys. A: Math. Gen. 33, L433– L438. Pisarchik, A. N. & Goswami, B. K. [2000] “Annihilation of one of the coexisting attractors in a bistable system,” Phys. Rev. Lett. 84, 1423–1426. Ullner, E., Zaikin, A., Garcia-Ojalvo, J., Bascones, R. & Kurths, J. [2003] “Vibrational resonance and vibrational propagation in excitable systems,” Phys. Lett. A 312, 348–354. Zaikin, A. A., Lopez, L., Baltanas, J. P., Kurths, J. & Sanjuan, M. A. F. [2002] “Vibrational resonance in a noise-induced structure,” Phys. Rev. E 66, 011106-4.
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International Journal of Bifurcation and Chaos, Vol. 18, No. 6 (2008) 1767–1773 c World Scientific Publishing Company
ANALYTICAL STUDY OF VIBRATIONAL RESONANCE IN AN OVERDAMPED BISTABLE OSCILLATOR V. N. CHIZHEVSKY B. I. Stepanov Institute of Physics, NASB, 220072 Minsk, Belarus Received February 1, 2007; Revised June 16, 2007 The results of analytical study of vibrational resonance (VR) occurring in overdamped bistable system driven by two periodic signals with very different frequencies are presented. Approximate solutions for responses at the low-frequency as a function of the amplitude, and the frequency of the additional high frequency modulation which describe well the main features of vibrational resonance are obtained. Scaling laws for the gain factor and the switching threshold in VR are also found. Analytical results are compared with results of the numerical simulation, showing a good agreement. Keywords: Bistability; vibrational resonance; biperiodic excitation.
1. Introduction An overdamped bistable oscillator is one of the basic models in studying different phenomena in real physical systems. Recently, much attention has been focused on investigations of dynamics of such a system excited by two periodic signals with highly different frequencies [Landa & McClintock, 2000; Gittermann, 2001; Zaikin et al., 2002; Ullner et al., 2003; Baltanas et al., 2003; Chizhevsky et al., 2003; Blekhman & Landa, 2004; Casado-Pascual & Baltanas, 2004; Chizhevsky & Giacomelli, 2004, 2005]. In these conditions the response to the lowfrequency (LF) periodic modulation passes through a maximum depending on the amplitude of the high-frequency (HF) modulation [Landa & McClintock, 2000]. Such a phenomenon has been named vibrational resonance (VR). An evidence of VR has been demonstrated in analog electric circuits utilized to model noise-induced structures [Zaikin et al., 2002], excitable systems [Ullner et al., 2003], an overdamped bistable oscillator [Baltanas et al., 2003], and in a vertical cavity surface emitting laser within the regime of polarization bistability for both
symmetric and strongly asymmetric quasipotentials [Chizhevsky et al., 2003]. Besides, the possibility to make use of VR for detection of weak signals has been also demonstrated [Chizhevsky et al., 2003]. Recently, the effect of noise on VR was studied both theoretically [Casado-Pascual & Baltanas, 2004; Chizhevsky & Giacomelli, 2004, 2005] and experimentally [Chizhevsky & Giacomelli, 2004, 2005]. In fact, it was shown that vibrational resonance can be used for the control of stochastic resonance. A theoretical explanation of the phenomenon was given in [Baltanas et al., 2003] on the basis of the analytical treatment of the problem. It has been shown that VR takes place in the vicinity of the critical point corresponding to transition from bistability to monostability controlled by the HF modulation. Because of the difference of time scales associated with LF and HF modulations, a rapidly oscillating double-well potential can be transformed into an effective potential with a parametric dependence on the amplitude and frequency of the HF modulation. In such a formulation, one can note that VR has a clear analogy with a parametric
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amplification of near-resonant perturbations near the first period-doubling bifurcation point in nonautonomous systems [Bryant & Wiesenfeld, 1986]. In spite of a number of theoretical papers [Landa & McClintock, 2000; Gittermann, 2001; Zaikin et al., 2002; Ullner et al., 2003; Baltanas et al., 2003; Blekhman & Landa, 2004] devoted to VR, there is no adequate analytical description of VR in overdamped bistable oscillator until now. Here we present the approximate analytical solutions for VR occurring in overdamped oscillator with symmetrical double-well potential that well describe the main features of VR. These solutions are in a good agreement with the numerical results. Besides, scaling laws relating the switching threshold and the maximum gain factor in VR with the amplitude of the LF signal have been also found.
2. Theoretical Model and Basic Relationships We consider the phenomenon of VR using the model of overdamped bistable oscillator driven by two forces. One of them is a slowly time-varying input signal f (t) with a characteristic time τL and the amplitude AL , where f (t) can be periodic or aperiodic functions. The second signal is a high frequency periodic modulation with an amplitude AH and a frequency ΩH so that ΩH τL 1. The dynamics of the system in this case can be described by the following equation ∂x = −V (x) + AL f (t) + AH sin ΩH t ∂t
(1)
where V (x) is the derivative with respect to x of a bistable potential function αx2 βx4 + (2) 2 4 with the local minima x± 0 = ± α/β, the barrier height ∆V0 = α2 /4β, the statical threshold of crossing the potential barrier µ0 = 4α3 /27β, and the relaxation time τr0 = 1/2α, where α and β are positive numbers. The parameter µ0 corresponds to a transition from the symmetric double-well potential to an asymmetrical one-well configuration. Taking into account the expression for the potential, the equation takes the form V (x) = −
dx = αx − βx3 + AL f (t) + AH sin ΩH t. dt
(3)
Generally, we have in the problem two timescales: the slow one, which is associated with the LF signal, and the fast one relating to the high frequency modulation. For the sake of clarity, we recall some of the results presented in [Baltanas et al., 2003]. We look for the solution as follows: x(t) = y(t) + η(AH , ΩH ) sin ΩH t,
(4)
where y(t) denotes a slow part of the solution, η(AH , ΩH ) is the coefficient. Substituting (4) into (3) and averaging over the period TH = 2π/ΩH we obtain the following equation governing the slow dynamics of the bistable system [Chizhevsky & Giacomelli, 2004] dY = α(1 − ξ 2 )Y − βY 3 + AL F (t), (5) dt where Y = y(t)TH , F (t) = f (t)TH . The ξ is a normalized amplitude of the HF modulation ξ = AH /Ac and Ac is the switching threshold Ac = (2α/3β)(Ω2H + (2α2 /9)) [Chizhevsky & Giacomelli, 2004]. Correspondingly, the effective potential Vξ (y) takes the form
α(1 − ξ 2 )y 2 βy 4 + . (6) 2 4 Obviously, after averaging operation, all parameters ±, of the potential such as the location of minima ym the barrier height ∆V , the crossing threshold µ and the relaxation time τr depend now on the parameter ξ in the following way: α(1 − ξ 2 ) ± , (7) ym = ± β Vξ (y) = −
α2 (1 − ξ 2 )2 , 4β 4α3 (1 − ξ 2 )3 , µ= 27β
∆V =
τr =
1 . 2α(1 − ξ 2 )
(8)
(9) (10)
For the approximate solution in the response at low frequency, some conclusions can be made from expressions (7)–(10). Figure 1 shows the view of potential Vξ (y) for different values of the normalized amplitude of the HF modulation ξ. The corresponding locations of the minima of Vξ (y) are depicted in Fig. 2. As ξ increases one can see the transition from the doublewell configuration of Vξ (y) to the one-well potential. This transition occurs for the critical value ξ = 1.
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Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
3. Adiabatic Approximation
2.5
First we consider the effect of the low-frequency periodic modulation F (t) = sin(ΩL t) when ΩL 2α. In this case we can neglect the time derivative in (5) and obtain the following equation: 4α3 sin(ΩL t) = 0. (11) α(1 − ξ 2 )Y − βY 3 + ε 27β
2 1.5 1 V(y)
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0.5 3
0
4
2
−0.5 1
−1 −1.5 −1.5
−1
−0.5
0 y
0.5
1
1.5
Fig. 1. View of the potential Vξ (y) for different values of ξ = 0 (1), 0.7 (2), 1 (3), 1.5 (4).
We here use the normalized LF amplitude ε = AL / 4α3 /27β. In this case, locations of extremums are determined by the following solutions: 1 1 − ξ2 cos arccos γ , (12) y1 = 2y0 3 3 y2 = −2y0
1 π 1 − ξ2 cos arccos γ − , 3 3 3
(13)
+ y0
y
0.5
yu 0
0 −0.5
−
y0
−1 0 Fig. 2.
0.5
ξ
1
1.5
The location of extrema y0+ , y0− and y0u versus ξ.
This means that driving the bistable system by the HF modulation, we place the system near the bifurcation point corresponding to the transition from bistability to monostability. At such conditions one can expect a high sensitivity of bistable systems to external perturbations, since nonlinear systems operating near simple critical points are known to be very sensitive to the effect of periodic perturbations and noise. It should be noted also that the relaxation time τr tends to infinity as ξ approaches the bifurcation point as it follows from (10). This can be considered as a manifestation of the general phenomenon of critical slowing down which is well known in the theory of bifurcations. On the other hand, one can also note that the periodic modulation with the amplitude larger than Ac leads to the elimination of bistability in the same sense as was found for the periodically modulated nonlinear systems possessing generalized bistability [Pisarchik & Goswami, 2000].
1 π 1 − ξ2 cos arccos γ + , (14) yu = 2y0 3 3 3 where y0 = a/b and γ = ε(1 − ξ 2 )−3/2 . Figure 3 shows a set of all real solutions corresponding to locations of two stable (y1 and y2 ) and one unstable (yu ) fixed points as a function of ξ. From the condition ε(1 − ξ 2 )−3/2 = 1 it follows that the critical amplitude ξc needed for the switching between two states depends on the normalized amplitude ε of the LF signal as follows: (15) ξc = 1 − ε2/3 . Analysis of the solutions (12)–(14) shows that the response of bistable system to the LF signal for ξ < ξc can be associated with the solution yu . As 1
y1
0.5
y
1
yu
0 y2 −0.5 −1 0
0.5
ξ
1
1.5
Fig. 3. Set of all real solutions obtained from (11) showing locations of extrema as a function of ξ (AL = 0.25).
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V. N. Chizhevsky
Now one can evaluate the gain factor GVR . We define the gain factor GVR as a ratio of responses of the bistable system to the LF periodic signal in the presence [RL (ΩL )] and the absence [R0 (ΩL )] of the HF modulation: RL (ΩL ) . (17) GVR = R0 (ΩL ) In the linear approximation AL , R0 = Ω2L + 4α2
(18)
and in the case of ΩL 2α the response R0 can be reduced to AL . (19) R0 ∼ = 2α In this case, one can find from (12) that 1/2 1/3 4α , (20) RL = ε 3β and the maximum gain Gmax VR is determined by the expression −2/3 . Gmax VR = 6ε
1
50 2
40 30
3 4
20 6
10 0
0.6
5
0.8
1.2
Fig. 4. Analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for different values of the LF amplitude AL = 0.05 (1), 0.1 (2), 0.2 (3), 0.33 (4), 0.5 (5), 0.75 (6).
α = β = 4. Depending on the context in the paper, along with the quantities AL and AH we use the dimensionless amplitudes ε and ξ defined previously as ε = AL /µL and ξ = AH /µH where µL and µH are the switching thresholds at the frequencies ΩL and ΩH , respectively, in the absence of any additional modulation. For a quantitative characterization of VR in the simulation we used the gain factor GVR (17), which was evaluated from the Fourier spectra of temporal responses of the system. Figure 4 shows the analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for different values of the amplitude AL of the LF signal for the case of low frequency (ΩL /2π = 10−3 in the numerical simulation) with respect to Ωr = 2α(ΩL ≪ Ωr ) using the expression 1.9
1
(21)
It should be noted such a scaling law has been experimentally observed in a nonautonomous system (a loss-modulated CO2 laser) for the gain factor in the amplification of near-resonant perturbations near the first period-doubling bifurcation [Corbalan et al., 1995]. In parallel, in order to check the analytical results we numerically integrated Eq. (3) with
ξ
1
log10Gmax VR
For ξ ≥ 1 the real solution y1 is determined by the expression 1 1 − ξ2 sinh arcsinh γ . y1 = 2y0 3 3
60
GVR
ξ increases from zero to ξc the response follows the solution yu until ξ reaches its critical value ξc where purely real solutions yu and y2 become the imaginary ones. This condition corresponds to the disappearance of bistability. At the point ξ = ξc the system response jumps across into solution y1 which results in the appearance of a discontinuity in the response. For ξ > ξc the response of the system follows the solution y1 which is purely real. Based on these speculations one can build the response of a bistable system to the LF modulation as a function of ξ by the following rule: yu for ξ < ξc (16) RL = y1 for ξ ≥ ξc
ξc
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15:58
0
0
0.5 ε (a)
1
0.9 −1.5
−1 −0.5 log10ε (b)
Fig. 5. (a) Switching threshold ξc and (b) the maximum gain Gmax VR as a function of ε. Solid lines — analytics, open circles — numerics.
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Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
(16) for RL . One can note that analytical results well agree with the numerical ones and well describe the main regularities of VR — the strong dependence of the response on the LF amplitude AL , the shift of the optimal value of ξ and the broadening of curves as AL increases. In Fig. 5 we compare our analytical results for the switching threshold (15) and the maximum gain Gmax VR (21) with the numerical results as a function of ε. One can note a good agreement between them.
4. General Case We now consider the general case for F (t) = sin(ΩL t) when one cannot neglect the time derivative in (5). In this case, we look for the approximate solution in the form y(t) = σ sin(ΩL t + ϕ). Substituting it into Eq. (5) and neglecting the trig function of argument 3ΩL , we obtain two equations for a determination of σ and ϕ as a function of AL and ξ. 3βσ 2 cos ϕ = 0 −ω sin ϕ + 4 − α(1 − ξ 2 ) 3βσ 3 sin ϕ − AL = 0 σω cos ϕ + 4 − ασ(1 − ξ 2 ) Resolving the system with respect to σ we get the following equation: 6
2
δ + pδ + q = 0 where δ=
(22)
σ2 = −
κ2 u + κ1 v + 8α(1 − ξ 2 ) 9β
− 8α(1 9β
− ξ2)
p=
16 3Ω2L − α2 (1 − ξ 2 )2 27 β2
q=
16 8α3 (1 − ξ 2 )3 + 72α(1 − ξ 2 )Ω2L − 81βA2L 729 β3
Equation (22) can be resolved with respect to δ giving the location of stable and unstable fixed points. The complete set of solutions σ of one sign can be represented as follows: u + v + 8α(1 − ξ 2 ) (23) σ1 = 9β κ1 u + κ2 v + 8α(1 − ξ 2 ) (24) σu = 9β
(25)
where
√ √ q q + D, v = − − D, u= − 2 2 √
p 3 q 2 (−1 ± i 3) D= . + , κ1,2 = 3 2 2 Figure 6 shows an evolution of solutions σ1 , σ2 and σu versus ξ as ΩL increases (from (1) to (4) in the figure). In fact, these solutions allow us to study the influence of the frequency of the LF signal on the response of a bistable system. Similar to the previous case (11) the response to the LF signal for ξ < ξc can be associated with the solution σu . But in this case ξc depends on the frequency ΩL as well. This critical value ξc can be found from the condition that the discriminant D = 0. As ξ increases, the response follows the solution σu until ξ reaches its critical value ξc . At the point ξ = ξc the response of the system jumps across to the solution σ1 leading to the appearance of a discontinuity in the response for ΩL Ωr . As ΩL approaches Ωr , σ1
1 0.5
0.5
σu
0
σ1
1
σu
0
−0.5
σ2
1771
−0.5 σ2
−1 0
0.5
ξ
σ2
−1 1
1.5
0
(a)
0.5
1
1.5
1
1.5
σ1
1 0.5
σu
0
ξ
(b)
σ1
1
0.5
σu
0
−0.5
−0.5 σ2
−1 0
0.5 (c)
ξ
σ2
−1 1
1.5
0
0.5
ξ
(d)
Fig. 6. Solutions σ1 , σ2 and σu as a function of ξ for different values of ΩL (ΩL /2π = 0.01 (a), 0.0825 (b), 0.1 (c), 0.15 (d), AL = 0.25).
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V. N. Chizhevsky
the discontinuity is replaced by a smooth transition in the response (Fig. 6). For ξ > ξc the response of the system follows the solution σ1 which is purely real. Thus, in this case the response is determined by same rule as (16): for ξ < ξc for ξ ≥ ξc
(26)
In Figs. 7 and 8 we compare analytical and numerical results for the gain factor GVR in the general case using RL in the expression (26). Figure 7 shows GVR for ΩL /2π = 0.075. In this case, the smooth shape of GVR is noted, also the weak dependence of GVR on AL as well as the small shift of Gmax VR on AL for the weak LF amplitudes in contrast with the previous case shown in Fig. 4. For the given amplitude of the LF signal, an increase of ΩL leads to a significant lowering of GVR , smoothing of its shape and to the location of Gmax VR close to the value ξ = 1 (Fig. 8). Also seen is a good agreement between analytical and numerical results in Figs. 7 and 8. It should be noted that there are some systematic quantitative discrepancies in the solutions for the values of parameter ξ less than the critical value corresponding to the switching threshold as well as some difference in the analytical shapes of GVR compared to numerical ones (Figs. 4, 7 and 8). Such discrepancies are conditioned by the approaches used for obtaining approximate solutions. Nevertheless, in spite of these discrepancies, approximate
1 30 GVR
Re σu RL = Re σ1
40
2
20
3 4
10
0
0.8
0.9
1 ξ
1.1
1.2
Fig. 8. Analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for different values of ΩL /2π = 0.025 (1), 0.05 (2), 0.075 (3), 0.1 (4). (AL = 0.1.)
solutions well describe all main regularities of vibrational resonance in overdamped bistable oscillator. Since ξ explicitly depends on AH and ΩH , the solutions (16) and (26) also contain the dependences of RL on ΩH . But in order to observe a resonancelike behavior depending on the frequency ΩH , the initial amplitude AH should be taken larger than µ0 . In fact, these solutions parametrically depend on the amplitude and the frequency of the additional HF modulation. From this point of view, VR can be considered as a parametric amplification near the onset of bistability controlled by the HF modulation.
30 2
GVR
5. Conclusions
4
20 15
1
3
25
5
10 5 0 0.6
0.8
ξ
1
1.2
Fig. 7. Analytical (heavy lines) and numerical (thin lines) gain factor GVR as a function of ξ for ΩL /2π = 0.075 and different values of the LF amplitude AL = 0.05 (1), 0.1 (2), 0.2(3), 0.33 (4), 0.5 (5).
To conclude, the analytical study of vibrational resonance showing a good agreement with the numerical results has been presented. These results are very general and can be applicable to bistable systems in different fields as well as to a broad class of nonautonomous systems displaying the perioddoubling bifurcation being subjected to the effect of near-resonant perturbations, since the normal form equation [Bryant & Wiesenfeld, 1986] coincides with the equation for overdamped bistable oscillator with a parametrically dependent potential.
Acknowledgment Author acknowledges the partial support from BRFFI (project F06-265).
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Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
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Chizhevsky, V. N. & Giacomelli, G. [2005] “Improvement of signal-to-noise ratio in a bistable optical system: Comparison between vibrational and stochastic resonance,” Phys. Rev. A 71, 011801(R)-4. Corbalan, R., Cortit, J., Pisarchik, A. N., Chizhevsky, V. N. & Vilaseca, R. [1995] “Investigation of a CO2 laser response to loss perturbation near perioddoubling,” Phys. Rev. A 51, 663–668. Gittermann, M. [2001] “Bistable oscillator driven by two periodic fields,” J. Phys. A: Math. Gen. 34, L355– L357. Landa P. S. & McClintock, P. V. E. [2000] “Vibrational resonance,” J. Phys. A: Math. Gen. 33, L433– L438. Pisarchik, A. N. & Goswami, B. K. [2000] “Annihilation of one of the coexisting attractors in a bistable system,” Phys. Rev. Lett. 84, 1423–1426. Ullner, E., Zaikin, A., Garcia-Ojalvo, J., Bascones, R. & Kurths, J. [2003] “Vibrational resonance and vibrational propagation in excitable systems,” Phys. Lett. A 312, 348–354. Zaikin, A. A., Lopez, L., Baltanas, J. P., Kurths, J. & Sanjuan, M. A. F. [2002] “Vibrational resonance in a noise-induced structure,” Phys. Rev. E 66, 011106-4.
