IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2011

Analysis of Simple Stochastic Resonancewith Various Noise Levels
Shota Ozaki

Shintaro Arai

Takaya Yamazato

Dept. of Telecommunications,
Dept. of Communication Network Eng.,
Inst. of Leberal Arts & Sciences,
Kagawa National College of Technology Kagawa National College of Technology
Nagoya University
Email: os.z0391@gmail.com
Email: arai@cn.kagawa-nct.ac.jp
Email: yamazato@nuee.nagoya-u.ac.jp

Yoshifumi Nishio
Dept. of Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

State of xxt1

I. I NTRODUCTIONStochastic resonance (SR) is a nonlinear phenomenon inwhich a responsiveness of a system is improved by adding
suitable noise to certain nonlinear systems. Recently, SR has
attracted a great deal of attention from a variety of researchers,
including nonlinear circuits and systems, semiconductor devices and biology[1]–[5]. In this paper, we consider that
SR is applied to a communication system which is one ofengineering systems. In standard communication systems, thesignal detection becomes difficult according to increase a noiselevel. We expect that it is possible to detect the signal, whichis influenced significantly by noise, by applying SR to thecommunication systems. For robustly detecting the signal in
the communication system using SR, a control of SR bynoise is very important task. Therefore, we consider that it isnecessary to investigate a relationship between SR and noise.
Based on the above research background, this paper focuseson an analysis of SR with various noise levels using simple
SR model. First, we analyze SR in a double-well potential
which is well-known as a typical bistable SR model. Thebistable system has two stable states (+1 or −1). Thus, we canregard the two stable states as 1bit data in the communication
systems. Next, we add a one dimension to the bistable SR
model and increase two stable states to the model, namely,
the system is extended to “four-stable state”. In this paper, wecall the extended SR model “SR in a quadruple-well potential”.
Here, four-stable states of the extended system are (−1, −1),(+1, −1), (−1, +1) and (+1, +1). Thus, we can regard thetwo stable states as 2bit data in the communication systems.
This paper describes an outline of SR in the quadruple-well

U(x)

U(x)

Abstract—In this paper, we analyze two simple stochastic
resonances with various noise levels. First, it is analyzed that
SR in a double-well potential which is well-known as a typicalbistable SR model. Second, we add a one dimension to the bistable
SR model and increase two stable states to the model, namely, thesystem is extended to “four-stable state”. Using these SR models,
we carry out computer simulations and analyze the state of SR
when the intensity of the noise is changed.

xt2(a)

(b)

U(x)

U(x)
t4

t3xx(d)(c)Fig. 1.

Mechanism of SR in bistable system.

potential. In addition, we carry out computer simulations andanalyze the state of SR when the intensity of the noise ischanged.
II. SR IN DOUBLE - WELL POTENTIAL
Here, we briefly explain SR in the double-well potential. InRef. [5], the bistable SR is performed by following equationsdx
dts(t)= f (x) + Dn(t) + s(t)(1)

= A sin(2πf0 t)(2)
dU0 (x)f (x) = −(3)dx11U0 (x) = − x2 + x4(4)24Where U0 (x) is a bistable potential having two local minimums, x is a state variable, s(t) is an input signal, A is an- 122 -A

B

22

1

1

00y+1yyState(x, y)-1-1-2-2-2-10x

1-22(a) Dx=Dy=0C-1
Fig. 2.

D=U0 (x) + xs(t)1
1− x2 + x4 + xA sin(2πf0 t)
24(5)= f (x) + Dx nx (t) + s(t)(6)

Where nx is noise added to the state x, Dx is an intensity ofnoise for the state x. Next, the equation of the new state y isdescribed as follows.

U0 (y)00= f (y) + Dy ny (t) + s(t)dU0 (y)dy
1 2 1 4= − y + y2
4= −(7)(8)(9)

Where ny is noise added to the state y, Dy is an intensity ofnoise for the state y. In this study, nx and ny are assumed
to be (AWGN). Additionally, the values of nx and ny are
different from each other. In other words, the nx and ny are
independent of each other.

12-1-2-10x1(c) Dx=Dy=30
Fig. 3.

In this study, we add a new state variable “y” to this bistablesystem and extend the system to SR in the quadruple-wellpotential. First, based on Eq. (1), the equation of the state x
is described as follows.

f (y)

1-2

III. SR IN QUADRUPLE - WELL POTENTIAL

dy
dt

1-2

Figure 1 shows a mechanism of SR in the bistable system.

dx
dt

2-1

amplitude of the input signal, n is assumed to be the additive
white Gaussian noise AWGN, D is an intensity of noise. An
effective potential U (x, t) is described as follows.
U (x, t) =2Four-ideal-stable states.

0x(b) Dx=Dy=10y+1y-1x-12-2-10x(d) Dx=Dy=60

12Output of SR in four-stable state (Dx = Dy).

IV. S IMULATION RESULTS
A. Simulation conditions
Using the above system, we carry out computer simulations.
In the simulation, the two states are expressed in (x, y) planesa coordinate (point (x, y)). Figure 2 shows four-ideal-stablestates of SR in the quadruple-well potential in (x, y) plane.
By changing Dx and Dy (i.e. noise level), we experiment
to control four-stable states in this plane. As parameters of
the simulation, the initial condition of x & y is “+1”, t =100, 000.
B. Results of same noise level (Dx = Dy).
Figure 3 shows simulation results for same noise level
Dx = Dy. First, in the case of Dx = Dy = 0 (Fig. 3(a)),
we can observed that the point (x, y) centers around (+1, +1)
since noise is not added to the system. Second, in the case ofDx = Dy = 10 (Fig. 3(b)), it can be confirmed that thepoint (x, y) moves the potential barrier and moves aroundthe four-stable states. Third, in the case of Dx = Dy = 30
(Fig. 3(c)), the point (x, y) continually hops the potential
barrier. However, we can see that the point (x, y) does notstay in one-stable state for a long time although the point(x, y) moves around the four-stable states. Finally, in the caseof Dx = Dy = 60 (Fig. 3(d)), it can be observed that the point(x, y) move around (x, y) plane, regardless of the four-stablestates.
C. Results of different noise level (fixed Dx, variable Dy).
Next, Fig. 4 shows simulation results for fixed Dx = 10 andvariable Dy. First, in the case of Dy = 0 (Fig. 4(a)), the point(x, y) moves a little in y-axis. On the other hands, the point
hops only in x-axis. Second, in the case of Dy = 5 (Fig. 4(b)),- 123 -211yy20

behavior of the point (x, y) , in this study. Therefore, thequantitative analysis of the SR in the quadruple-well potential,
such as a signal-to-noise ratio (SNR), is our future work.

0-1R EFERENCES-1-2-2-2-10x

1-22-10x10x10x12(b) Dx=10, Dy=52211yy(a) Dx=10, Dy=00-10-1-2-2-2-10x12-2(c) Dx=10, Dy=10-12(d) Dx=10, Dy=1522

11yy[1] F. Chapeau-Blondeau and X. Godivier, “Theory of stochastic resonancein signal transmission by static nonlinear systems,” Physical Review E,
vol. 55, no. 2, pp. 1478-1495, Feb. 1997.
[2] L. Gammaitoni, P. Hänggi, P. Jung and F. Marchesoni, “Stochasticresonance,” Reviews of Modern Physics, vol. 70, no. 1, pp. 223-287,Feb. 1997.
[3] H. Fujisaka, T. Kamio, K. Haeiwa, “Stochastic resonance in coupledsynchronization loops,” IEICE Trans. Fundamentals, vol. J90-A, no. 11,pp. 806-816, Nov. 2007. (in Japansese)[4] A. Utagawa, T. Asai, K. Yoshida and Y. Amemiya, “Stochastic resonancein a simple electrical circuits having a double-well potential,” Technical
Report of IEICE, vol. 109, no. 458, NLP2009-171, pp. 75-80, Mar. 2010.
(in Japanese)[5] H. Nishimura, “Stochastic and Chaotic Resonances in Nonlinear Systems
with Fluctuations,” J. Society of Instrument and Control Engineers,
vol. 49, no. 4, pp. 244-249, Apr. 2010. (in Japanese)

0-10-1-2-2-2-10x1(e) Dx=10, Dy=20Fig. 4.

2-2-12(f) Dx=10, Dy=30

Output of SR in four-stable state (fixed Dx, variable Dy).

we can observe that the point (x, y) frequently hops in x-axis
since the noise level of x (Dx) is stronger than that of y (Dy).
Third, in the case of Dy = 10 (Fig. 4(c)), the point (x, y)
symmetrically hops in (x, y) plane because Dx and Dy aresame values. Fourth, in the case of Dy = 15 (Fig. 4(d)) and
Dy = 20 (Fig. 4(e)), we can find that the number of the point’s
move increases. In other words, the point (x, y) actively hopsin (x, y) plane. Finally, in the case of Dy = 30 (Fig. 4(f)),
the point’s move around the origin is not confirmed in (x, y)plane. Here, we compare the results of the same noise level(Dx = Dy) and the different noise level (fixed Dx, variable
Dy). By setting the different noise level, we can make thedifference point’s between x-axis and y-axis and can confirm
that the point move linearly. Therefore, to use the different
noise levels is effective for controlling the state of the SR in
the quadruple-well potential.
V. C ONCLUSIONSIn this paper, we have analyzed SR in the quadruple-wellpotential which extended the SR in a double-well potential. As
results of computer simulations, it have been confirmed thatthe various point’s moves by changing noise levels. However,
we have performed only the qualitative analysis, such as the- 124 -