2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

Analysis of Stochastic Resonance in Quadruple-Well Potential
Shintaro ARAI† , Shota OZAKI† , Takaya YAMAZATO‡ and Yoshifumi NISHIO∗†Kagawa National College of Technology
551 Kohda, Takuma-cho, Mitoyo, Kagawa, JAPAN
Phone:+81-875-83-8617
Email: arai@cn.kagawa-nct.ac.jp‡Nagoya University
Furo-cho, Chikusa-ku, Nagoya, JAPAN
Phone:+81-52-747-6596
Email: yamazato@nuee.nagoya-u.ac.jp* Tokushima University
2-1 Minami-Josanjima, Tokushima, JAPAN
Phone:+81-88-656-7470
Email: nishio@ee.tokushima-u.ac.jp

which is a center of SR in Fig. 1(a), is changed by injecting the noise to SR model, and the particle hops from one
This paper analyzes a stochastic resonance (SR) in a potential well to the other one, as shown in Fig 1(b). Thus,
quadruple-well potential, which extended the SR in a double- we consider that it is important to analyze the state of SR bywell potential, to control of SR. For achieving the control of injecting the noise for achieving the control of SR.
SR, it is important to analyze the state of the SR by injecting
In this study, we therefore calculate existence probabilities
the noise. In this study, we calculate existence probabilities of the state (particle) to analyze how the state changes by theof the SR’s state to analyze how the state changes by a noise noise intensity. For analyzing the existence probabilities ofintensity.
the particle, this paper uses a SR in a quadruple-well potential which added one dimension in SR model in a double-wellpotential. By analyzing the existence probabilities of the par1. Introduction
ticle in the SR with various noise intensities, we discuss theStochastic resonance (SR) is a nonlinear phenomenon in capability of the control of SR by the noise.
which a responsiveness of a system is improved by injecting suitable noise to certain nonlinear systems. Recently, 2. SR in double-well potential
SR has attracted a great deal of attention from a variety ofWe brieﬂy explain SR in the double-well potential. In
researchers[1]–[7]. In this paper, we consider that SR is apRef.
[7], the bistable SR is performed by following equations.
plied to a communication system which is one of engineeringsystems. In standard communication systems, the signal dedx
tection becomes difﬁcult according to increase a noise level.
= f (x) + Dn(t) + s(t),(1)dt
By applying SR to the communication systems, we expects(t) = A sin(2πf0 t),(2)that it is possible to detect the signal which is inﬂuenced sigdU0 (x)niﬁcantly by noise. For robustly detecting the signal in the,f (x) = −(3)dxcommunication system using SR, a control of SR by noise is1
1very important task. Therefore, we consider that it is necesU0 (x) = − x2 + x4 .
(4)24sary to investigate a relationship between SR and noise.
Based on the above research background, this paper ana- Where U (x) is a bistable potential having two local mini0lyzes SR with various noise intensities. Especially, we focus mums (i.e., two potential wells), x is a state variable, s(t) is
on a state of SR depending on the noise intensity. In Fig. 1, an input signal, A is an amplitude of the input signal, n is
we show a mechanism of SR model in a double-well poten- assumed to be the additive white Gaussian noise (AWGN),
tial which is well-known as a typical bistable SR model. Hear, D is an intensity of noise. An effective potential U (x, t) isthe ball (particle) in this ﬁgure is a state of SR. In this model, described as follows.
the particle stays in the one potential well unless the noise, as
shown in Fig. 1(a). However, a height of a potential barrier,
Abstract- 60 -U(x)U(x)

yState of xxt1

Ax

S ta te( x , y)Bt2(a)

(b)

U(x)

U(x)
t4

x

t3Cxx(d)DFigure 2: Four-ideal-stable states of SR in quadruple-well potential.

(c)Figure 1: Mechanism of SR in double-well potential.

U (x, t) =U0 (x) + xs(t)11= − x2 + x4 + xA sin(2πf0 t).
24(5)The bistable system has two stable states (1 or −1). Thus, we
can regard the two stable states as 1bit data in the communication systems.

Where ny is noise injected to the state y, Dy is an intensity
of noise for the state y.
SR in the quadruple-well potential has four local minimums, i.e., four potential wells ((1, 1), (−1, 1), (1, −1) and(−1, −1)). Thus, we can regard the two stable states as 2bit
data in the communication systems.
In this study, nx and ny are assumed to be AWGN. Additionally, nx and ny are different from each other. In otherwords, nx and ny are independent of each other.
4. Simulation results and discussions

3. SR in quadruple-well potential
4.1. Simulation conditions
As described in Sec. 1, this paper uses the SR in thequadruple-well potential, which added one dimension in SR
model in a double-well potential, for analyzing the existence
probabilities of the particle. In this study, we add a new statevariable “y” to the bistable system and extend the system toSR in the quadruple-well potential.
Here, we explain equations of SR in the quadruple-wellpotential. First, based on Eq. (1), the equation of the state x
is described as follows.
dxdtf (y)U0 (y)(1) Same noise intensity (Dx = Dy ),(2) Different noise intensity (Dx = 10 (ﬁxed), Dy : variable).

= f (x) + Dx nx (t) + s(t).

(6)

Where nx is noise injected to the state x, Dx is an intensity
of noise for the state x. Next, the equation of the new state yis described as follows.
dydt

Using the above system, we carry out computer simulations. In the simulation, the two states (particles) are expressed in (x, y) planes as a coordinate (point (x, y)). As
parameters of the simulation, the initial condition of x and yis “1”, t = 10, 000, dt = 0.01. In addition, we use two typesof the noise parameter for the simulations:= f (y) + Dy ny (t) + s(t),dU0 (y),= −dy11= − y2 + y4 ,24Figure 2 shows four-ideal-stable states of SR in thequadruple-well potential in (x, y) plane. By changing thenoise intensity, we observe various behaviors of the particles
and calculate the existence probability of the particle in eachpotential well.

(7)
4.2. Results of same noise intensity (Dx = Dy )(8)
(9)

Table 1 shows existence probabilities of SR in thequadruple-well potential for Dx = Dy . In addition, Fig. 3
shows some outputs of the SR based on Tab. 1. From these- 61 -

2

2

23.0%

1.5

16.1%

2

33.2%

1.5

1

1

0.5

0.5

0.5

000y

1

yy

1.5

25.2%

-0.5

-0.5

-0.5

-1

-1

-1

-1.5
-2
-2

32.0%
-1.5

19.8%
-1

-0.5

0x

0.5

1

1.5

2-1.5
-2
-2

37.3%
-1.5

13.4%
-1

(a)

-0.5

0x(b)

0.5

11.5

2-1.5
-2-2

7.0%

41.1%

43.6%
-1.5

8.3%
-1-0.5

0x

0.5

1

1.5

2(c)

Figure 3: Outputs of SR in quadruple-well potential (Dx = Dy ): (a) Dx = Dy = 3, (b) Dx = Dy = 10, (c) Dx = Dy = 30.

Table 1: Existence probabilities of (x, y) (Dx = Dy )Existence probabilities of (x, y) [%]Dx = Dy (1, 1) (−1, 1) (1, −1) (−1, −1)
0100
0003
23.0
25.2
19.8
32.0
726.2
22.1
17.0
34.7
1033.2
16.1
13.4
37.3
20
41.1
7.5
8.9
42.5
30
41.1
7.0
8.3
43.6

Table 2: Existence probabilities of (x, y) (Dx = 10 (ﬁxed),
Dy : variable).
Existence probabilities of (x, y) [%]Dy (1, 1) (−1, 1) (1, −1) (−1, −1)
0
46.6
53.4
0
0
3
25.7
22.5
20.9
30.9
7
29.1
19.2
17.5
34.2
20
35.3
13.3
11.3
40.1
30
31.8
16.3
14.8
37.1

results, we can see that the particles (points (x, y)) are almost
evenly distributed in four potential wells when Dx and Dy aresmall. However, the existence probability is biased according
to increasing the noise intensity. Especially, the particles converge on two potential wells ((1, 1) and (−1, −1)) , as shownin Fig. 3(c). In other words, the particles only pass throughthe wells of (−1, 1) and (1, −1) when Dx and Dy are large.
Next, we observe the motion of the particles from Fig. 3. As
one can see, although the motion of the particles is small ineach potential well when Dx and Dy are small (Fig. 3(a)), theparticles actively move in the wells of (1, 1) and (−1, −1)
according to increasing the noise intensity (Fig. 3(c)). Therefore, it can be said that the particles do not stay for long termin one potential well when Dx and Dy are large, and the statetransition occurs frequently in the wells.

Tab. 2. As a matter of course, it can be observed that the existence probabilities of Tab. 2 are different from that of Tab. 1.
Especially, we focus on the larger Dy (Dy = 20 and 30). The
existence probabilities of (−1, 1) and (1, −1) when Dy = 20
and 30 in Tab. 2 are larger than that of Dx = Dy = 20 and
Dx = Dy = 30 in Tab. 1. Since the inﬂuence of the noisethat the particle of x-component receives is smaller than thatof y-component, the probability that the two-state transition
occurs simultaneously becomes small. Thus, we consider that
it is easy to change the existence probabilities using the different noise intensity. In other words, it is possible that to use thedifferent noise intensity is effective for controlling the stateof the SR in the quadruple-well potential. However, it canbe also conﬁrmed that the existence probability in Tab. 2 isbiased in two potential wells ((1, 1) and (−1, −1)) according
to increasing the noise intensity in common with Dx = Dy .
We discuss the reason for this in Sec. 4.4.

4.3. Results of different noise intensity (Dx = 10 (ﬁxed),Dy : variable)

4.4. Discussions
Table 2 shows existence probabilities of SR in theHere, we discuss that the reason for the biased existence
quadruple-well potential when Dx = 10 and Dy is variable.
In addition, Fig. 4 shows some outputs of the SR based on probability according to increasing the noise intensity. The- 62 -

2

2

25.7%

1.5

13.3%

235.3%

1.5

11

0.5

0.5

0.5

000y

1

yy

1.5

22.5%

-0.5

-0.5

-0.5

-1

-1

-1

-1.5
-2
-2

30.9%
-1.5

20.9%
-1

-0.5

0x

0.5

1

1.5

(a)

2

-1.5
-2
-2

40.1%
-1.5

11.3%
-1

-0.5

0x(b)

0.5

11.5

2-1.5
-2
-2

16.3%

31.8%

37.1%
-1.5

14.8%
-1-0.50x

0.5

1

1.5

2(c)

Figure 4: Output of SR in four-stable state (Dx = 10 (ﬁxed), Dy : variable): (a) Dy = 3, (b) Dy = 20, (c) Dy = 30.

major reason for this is that s(t) dose not depend on the stateof x and y, as shown in Eq. 2. The states of x and y in SRchange independently of each other depending on each input
signal and each noise. Thus, the particles are biased to thewells of (1, 1) and (−1, −1). Therefore, we consider that toanalyze the SR using the input signal which x and y inﬂuencemutually is our future work.
5. Conclusions[3] D. G. Luchinsky, R. Mannella, P. V. E. McClintock
and N. G. Stocks, “Stochastic resonance in electricalcircuits–I: Conventional stochastic resonance,” IEEE
Trans. CAS–II, vol. 46, no. 9, pp. 1205-1214, Sep. 1999.
[4] D. G. Luchinsky, R. Mannella, P. V. E. McClintock
and N. G. Stocks, “Stochastic resonance in electrical circuits–II: Nonconventional stochastic resonance,”
IEEE Trans. CAS–II, vol. 46, no. 9, pp. 1215-1224,
Sep. 1999.

In this study, we have analyzed existence probabilities ofthe particle in SR in the quadruple-well potential with various noise intensities. As results of computer simulations, wehave found that it is possible that to use the different noise
intensity is effective for controlling the state of the SR in thequadruple-well potential. In addition, it has been conﬁrmedthat the existence probability is biased in the wells of (1, 1)
and (−1, −1) due to s(t) which dose not depend on the stateof x and y.

[5] H. Fujisaka, T. Kamio, K. Haeiwa, “Stochastic resonance in coupled synchronization loops,” IEICE
Trans. Fundamentals, vol. J90-A, no. 11, pp. 806-816,
Nov. 2007. (in Japanese)

Acknowledgment[7] H. Nishimura, “Stochastic and chaotic resonances innonlinear systems with ﬂuctuations,” J. Society of Instrument and Control Engineers, vol. 49, no. 4, pp. 244249, Apr. 2010. (in Japanese)

The authors would like to thank Dr. Yukihiro TADOKORO
of Toyota Central R&D Labs., Inc..

[6] A. Utagawa, T. Sahashi, T. Asai and Y. Amemiya,“Stochastic resonance in an array of locally-coupled
McCulloch-Pitts neurons with population heterogeneity,” IEICE Trans. Fundamentals, vol. E92-A, no. 10,
pp. 2508-2513, Oct. 2009.

References[1] F. Chapeau-Blondeau and X. Godivier, “Theory ofstochastic resonance in signal transmission by static
nonlinear systems,” Phys. Rev. E, vol. 55, no. 2,
pp. 1478-1495, Feb. 1997.
[2] L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni,“Stochastic resonance,” Rev. Mod. Phys., vol. 70, no. 1,
pp. 223-287, Jan. 1998.

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