• No results found

Vibrational resonance in a harmonically trapped potential system with time delay

N/A
N/A
Protected

Academic year: 2022

Share "Vibrational resonance in a harmonically trapped potential system with time delay"

Copied!
12
0
0

Loading.... (view fulltext now)

Full text

(1)

Vibrational resonance in a harmonically trapped potential system with time delay

ZHENGLEI YANG and LIJUAN NING

School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China

Corresponding author. E-mail: ninglijuan@snnu.edu.cn

MS received 11 July 2018; revised 2 November 2018; accepted 13 November 2018;

published online 30 March 2019

Abstract. This paper is focussed on investigating the effect of linear time delay on vibrational resonance of a harmonically trapped potential system driven by a biharmonic external force with two wildly different frequencies ωandwithω . Firstly, the approximate analytical expression of the response amplitude Q at the low- frequencyωis obtained by means of the direct separation of the slow and fast motions, and then we verified the numerical simulation by using the fourth-order Runge–Kutta method and found that it is in good agreement with the theoretical analysis. Next, the influence of the time-delay parameters on the vibrational resonance are discussed.

There are some meaningful conclusions. Ifτis a controllable parameter, the response amplitudeQnot only exhibits periodicity but also can be amplified via the cooperation ofF andτ. If the time-delay intensity parameterr is a controllable parameter, the response amplitudeQis found to be much larger than that in the absence of time delay.

Moreover, adjustingrcan result in a better response than adjustingτ. This undoubtedly gives us a superior way to amplify the weak low-frequency signal.

Keywords. Vibrational resonance; time delay; signal amplification; harmonically trapped potential.

PACS Nos 05.45.−a; 02.30.Ks; 05.90.+m; 46.40.Ff 1. Introduction

Over the past few years, the analysis of the dynamic characteristics induced by two different frequency signals has attracted lots of attention in commuta- tion technology [1], acoustics [2], neuroscience [3], laser physics [4], etc. This feature mainly reflects the phenomenon that the response amplitude of the system at the low-frequency ω can be amplified excellently by adjusting the relatively high-frequency input signal. This meaningful phenomenon is called vibrational resonance (VR), which is a phenomenon originally discovered by Landa and McClintock [5] in a bistable system. Owing to their special nature, the biharmonic signals are pervasively found in many fields of science and engineering, and they have been developed from theoretical, numerical, and experi- mental studies [5–7] in many systems, such as the delayed system [8–13], fractional-order system [14,15], complex network [16–18], signal processing research [5], convective system [19] and so on. In addition, VR is also used in the field of ecology [20].

Time delay, a factor that cannot be ignored, is ubiquitous in many systems. It can more realistically reflect the characteristics of many systems in nature.

In many cases, time delay can reflect the transmission of information and energy, and conversion of mate- rial related to the transport of the system. It can also reflect the finite conversion speed of the amplitude, the limited reaction time, the memory effect, etc. More than this, time delay can also lead to many interest- ing phenomena in nonlinear dynamical systems. The characteristic of VR with time delay can be reflected in nature more authentically. Therefore, studying the effect of time delay on the output signal has great significance, especially for VR, which can increase the output of the low-frequency signal. So far, the research on the VR in time-delay system have experienced great develop- ment. Yang and Liu [8,9] found that in some nonlinear dynamic systems, with the introduction of time delay, the response amplitude of the low-frequency signal can present periodical or quasi-periodical phenomenon with the change of the time-delay parameter. With the aim of achieving effective information, which is usually car- ried by low-frequency signals, with the help of the

(2)

time-delay parameter, not only the VR of the system can be effectively controlled, but also the response of the system at the low-frequency signal can be further improved when the VR phenomenon occurs [21,22].

This provides an effective approach to control the signal and amplify the weak low-frequency signal in the field of science and signal processing. The VR in a harmon- ically trapped potential system driven by a biharmonic external force has been studied by Abiramiet al[23]. It is a novel multistable potential system. However, as far as we know, much attention was not paid to the effect of time delay on the dynamic behaviour of the system.

Hence, some new problems have arisen. Does the time- delay parameter enlarge the weak low-frequency signal in the harmonically trapped potential system? Can the system generate some of the above special phenomena by adjusting the time-delay parameters? Motivated by the above consideration, we shall investigate the effect of time delay on this system in more details. We hope our conclusions will play a potential role in the field of signal processing.

The structure of the paper is organised as follows:

In §2, the motion equation of the harmonically trapped potential system with time delay is introduced. Mean- while, we analyse the potential of the system and give the image of the numerically computedxas a function ofβ. In §3, we derive an approximate analytical expres- sion of the response amplitude at the low-frequency ω via the direct separation of slow and fast motions. Next, we use numerical simulation to prove the theoretical pre- diction. In §4and5, we show the influence of discrete time delay and continuous time delay on the response amplitude of the system with monostable (β =3) and tristable (β = 6) potentials, respectively. We present the effect of time-delay intensity parameter r on the response amplitude of the system in §6. Finally, conclu- sions are summarised in §7.

2. Model

When ω , the method of direct separation of slow and fast motions is adopted to obtain approximate response amplitude of the system at the low-frequency ω. In this section, we shall use this technique to theoreti- cally predict the response amplitude in the harmonically trapped potential system with time delay. In this paper, the motion of a particle in a harmonically trapped poten- tial system with time delay can be described by the following equation:

dx2(t)

dt2 +ddx(t)

dt +ω20x(t)+βsinx(t)+r x(tτ)

= f cos(ωt)+Fcos(t) (1)

and the biharmonic frequency signals are represented by f cos(ωt) and Fcos(t), where fcos(ωt) and Fcos(t)are respectively the low-frequency input sig- nal and high-frequency signal when frequencyω. d is the coefficient of linear damping which is greater than zero. τ is the time delay andr is the strength of the time delay term. The harmonically trapped potential satisfies the following form in the absence of time delay:

V(x)= 1

2(r +ω20)x2βcosx, (ω20, β >0). (2) The odd number of potential wells of different depths can be generated by adjusting the parameters ω20 and β in eq. (2). A system with an odd number of poten- tial wells can be well described by it. Figure1d shows the plot of the numerically computed x (equilibrium points) vs. the parameterβof system (1) in the absence of biharmonic force forω20 =1 andr =0.1 and we can see that the shape of the potential changes withβ.The stable equilibrium points are represented by continuous lines whereas the unstable points are shown by dashed curves. It can be seen clearly from figures1a–1c that the number of potential wells changes from one to three, to five with the increase of the parameterβ from 3 to 6, then to 15. More odd potential wells can be generated by further increasingβ(figure1d).

Here, in order to better describe the role of the time delay parametersτ andr on VR, we choose the poten- tial with the single (β =3) and triple (β =6) potential wells, respectively. Applying a theoretical treatment, we obtain an approximate analytical expression of the response amplitudeQat the low-frequencyω.The the- oretical prediction is found to be in good agreement with the numerically computedQ.

3. Analytical expression for the response amplitude When the condition ω is satisfied, the method of direct separation of slow and fast motions can be successfully used to get the theoretical predictions of the response amplitude to the system at the low- frequencyω.

3.1 Approximate theoretical expression for the response amplitudeQ

Whenω, we can reasonably assume that the solu- tion of eq. (1) can be decomposed into slow and fast motions. Therefore, the solution of eq. (1) can be given in the following form:

x(t)= X(t)+(t), (3)

(3)

x

-10 -5 0 5 10 15 20 25 30

V(x)

=3

(a)

x

-10 -5 0 5 10 15 20 25 30

V(x)

=6

(b)

x

-20 0 20 40 60 80 100

V(x)

=15

(c)

-10 -5 0 5 10 -10 -5 0 5 10

-20 -10 0 10 20 0 5 10 15 20 25

-20 -15 -10 -5 0 5 10 15 20

x*

(d)

Figure 1. (a)–(c) Shapes of the potential equation (2) with three different values ofβforω02=1,r =0.1 and (d) shape of the equilibrium pointsxas a function ofβ.

where X(t)shows the slow motion with period 2π/ω and(t)stands for the fast variable with period 2π/: X¨ + ¨+dX˙ +d˙ +ω02X+ω20+βsin(X+)

+r(X(t−τ)+(t−τ))=f cos(ωt)+Fcos(t).

(4) By averaging over the fast time in one of its period, we can obtain the following equations for slow and fast motions, respectively:

X¨ +dX˙ +ω20X +βcossinX +βsincosX +r X(tτ)= f cos(ωt), (5) ¨ +d˙ +ω02+β(cos− cos)sinX

+β(sin−sin)cosX+r(t−τ)=Fcos(t), (6)

where cos =

2π

/2π

0

cos(t)dt, (7)

sin = 2π

/2π 0

sin(t)dt. (8)

According to the assumption that is a rapidly changing periodic oscillation, the terms containing in the evolution equation ofX can be averaged out over its period. Thus, eq. (6) can be approximated as

¨ +d˙ = Fcos(t). (9) can be achieved as follows:

=θ cos(t+φ), (10)

where

φ =tan1(d/), θ = −F/(

2+d2). (11)

(4)

By calculation, we find thatcos=J0(θ)andsin

= 0, where J0 is the zeroth-order Bessel function [24,25]. Throughout our analysis on underdamped sys- tem, d is satisfied. Ignoring d, J0(θ) can be approximated as J0(−F/2). Then eq. (5) becomes

X¨+dX˙+ω02XJ0sinX+r X(tτ)=f cos(ωt).

(12) The corresponding effective potential of eq. (12) without time delay is given as

Veff(X)= 1

2(r +ω02)X2βJ0cosX. (13) Slow motion can occur around the stable equilibrium points of the system in the absence of period force and time delay. The equilibrium point Xsatisfies the fol- lowing equation:

(r +ω20)X+βJ0sinX =0. (14) Due to the slow motion oscillating aroundX, the devi- ationY = XXis introduced and satisfies

Y¨+dY˙ +ωr2Y +r Y(tτ)= fcos(ωt), (15) where

ω2r =ω20+βJ0cosX. (16) It is obvious thatωris the resonant frequency of oscil- lation of the slow variable. For a long time, the solution of eq. (15) is given asY = ALcos(ωt+σ)in which

AL = f

2r−ω2+rcos(ωτ))2+(rsin(ωτ)−dω)2 (17) and

σ =tan1

rsin(ωτ)

ω2rω2+rcos(ωτ)

. (18)

We can define the response amplitude of the system as Q = AL

f = 1

S, (19)

with

S=((ω2r−ω2+rcos(ωτ))2+(rsin(ωτ)−dω)2. (20) Apparently, the resonance behaviour can be analysed.

Here, Q is a quantitative indicator that signifies the extent of amplification of the weak input signal through the nonlinear system. VR occurs when Q reaches the local maximum, namely,Sarrives at its local minimum.

According to eqs (19) and (20), the resonance can appear fixing other parameters when

ω2r =ω2rcos(ωτ). (21)

3.2 Numerical calculation of the response amplitudeQ

The fourth-order Runge–Kutta method is used to calcu- lateQin eq. (1) with step size(2π/ω)/1000. Neglecting a sufficient transient, one can obtainQ, which is given by

Q=

Bs2+Bc2

f , (22)

where Bs = 2

mT

mT

0

x(t)sin(ωt)dt, (22a) Bc= 2

mT

mT

0

x(t)cos(ωt)dt (22b) withT =2π/ωandmis a positive integer.

4. Analysis of the response amplitude Qwith time delay for the monostable (β=3) potential

4.1 Analysis of the response amplitude Qwith fixed delayβ =3

First, figures2c and2d show that the theoretical pre- dictions of the response amplitude are in very close agreement with the numerical simulations with different time-delay values ofτ ford =0.5, f =0.1,r =0.1, ω20 =1, ω=1, =10.It is obvious that the response amplitudeQis maximum when|ω2rω2+rcos(ωτ)|

is minimum or J0 is maximum (i.e. J0 = 0). It is an interesting phenomenon that the effect of F on Q and

r2ω2 +rcos(ωτ)| is quite different. Q does not decay to 0 whenF → ∞and the height of the resonant is hardly changed. Q tends to a non-zero value with the increase of F. However, the oscillation amplitude of|ω2rω2+rcos(ωτ)|gets smaller and smaller, and its amplitude eventually becomes zero when F → ∞. Below, we shall explain the variation of Q with F by means of the quantities|ω2rω2+rcos(ωτ)|and J0. In figures 2c and 2d, not only can we see some spe- cial values of F at which J0 = 0, but also we can see that J0 → 0 when F → ∞. For the finite value of F, we find thatωr2 = ω2rcos(ωτ) when J0 = 0.

Under the circumstance of eqs (19) and (20), we have Q=1/(dωrsin(ωτ)),i.e. the maximum value ofQ is 1/(dω−rsin(ωτ))and it appears when the resonance frequencyω2r matches withω02rcos(ωτ)(here,ωis the low frequency). J0 can pass many times before it reaches its limit. Several resonance peaks ofQ can be found in the figure because of this reason.

(5)

F 0

1 2 3 4

Q, | r2 -2 |

(a)

F -1

0 1 2 3

Q, J 0

=3 Q

=3

Q

(b)

J0

| r2- 2|

- 02/

F 0

1 2 3 4

Q, | r2 -2 +rcos() |

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800 F

-1 0 1 2 3

Q, J 0

Q

| r2- 2+rcos( ) |

Q

=3

=3

J0

-(r+ 02)/

(c)

(d)

Figure 2. Response amplitudeQvs. the control parameterFforβ=3,d=0.5, f =0.1, ω20=1, ω=1, =10.(a) and (b) show the absence of time-delay term and (c) and (d) show the time-delay parametersr =0.1, τ =1.5. The continuous curves represent the variation of Qwithr2ω2|in (a),Qwith2r ω2+rcos(ωτ)|in (c),Qwith J0in (b) and (d), respectively. The dashed lines represent the numerically calculated values ofQ.

1800 1200

600 0

F -1.5

-1 -0.5 0 0.5 1 1.5

X*

c1 b1 a1

=3

Figure 3. Plot ofXas a function ofFforβ =3, ω20=1,τ =1.5,d =0.5 andr =0.1 of the unforced case of system (12).

Here, what we are more interested in is the second resonance in figures2c and2d. It is easy to find from the graph that the second resonance peak is lower than the

other ones. Consider the situation in which resonance occurs in the interval F ∈ [a1,b1] (figure 3). In this intervalF, the quantityJ0is less than−(r+ω20)/β, and

(6)

1620 8 12 0

1800 4

1

F

1500 1200

Q

900 2

600 300 0 0

3

0.5 1 1.5 2 2.5

Figure 4. Three-dimensional plot ofQvs. the parametersFandτ forr =0.1.

there exist three equilibrium points: X0 =0, X± = 0.

X± = 0 are stable while X0 = 0 is unstable. This suggests that slow oscillation happens around X± =0.

Furthermore, it is easy to find X = 0 from figure 3 and J0 = −(r +ω20)/β from figure2d when F = a1

andb1. Meanwhile,|ω2rω2+rcos(ωτ)|has a non- zero local maximum, but Q has a local minimum as shown in figure2c. When F = c1 (a1 < c1 < b1) in figure3, the quantity|ωr2ω2+rcos(ωτ)|has a non- zero local minimum butQhas a local maximum, which can be seen from figure2c. Based on the above analysis, we can draw a conclusion that whenJ0 <−(r+ω20)/β and|ωr2ω2 +rcos(ωτ)|has a local minimum with a value which is not equal to zero, then there will be a resonance with a value less than 1/(dωrsin(ωτ)).

4.2 The response amplitudeQvs. the delay parameterτ

Figure4depicts a three-dimensional evolution diagram of the resonance amplitudeQcorresponding to the delay parameterτand the high-frequency signal amplitudeF. In this figure, one can clearly see that when F or the time-delay parameters increase, the resonance peaks appear. An important observation from the curved sur- face ofQis that it has a periodicity dependence onτwith period 2π/ω. This just verifies the correctness of the the- oretical value. Figure5a shows the maximum value of the response amplitudeQwith a certain range of values ofτ. It is incredible that the maximal value Qmaxis a periodic function of time delay. This is mainly because, in a long time, with the variation of the delay,Qpresents a periodic 2π/ω(see figures4and6). In figure5b, Fm

is the critical amplitude of the high-frequency input sig- nal that makes the response amplitude Q achieve the maximum, and it is also a periodic function.

Figure 6 demonstrates the variation of response amplitudeQwith time delay under different fixed values of F. With the increment ofτ, the response amplitude

1.6 1.8 2 2.2 2.4 2.6

Q max

(a)

0 5 10 15 20

0 5 10 15 20

200 400 600 800 1000 1200 1400 1600 1800 2000

F m

(b)

Figure 5. (a)Qmaxand (b)Fmvs. the delay parameterτ.

of the system at the low-frequency signal shows peri- odicity. Moreover, by comparing figures6and2a, it is found that the response amplitudeQof the system with time delay is stronger than that without time delay even if the value ofFis the same. This means that by adjust- ing the time-delay parameterτ, the response amplitude of the system can be amplified. When F changes in

(7)

1 1.2 1.4 1.6

Q

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

1.5 2 2.5

Q

2 /

2 /

F=200

F=550

(b) (a)

Figure 6. The change of delay parameterτ causes periodic change of the response amplitude gain of the system at the low-frequency signal forβ =3 and=10.

1800 1200

600 0

F

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

X*

a2 b2 c2 d2 m2 n2

=6

Figure 7. Plot ofXas a function ofF for the tristable potential (β =6), ω02=1, τ =1.5,d =0.5 andr =0.1 of the unforced case of system (12).

the same interval, the effect of the time-delay parameter on the amplification of the amplitude is also shown in figure2. Accordingly, as discussed before, a conclusion can be drawn that the system can achieve stronger res- onances not only by adjusting the time-delay parameter τ, but also by the interaction ofτandF. In other words, employing these methods, the weak low-frequency sig- nal can be amplified.

5. Analysis of the response amplitude Qwith time delay for the tristable (β=6) potential

In this section, we shall analyse the occurrence of vibra- tion resonance for the case of the potential with three wells whenβ =6 (figure1b).

5.1 Analysis of the response amplitude Qwith fixed delay forβ =6

As stated in figures8c and8d, the theoretical analysis of the response amplitude with time delay (the continu- ous lines) is in very good agreement with the numerical simulation (the dashed line). For the tristable potential (β =6), the potentialV(x)has a three-well shape which can be seen in figure1. One can observe thatQpresents the number of resonance peaks vs.Fand approaches the limiting value 1/(dωrsin(ωτ)). For a better under- standing, we focus on the sixth and the ninth resonance peaks which have much smaller resonance peaks than the other resonance peaks. This phenomenon takes place at the local minimum ofJ0withJ0<−(r+ω20)/βand

r2ω2+rcos(ωτ)|becomes a local minimum with a value not equal to zero.

(8)

F 0

2 4

Q, | r2-2 |

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

F -1

0 1 2 3

Q, J0

| r2- 2 |

=6

=6

Q

J0 Q

- 02/

(b) (a)

F 0

2 4

Q, | r2-2+rcos() |

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

F -1

0 1 2 3

Q, J 0

| r2- 2+rcos( ) |

Q

=6 Q

=6

J0

-(r+ 02)/

(d) (c)

Figure 8. Response amplitudeQvs. the control parameterFfor the tristable potential (β=6),d =0.5, f =0.1, ω20=1, ω=1, =10.(a) and (b) show the absence of time-delay term and (c) and (d) show the time-delay parametersr =0.1, τ =1.5. The continuous curves represent the variation ofQwith2r ω2|in (a),Qwith2r ω2+rcos(ωτ)|in (c),Q withJ0in (b) and (d), respectively. The dashed lines represent the numerically calculated values ofQ.

20 16 12 0 8

1800 1500 4

F

1200 900 1

600 300 0 0

Q

2 3

0.5 1 1.5 2 2.5

Figure 9. Variation of the response amplitudeQwithFandτ whenr =0.1.

Additionally, by comparing figures7and8c, we also find that Q displays double resonances when F ∈ [a2,b2], and with the peak valueQ=1/(dω−rsin(ωτ)). Unlike the interval F ∈ [a2,b2], for the intervals

F ∈ [c2,d2] and F ∈ [m2,n2], the response ampli- tude Q shows a single resonance (the sixth or the ninth resonance) and the peak value is much less than 1/(dω−rsin(ωτ)).

(9)

1.6 1.8 2 2.2 2.4 2.6

Qmax

(a)

0 5 10 15 20

0 5 10 15 20

200 400 600 800 1000 1200 1400 1600 1800 2000

Fm

(b)

Figure 10. (a)Qmaxand (b)Fmvs. the delay parameterτ.

5.2 Analysis of the response amplitude Qvs. the delayτ

Figure 9 shows the three-dimensional diagram of the variation of response amplitude Q and the high- frequency signal F with delay parameter τ. Similar to the monostable potential (β = 3) (figure 4), the resonances occur in turn with the increment of F or τ. At the same time, it is also observed that with the variation of time-delay parameterτ,Qis still a periodic function with 2π/ω, no matter how F changes. This result is consistent with the counterpart of the theoretical verification. Figure10a shows the curve of the maximum value of Q with respect to a certain range ofτ. Qmax is also a periodic function ofτ with period 2π/ω (see figures9 and11). In figure 10b, withτ changing, Fm

still exhibits periodicity.

For different fixed values ofF, the changing image of the response amplitude Qwithτ is given in figure11.

Obviously, the response amplitudeQis periodical when τ changes from zero, which is similar to the case of monostable potential (β =3) where the period is 2π/ω too. In addition, making a comparison between fig- ures11and8a whenFremains unchanged, the value of Qin the presence of time-delay is much larger than that in the absence of time delay. WhenF is continuously changed, the response amplitude Q can be further enlarged by the cooperation of time-delay parameters τ and F. This reveals that the weak low-frequency signal can be amplified effectively by adjusting the parameters.

1.5 2 2.5

Q

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0.54 0.56 0.58 0.6 0.62

Q

(a)

F=550

F=700 2 /

2 /

(b)

Figure 11. The change of delay parameterτ causes the periodic change of the response amplitude gain of the system in the low-frequency signal forβ =6 and=10.

(10)

F

0 1 2 3 4

Q

(a)

r=0.05 r=0.1 r=0.15 r=0.2

0 100 200 300 400 500 600

0 100 200 300 400 500 600

F

0 1 2 3

Q

(b)

r=0.05 r=0.1 r=0.15 r=0.2

Figure 12. Plot of the response amplitude Qas a function of F forr = 0.05,r =0.1,r =0.15,r =0.2 withτ =1.6 when (a)β =3 and (b)β =6. The continuous lines represent theoretical Q, while the dashed lines represent numerically calculatedQ.

0 2 4 6 8 10 12 14 16 18 20

1 1.5 2 2.5 3 3.5

Q

r=0.05 r=0.1 r=0.15 r=0.2

10 15 20

0.05 10 2

r

0.1 5

Q

0.15 3

0.2 0 4

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

(a)

(b)

Figure 13. (a) Variation of the response amplitude Qvs.τ for different fixed values ofr. The continuous lines represent theoreticalQ, while the dashed lines represent numerically calculatedQand (b) the three-dimensional plot ofQvs. parameters randτ forF =550 whenβ =3.

(11)

0 5 10 15 20 1

1.5 2 2.5 3 3.5

Q

r=0.05 r=0.1 r=0.15 r=0.2

1

20 0

0.04 16

2

0.08 12

r

Q

0.12 8

3

0.16 0.20 0 4

4

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4

(a)

(b)

Figure 14. (a) The change of the response amplitudeQvs.τ for different fixed values ofr. The continuous lines represent theoreticalQ, while the dashed lines represent numerically calculatedQand (b) the three-dimensional plot ofQvs. parameters randτ forF =550 whenβ =6.

6. Effect of time-delay strength parameter ron VR The influence of time-delay intensity parameterron VR will be explored in this section, and some meaningful results can be seen.

6.1 Influence of time-delay strengthr on the response amplitudeQwith changes in F

Figure12shows the plots of the response amplitude Q vs. high-frequency signal F under different time-delay strength values ofrwhenβ =3 (monostable potential) (figure12a) andβ =6 (tristable potential) (figure12b) respectively. It is not difficult to draw a conclusion from the diagram that the response amplitude Qhas greatly improved with the increment ofr. In other words, the amplification effect of the response amplitude obtained by adjustingr and F is much better than that obtained by adjustingτ andF. This indicates that for the ampli- fication of the signal, changing the parameterr is more efficient than adjusting the parameterτ. That is to say, a superior way to magnify the weak low-frequency signal is born.

6.2 Influence of time-delay strengthr on response amplitudeQwith changes inτ

By comparing figures13 and14, we can come to two main conclusions. First, by changing τ, the response

amplitude Q shows periodic change in 2π/ω and secondly the value of the response amplitude gradu- ally increases by increasing the strength of time-delay parameterr, irrespective of whetherβ =3 (monostable potential) or β = 6 (tristable potential). What is the most remarkable is that, in a suitable range, the response amplitude Q can be magnified with the increase ofr.

This can be confirmed from the three-dimensional dia- gram. As stated above, the weak low-frequency signal can be amplified by the synergy between the parameters r andτ. This provides an effective way to amplify the signal.

7. Conclusion

The effect of time delay on VR in a harmonically trapped potential system is studied in this paper. We discussed the role of time delay in the response amplitudeQwhen β = 3 (monostable potential) and β = 6 (tristable potential). Some significant conclusions are derived.

First, when the strength of time-delay parameterr was regarded as a constant, the system with time delay shows a stronger resonance than that without the time delay even if Fis in the same intervals. Withτ changing, the response amplitude shows a periodic behaviour, and the resonance phenomenon appears at some special values ofτ. This means that the value of Q can be amplified

(12)

by adjustingτ. Secondly, for different values of high- frequency amplitude F, response amplitude will have different amplification with the same τ. This reveals that with the cooperation of F and τ, the response amplitude of the system can also be magnified. Fur- thermore, the influence of the strength parameterr on the response amplitude is discussed. Based on the anal- ysis of the impact of the discrete value of r and the continuous value ofr on the response amplitude, it is found that in a certain range, by adjusting the time- delay strength parameterr, the response amplitude can be increased, and more importantly, the amplitude of response obtained by adjusting r is much larger than that by adjustingτ. This reminds us of an optimal way to magnify the weak low-frequency signal. We hope our conclusions will play a potential role in the field of signal processing.

Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No.

GK201701001).

References

[1] V Mironov and V Sokolov,Radiotekh Elektron.41, 1501 (1996)

[2] A Maksimov,Ultrasonics.35, 79 (1997)

[3] J D Victor and M M Conte,Visual Neurosci.17, 959 (2000)

[4] D Su, M Chiu Victor and C Chen,Soc. Precis. Eng.18, 161 (1996)

[5] P S Landa and P V E McClintock,J. Phys. A33, L433 (2000)

[6] J P Baltanas, L Lopez, I I Blekhman, P S Landa, A Zaikin, J Kurths and M A F Sanjuan,Phys. Rev. E 67, 066119 (2003)

[7] S Rajasekar, K Abirami and M A F Sanjuan,Chaos21, 061129 (2011)

[8] J H Yang and X B Liu,J. Phys. A43, 12001 (2010) [9] J H Yang and X B Liu, Acta Phys. Sin. 61, 010505

(2012)

[10] J H Yang and X B Liu,Chaos20, 033124 (2010) [11] J H Yang and X B Liu,Phys. Scr.83, 065008 (2011) [12] D Hu, J Yang and X Liu, Commun. Nonlinear Sci.

Numer. Simulat.17, 1031 (2012)

[13] C Jeevarathinam, S Rajasekar and M A F Sanjuan,Chaos 23, 013136 (2013)

[14] J H Yang and H Zhu,Chaos22, 013112 (2012) [15] T Q Qin, T T Xie, M K Luo and K Deng,Chin. J. Phys.

55, 546 (2017)

[16] B Deng, J Wang, X Wei, K M Tsang and W L Chan, Chaos20, 013113 (2010)

[17] Y Qin, J Wang, C Men, B Deng and X Wei,Chaos21, 023133 (2011)

[18] H Yu, J Wang, C Men, B Deng and X Wei,Chaos21, 043101 (2011)

[19] A Jeevarekha and P Philominathan,Pramana – J. Phys.

86, 1091 (2016)

[20] C Jeevarathinam, S Rajasekar, M A F Sanjuan,Ecol.

Complex15, 33 (2013)

[21] J H Yang and H Zhu,Commun. Nonlinear Sci. Numer.

Simulat.18, 1316 (2013)

[22] C Jeevarathinam, S Rajasekar and M A F Sanjuan,Phys.

Rev. E83, 066205 (2011)

[23] K Abirami, S Rajasekar and M A F Sanjuan,Commun.

Nonlinear Sci. Numer. Simulat.47, 370 (2017) [24] M L Boas,Mathematical methods in the physical science

(Wiley, New Delhi, 2006)

[25] K T Tang, Mathematical methods for engineers and scientists 3: Fourier analysis, partial differential equa- tions and variational methods (Springer, Heidelberg, 2007)

References

Related documents

In summary, compared with what is happening in the rest of the world, where the lockdown measures and the economic crisis are driving the decrease in energy demand, the general

Percentage of countries with DRR integrated in climate change adaptation frameworks, mechanisms and processes Disaster risk reduction is an integral objective of

The Congo has ratified CITES and other international conventions relevant to shark conservation and management, notably the Convention on the Conservation of Migratory

Although a refined source apportionment study is needed to quantify the contribution of each source to the pollution level, road transport stands out as a key source of PM 2.5

These gains in crop production are unprecedented which is why 5 million small farmers in India in 2008 elected to plant 7.6 million hectares of Bt cotton which

INDEPENDENT MONITORING BOARD | RECOMMENDED ACTION.. Rationale: Repeatedly, in field surveys, from front-line polio workers, and in meeting after meeting, it has become clear that

With an aim to conduct a multi-round study across 18 states of India, we conducted a pilot study of 177 sample workers of 15 districts of Bihar, 96 per cent of whom were

With respect to other government schemes, only 3.7 per cent of waste workers said that they were enrolled in ICDS, out of which 50 per cent could access it after lockdown, 11 per