Stochastic resonance in symmetric double well: higher harmonics
The work deals with the phenomenon of Stochastic Resonance in its genuine model, proposed by B. McNamara and K. Wiesenfeld for explanation of long-term climatic changes on Earth. It is shown that in two state model the higher harmonics behave in a non-monotonous way with increase of the noise level,...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Zitieren: | Stochastic resonance in symmetric double well: higher harmonics / A.V. Zhiglo // Вопросы атомной науки и техники. — 2001. — № 6. — С. 251-254. — Бібліогр.: 8 назв. — англ. |
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| author_facet | Zhiglo, A.V. |
| citation_txt | Stochastic resonance in symmetric double well: higher harmonics / A.V. Zhiglo // Вопросы атомной науки и техники. — 2001. — № 6. — С. 251-254. — Бібліогр.: 8 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The work deals with the phenomenon of Stochastic Resonance in its genuine model, proposed by B. McNamara and K. Wiesenfeld for explanation of long-term climatic changes on Earth. It is shown that in two state model the higher harmonics behave in a non-monotonous way with increase of the noise level, possessing one or more maxima. Explicit formulae for third and fifth harmonic amplitudes and corresponding SNR are obtained. Studied by other authors peculiarities, like dips and sharp peaks in output signal do not occur in two state model, thus they only exist in systems with continuous configuration space.
|
| first_indexed | 2025-12-07T18:38:35Z |
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STOCHASTIC RESONANCE IN SYMMETRIC DOUBLE WELL: HIGH-
ER HARMONICS
A.V. Zhiglo
Institute for Theoretical Physics,
National Science Center"Kharkov Institute of Physics and Technology", Kharkov, Ukraine
The work deals with the phenomenon of Stochastic Resonance in its genuine model, proposed by B. McNamara
and K. Wiesenfeld for explanation of long-term climatic changes on Earth. It is shown that in two state model the
higher harmonics behave in a non-monotonous way with increase of the noise level, possessing one or more maxima.
Explicit formulae for third and fifth harmonic amplitudes and corresponding SNR are obtained. Studied by other au-
thors peculiarities, like dips and sharp peaks in output signal do not occur in two state model, thus they only exist in
systems with continuous configuration space.
PACS: 05.40. +j
1. INTRODUCTION
Stochastic resonance (SR) has been studied for about
20 years. The principal manifestation of the phe-
nomenon is a strong reaction of different output charac-
teristics of the system (like component with initial fre-
quency in residual-time distribution, signal/noise ratio
etc.) to a weak periodic signal. This reaction grows with
increasing of the noise level up to certain extent. The
motivations for study of the phenomenon as well as the-
oretical model are rather naturally presented in [1,2],
early works describing SR (initially put forward for ex-
planation of the correlation between glacial periods on
Earth with the periodic changing of the Earth orbit ec-
centricity). Now SR constitutes an important subfield of
non-linear physics. According to usual understanding of
stochastic resonance phenomenon as non-monotonous
dependence of the output as a function of noise intensi-
ty, the majority of investigations (both theoretical and
experimental) in the case of monochromatic input ana-
lyze the component of output with initial frequency.
There are several works where higher harmonics are
investigated as well as the first one — using different
models in continuous configuration space (most com-
mon approaches are linear-response theory, numerical
integration of Fokker-Planck equation, matrix continued
fraction technique) they study the problems of optimal
generation or suppressing of higher harmonics [3,4,5];
the strengths of higher harmonics show various peculi-
arities such as extremely sharp peaks and resonance-ab-
sorption like dips at certain noise intensities. The de-
pendence of non-zero intensity of even harmonics on the
potential asymmetry is studied in [6]. Different methods
used for study of SR-like phenomena and further numer-
ous references can be found in [7].
We will demonstrate some qualitative aspects of the
problem in two state model, intending to find out what
of studied features of higher harmonics can be observed
in it. We follow the notation from the work [1]; we will
propose some useful representations of higher harmon-
ics intensities — absolute and in relation to noise power
densities at corresponding frequencies.
Let us consider a Brownian particle in an external
smooth potential )(xU with a barrier between two
wells, subjected to a strong friction. If the noise intensity
D (we consider a white noise that provides a term
2 ( )D tξ in the Langevin equation (3); =′)()( tt ξξ
)( tt ′−δ ) is much smaller than the height of the barrier
between wells U∆ , the average reciprocal transition
time through the barrier is given by Kramers formula:
1 (0) ( ) exp
2 m
Ur U U x
Dπ
∆ж цўў ўў= −з ч
и ш
;
0 and mx are positions of potential maximum and mini-
mum.
We consider a symmetric potential (with dimen-
sionless x )
42
)(
42 xxxU +−= (1)
thus
−=
D
r
4
1exp
2
1
π
. (2)
Let us assume that beyond the chaotic force 2D tξ ( ) ,
a friction and − ∂ ∂U x/ the particle is subjected to an
external periodic force A tcosω . For overdamped sys-
tem the dependence x t( ) is to be found from
cos ( )x x x A t D t= − + +3 2ω ξ . (3)
Under these conditions ( )x t is a periodic function
having period 2 /T π ω= (providing 0 1/t t r− ? ); the
point is to study the behavior of its different harmonics
as functions of D .
We mainly study the power spectrum, being obtained
as a Fourier transform of autocorrelation function
( ) ( )x t x t τ+ . ( )kSNR D , defined as a ratio of signal
power in -spike at kωΩ = to noise power density at
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 251-254. 251
this frequency is also investigated; non-monotonous
1( )SNR D dependence is a usual manifestation of SR.
Two state model, in which there are only two states
for the system to occupy ( mx x= ± ), may be considered
a limit of described above dynamical system in which
the transition time is much greater than the relaxation
time within a well. It is natural to assume minima of
( )U x as mx± ; for the potential (1) 1mx = .
In such an approximation the distribution ( , )p x t re-
duces to ( )n t± — probabilities of location near mx±
(within right or left well) ( )1n n+ −+ = :
( , ) ( ) ( ) ( ) ( )m mp x t n t x x n t x xδ δ+ −= − + + . (4)
Their evolution is given by the rate equation:
( ) ( )
dn n W t n W t
dt
±
± ±= − +m m , (5)
where ( )W t± are normalized probabilities of transition
into state.
In the work [1] the following expression was pro-
posed:
( ) exp cosmAx
W t r t
D
ω±
ж ц= ±з ч
и ш
(6)
with , , r A D introduced above.
The solution of (5) (for sign +)
2 cosh cos exp cosm mAx Axdn
n r t r t
dt D D
ω ω+
+= − +ж ц ж ц
з ч з ч
и ш и ш
( ) ( )P t n Q t+− +є (7)
is
( ) exp ( ) exp ( )
t t t
n t P dt C Q t P d dtτ τ+
− − −Ґ Ґ Ґ
ж цж ц ж ц
= − + −з чз ч з чз чи ш и ши ш
т т т
(8)
The integrals in (8) cannot be calculated in terms of
known functions. In [1] integrals were evaluated with
accuracy /mAx Dε =: , considered to be a small pa-
rameter. As a result ( )n t+ only contained the first har-
monic that was ~ ε (comparing with the constant com-
ponent of distribution at t = t0+1/r).
Using more precise expansion in small ε it is possi-
ble to account for higher harmonics and to calculate
their amplitudes with desired accuracy. However, the
problem is rather complicated especially if the aim is to
obtain kSNR – because in such a way it is inevitable to
take into account terms with different frequencies (not
only kω ). In the work [8] the authors obtained results
for higher harmonics in similar model considering hop-
ping between the wells as process discrete in time (i.e.
not only configuration space, but also the time scale was
treated as discrete). They also studied interesting fea-
tures connected with modulation of equilibrium posi-
tions ( )mx t ; this resulted in (rather weak) peaks at even
harmonics in power density.
We will obtain sum representation of amplitudes in
power spectrum; these sums:
1) have sense at all ε
2) can be used for evaluating the largest terms
(those are 2~ for k
kSNRε ) at small ε .
2. POWER SPECTRUM
Using exp( cos ) I ( ) cosn
n
t n t
Z
e w e w
О
= е , where In
are modified Bessel functions we can obtain
0 20
1
2 sin 2t
n
n
P dt r t n tε ω
Ґ
=
= + ет , (9)
where
r r r
nn
n
0 0 2
22= =I ( ) I ( )ε ε ε
ω
, . (10)
So, using (8)
(2 0 0 0sin 2 2 2 2 cos( ) n tn t r t r t r t tn t e Ce e dt r e eε ω ε ω− − −
+ − Ґ
е= + т
) ( )2 0sin 2 2( ) ( )n n t r te t Ce V tε ω π −е+ґ є . (11)
Here ( / ) ( )V t V tπ ω+ = .
For the conditional probability ( )0 0| , :n t x t+
( ( )
00 0 0| ,
mx xn t x t δ+ = )
( )0 0
0
2 ( ) 1
0 0( ) ( ) ( ) ( ) ( )
m
r t t
x xn t t V t e V t tπ δ π− − −
+
й щ= + − +л ы .
(12)
The amplitudes of different harmonics in output power
are connected with the coefficients kG in
0
0 0( ) ,
( ) lim 2 ( ) ( ) 1
t m
x t x t
H t t V t
x
π
→ − Ґ
= = −
0
0
cos( )k k
k
G G k tω ψ
Ґ
=
+ +є е . (13)
The periodicity of ( )H t is obvious; the autocorrel-
tion function 0 0( ) ( ) ( , | . )x t x t K t x tτ τ+ =
0 0( ) ( ) ,x t x t x tτ+
0 0( , | , ) ( , | , )xy dx dy p x t y t p y t x tτ= +тт (14)
is periodic as a function of t in the limit 0t → − Ґ . On
averaging over random initial phase in external force
(or, equivalently, over the period of t, taking 0t t− and
τ constant)
0
2
0 0lim ( ) ( ) , ( ) ( )mt
x x t x t x t H t H tτ τ−
→ − Ґ
+ = +
( )02 2 ( )1 ( )
( )
r te H t
t
τ π τ
π
− ++ − . (15)
The first term (see (13)) gives (after averaging)
2 2
0
1
1( ) ( ) cos
2 kt
k
H t H t G G kτ ω τ
Ґ
=
+ = + е . (16)
Fourier transform of (16) only contains δ-spikes at
kωΩ = :
252
2
2
0
1
2 ( ) ( ) cos
2
i t k
S
k
GG K e k t dtπ δ ω
ҐҐ
− Ω
= − Ґ
− Ω + Ω = е т
[ ]2
1
( ) ( )
2 k
k
G k kπ δ ω δ ω
Ґ
=
= Ω − + Ω +е . (17)
Using the same procedure one can see that the 2nd term in
(15) corresponds to noise component of output power since
( )0 02 22( ) 1 ( ) ( )
( )
r r
t
te H t e U tτ τπ τ
π τ
− −+ − = . (18)
with ( / ) ( )U t U tπ ω+ = . Fourier transform ( )NK Ω of
(18) (i.e. the second term in (15)) is a regular function
with Gaussian maxima near 2 (n n N −w=W О positive
integers). So, the useful signal is given by (17); in part,
if some 0kG = the power spectrum is regular at
kw=W (because of the symmetry of the potential all
even harmonics vanish: 2 0lG l N= " О ). According to
the definition
2
4 ( )
k
k
N
GSNR
K k
=
W
. (19)
3. HARMONICS AMPLITUDES EVALUA-
TION
Let us represent V(t) (see(11)) as a sum:
0 02 2
2
1
( ) exp( cos ) exp sin 2
t
r t r t
k
k
V t e t e dt k te w e w
Ґ
=- Ґ
= ет
( )
( )
0 22
2
, 0
I ( ) I ( )
2 2
k
k
k
k
i t n knr t
n
n n k
kn n k
e ei
r i n kn
w
e e
w
+
-
е
е= Ч
+ +
е Х
е
.
(20)
Here
kn
е means a sum over all 1{ }k kn ZҐ Ґ
= О – conse-
quences of integers (with finite number of non-zero ele-
ments). Let us obtain representation of ( )H t as a Fouri-
er sum:
,
( ) 2 ( ) ( ) 1 I ( )
k k
i t
n
n l
H t t V t e ws
s
p e
Ґ
=- Ґ
= - = е е
( )
( )
2 2
0
2I ( )I ( ) 1
2
k k
k
k k
n l
n k l k
k
ri
r im
e e
w
- -е
-ґ Ч
+
Х
i tH e wssє е . (21)
In the last equation we have introduced , nZs s =О
12 ( )k kk k n lҐ
=+ +е , thus n is to be calculated from
here; 2 k
k
m kl
N
s
О
= - е .
Hs are simply connected with :kG
cos( ) , 2i kt i kt
k k k k k kG k t H e H e G Hω ωω ψ −
−+ = + =
For 1e= (21) enables us to obtain approximations for
Hs : below
/ ,z r w= { } { }
( )
0
( , ) ,
2
k kl n
k k
iB B n l
r imw
-е
=є
+
!N means !N
( )
2
2
1,
( / 2) ( / 2) 1 ( )
2 ! (2 )! ! !
k k
k k
n ln k
kn l k k
H z B O
r n k k n l
s e e e
+
Ґ
=
ж цчз чз= +чз ччззи ш
е Х
( )2
1, 2 (2 )! ! !
k kk k
k k
n ln k n l
kn l k k
z B
k k n l
e
++ + Ґ
=
е ж цж ц чзчз ч» ч зз ччч зз чзи ш и ш
е Х . (22)
It is obvious that the terms containing the lowest power
of e correspond to those with equal-signed , kn n and
kl . There is a finite number of such terms. Using only
them we get for 0s >
( )
( )
0, , 0
2
2 2 2
k k
k k
k k
l n
n n l
n k l n
H i
r r im
s
s
s
e
w
-
і
+ + =
е
е
ж цчз= чз ччзи ш +е
1
1
(2 )! ! !
k kn l
k k k
z
k k n l
+Ґ
=
ж цчз чґ з чз чзи ш
Х . (23)
The calculations give for first non-zero amplitudes kG :
( ) ( )
2
5 3
1 2 2 2 2
1 / 4 22
4 1 2 4 1
zG z O O
z z z
ee e e
e
+
= + = +
+ + +
3 2
3 2 2
1/16
3 (4 1)(4 9)
z zG
z z
e +
=
+ +
(24)
( ) ( )
( )( )( )
2 225
5 2 2 2
64 / 3 1 14
320 4 1 4 9 4 25
z zzG
z z z
e - +
=
+ + +
.
Using the terms with the lowest power of e in NK one
can obtain from (19) expressions for kSNR applicable
for small e :
( )( )
2
2 2 2 4
0
0
4 1 ( )
8
k
k
GSNR r k O
r
p
w e= + + . (25)
Thus
( )
2 2
2 6
1 2
21
2 4 1
zSNR z O
z
p ew e e
ж цчз чз= + +чз ччзз +и ш
2
6
3 2
1/16
72 4 1
zSNR z
z
p w e +
=
+
(26)
( ) ( )
( )( )
2 2210
5 2 13 2 2
64 / 3 1 14
10 2 4 1 4 9
z zzSNR
z z
pw e - +
=
Ч + +
.
The dependence ( )kSNR w has non-monotonous charac-
ter. Unless A or e is large kSNR attains its maximum at
1e< where our approximation is valid. So, maximum
positions can be calculated from the above expressions.
From expansion of I ( )n e one can derive the following
structure of the main term of SNR:
( )2 2/ 2 ( )k
k kSNR z Q zw e= ,
where kQ is some rational function, 2 (0) 0kQ № . There-
fore, for small z (i.e. large w or low noise, though the
last alternative is restricted by the requirement
/ 1mAx D e<є ) one can evaluate the position of prin-
cipal maximum of kSNR :
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 251-254. 253
2
max
0
( )
0 0
2
k k
z
dQ z d SNR UD
d d ke e
=
м ьDп пп п= = « =Юн эп пп по ю
.
Corresponding graphics of kSNR are depicted in Fig. 1–3.
Maxima in Fig. 1 are located at predicted values.
Figs. 2,3 show deviation from description based on ne-
glecting of intricate z-containing factor behavior – they
correspond to low w , when maxima are situated in the
region of D, where 1e> . Fig. 3 shows anomalous rela-
tive heights and position of multiple maxima of the first
three non-zero harmonics.
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.00
0.05
0.10
0.15
0.20 SNR1,3,5
D
Fig. 1. 1,3,5 0.1SNR ω = – regular situation. Crosses
correspond to 1SNR , the light curve to 3SNR , and the
dark one to 5SNR
0.0 0.2 0.4 0.6
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
D
SNR3
Fig. 2. 3 0.018SNR ω = – low frequency, 2 maxima
with equal heights
0.0 0.2 0.4 0.6 0.8
0.00
0.05
0.10
0.15
0.20
SNR1,3,5
D
Fig. 3. 1,3,5 0.003SNR ω =
The main difference from corresponding characteris-
tics of the first harmonic is due to more complicated role
of z in the last expressions. This makes possible multiple
maxima, which are really observed at such w that the
"resonant" D corresponds to 1z » (or r w» ). In this
case the behavior of ( )z D is necessary to be accounted
for. Due to the different locations of maxima of different
factors containing z in kG and kSNR one can find spe-
cial values of w that provide several maxima of compa-
rable heights for these functions, though universally
these functions possess only one maximum in the region
of lower and higher frequencies.
For 1e> one should use greater number of terms in
(21). The number of required terms increases rapidly
when e becomes greater than 1. So, in order to find out
whether there exist other peculiarities in output charac-
teristic for considered system, it is more convenient to
use numerical calculation of different harmonics of
( )H t . Such numerical integration shows that the correc-
tions to (24) in the vicinities of maxima are not signifi-
cant. These precise results do not change the non-
monotonous character of both curves Gk ( )ε and
SNRk ( )ε (or, equivalently G Dk ( ) and SNR Dk ( ) ). The
only difference I would like to mention is less rapid van-
ishing of both functions at e® Ґ ( )0DЫ ® – these
vanish exponentially according to (24,26) and less rapidly
following the numerical results.
I am glad to acknowledge the participation of
Yu.L. Bolotin in both the putting of the problem and fur-
ther elaboration of questions needed consideration as
well as for his support in search for relevant literature
and inspiring discussions.
REFERENCES
1.B. McNamara, K. Wiesenfeld. Theory of Stochastic
Resonance // Phys. Rev. A. 1989, v. 39, p. 4854-4869.
2.R. Benzi, A. Sutera, A. Vulpiani. The mechanism
of stochastic resonance // J. Phys. A: Math.Gen.
1982, v. 34, p. 10-16.
3.L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni.
Stochastic Resonance // Rev. Mod. Phys. 1998,
v. 70, p. 223-287.
4.P. Jung, P. Hänggi. Stochastic Nonlinear Dynamics
Modulated by External Periodic Forces // Europhys.
Lett. 1989, v. 8 (6), p. 505-510.
5.P. Bartussek, P. Hänggi, P. Jung. Stochastic Reso-
nance in Optical Bistable Systems // Phys. Rev. E.
1994, v. 49, №5, p. 3930-3939.
6.P. Jung, P. Talkner. Suppression of Higher Har-
monics at Noise Induced Resonances // Phys. Rev.
E. 1995, v. 51, №3, p. 2640-2644.
7.A.R. Bulsara, M.E. Inchiosa, L. Gammaitoni.
Noise-Controlled Resonance Behavior in Nonlinear
Dynamical Systems with Broken Symmetry // Phys.
Rev. Lett. 1996, v. 77, №11, p. 2162-2165.
8.V.A. Shneidman, P. Jung, P. Hänggi. Power Spec-
trum of a Driven Bistable System // Europhys. Lett.
1994, v. 26, №8, p. 571-576.
254
A.V. Zhiglo
Institute for Theoretical Physics,
National Science Center"Kharkov Institute of Physics and Technology", Kharkov, Ukraine
PACS: 05.40. +j
1. INTRODUCTION
2. POWER SPECTRUM
3. HARMONICS AMPLITUDES EVALUATION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-79900 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:38:35Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Zhiglo, A.V. 2015-04-06T16:42:22Z 2015-04-06T16:42:22Z 2001 Stochastic resonance in symmetric double well: higher harmonics / A.V. Zhiglo // Вопросы атомной науки и техники. — 2001. — № 6. — С. 251-254. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 05.40. +j https://nasplib.isofts.kiev.ua/handle/123456789/79900 The work deals with the phenomenon of Stochastic Resonance in its genuine model, proposed by B. McNamara and K. Wiesenfeld for explanation of long-term climatic changes on Earth. It is shown that in two state model the higher harmonics behave in a non-monotonous way with increase of the noise level, possessing one or more maxima. Explicit formulae for third and fifth harmonic amplitudes and corresponding SNR are obtained. Studied by other authors peculiarities, like dips and sharp peaks in output signal do not occur in two state model, thus they only exist in systems with continuous configuration space. I am glad to acknowledge the participation of Yu.L. Bolotin in both the putting of the problem and further elaboration of questions needed consideration as well as for his support in search for relevant literature and inspiring discussions. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Anomalous diffusion, fractals, and chaos Stochastic resonance in symmetric double well: higher harmonics Стохастический резонанс в симметричной двойной яме: высшие гармоники Article published earlier |
| spellingShingle | Stochastic resonance in symmetric double well: higher harmonics Zhiglo, A.V. Anomalous diffusion, fractals, and chaos |
| title | Stochastic resonance in symmetric double well: higher harmonics |
| title_alt | Стохастический резонанс в симметричной двойной яме: высшие гармоники |
| title_full | Stochastic resonance in symmetric double well: higher harmonics |
| title_fullStr | Stochastic resonance in symmetric double well: higher harmonics |
| title_full_unstemmed | Stochastic resonance in symmetric double well: higher harmonics |
| title_short | Stochastic resonance in symmetric double well: higher harmonics |
| title_sort | stochastic resonance in symmetric double well: higher harmonics |
| topic | Anomalous diffusion, fractals, and chaos |
| topic_facet | Anomalous diffusion, fractals, and chaos |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79900 |
| work_keys_str_mv | AT zhigloav stochasticresonanceinsymmetricdoublewellhigherharmonics AT zhigloav stohastičeskiirezonansvsimmetričnoidvoinoiâmevysšiegarmoniki |