ls1b ws50">txt xtyt (47)
t
  
 
1
dd
=
dd
xt ytt
t1
xt yt
t
2
11
x
tytxtxt


The new correlator

1
txtyt
can be found
by using Equation (28ding to ) lea
   
   
    
11
2
2
11
2
11
d=;
d
dd
dd
=0
txtxttxtyt
t
txt xtxt yt
tt
txtyttxtxt
 
 
 








(48)
nd (48), one can
e correlation function
From Equations (47) afind the fourth-
order differential equation for th
1,
x
txt which, due to the linearity of this equation,
coincides with Equation (32) for the first moment.
For dichotomous noise, the correlation function shows
a non-monotonic dependence on both the noise strength
2
and the inverse correlation time 1.
4.2. Polynomial Dichotomous Noise
Pr
ost general
owing
eviously we treated linear and quadratic dichotomous
fluctuations of the oscillator mass. Here we consider the
m case of polynomial dichotomous noise [42],
which transforms the oscillator equation to the foll
form
 
2
=1 2
dd
1
d
d
nk
kk xx
at t
t




2=
x t

(49)
where
t
is white noise

11212
=Dtt
tt

(50)
For asymmetric dichotomous noise
t
, one gets
[42]
 
=
kkk
bc
tgt gt
tgt
(51)
where
 
=;=
k
kk
k
kk
A
B
bc
BAAB
A
BAB
 
(52)
It is easy to check that for Equation (51) re-
du the
by
=2k
ces to Equation (18), after multiplying latter equ-
ation
g
t and averaging. ogous to the pre-
vious analysis, Equation (49) written as
order differential equations,
Anal
can be retwo first
 
2
=1
dx=;
d
dd d
=
dd d
nk
kk
y
t
yy y
at yxt
tt t
(53)
 
 
Multiplying the first equation by 2
x
and the second
by 2
y
aneraging one gets after using Equations (51)
and (23)
d av
Copyright © 2012 SciRes. WJM
M. GITTERMAN
Copyright © 2012 SciRes. WJM
118

2
d=2 ;
d
xxy
22 2
2
22
=1 1
dd d
d
=222
dd d
n
kk k
t
yy y
ab cD
xy
ty y
ttt






(54)
Equation (54) contains the new correlator
d
k
t
2
y
.
Equa- One can calculate this correlator by multiplying
tions (53) by 2
x
and 2,
y
which gives after ave-
raging

2
2
12
22
=1
d=2
d
dd
2= 22
dd
nk
kk
xy
x
t
a
x
y
ty
yy
tt





 
 
 
 
(55)
last equation in Equation (55),
or, by inserting (51) with k replaced by 1k into the
2
22
=11 1
d
2d
d
=d
n
kkk k
y
t
abc
yy
t






 


 (56)
iplying Equations (53) by y and x, respectively,
and summing these equations one gets by using Equation
(23),
Mult


22
=1
d
2d
d
=2 22nkk
kk
xy
t
ya xyyxyx
 
dt


 


(57)
using Equation (51),


2
2
22
xy
y


which yields, after averaging and

2 2
22 2
=1
dd
2=2
dd
n
kkkk kk
yab cbc
xyxy xyxy
yy
x
tt



 




(58)
Multiplying Equation (57) by
and averaging, one obtains


2
222 2
=11 111
dd
2=2
dd
n
kkk kkk
abcbc
xyxy xyxy
yyy
x
tt
 


 

 

 

 

(59)
In this way we obtain six equations, (54)-(56) and
(58)-(59), for the six variables
2,
x
2,y ,
x
y
2,
x
2,y
and .
x
y
W these e will not solve
cumbersome dynamic equations, but prese
solution for the stationary
nt only the
(dd =0t) second moment
2,
x



22
143 2
1
23
2
22
=2
2
SS WS
DS
xUW V
 


 



and
(60)
where
=1=11 2
nn
kk kkk
abab S
 
(61)
1
=13=11 4
;
k
nn
kkk kkk
S
ac SacS

 


2
2
2
14
2
14
2
22
=;
=2 ;
2
=22 2
S
U
SS
VS
SS
WS
  








(62)
As one can see from Equation (60), the polynomial
the oscilladichotomous fluctuations of tor mass can lead
to instability, 2<0,
x for sompara-
s
e values of the
meter.
5. Resonance Phenomena
The simplest example of mechanical resonance is a har-
monic oscillator subject to a periodic force, where the
steady-state amplitude of the oscillator approaches in-
finity when the external force frequency approaches the
eigenfrequency of the oscillator. This phenomenon was
probably already known to the ancient Egyptians who
invented the water clock, but the classical demonstration
of dynamic resonance are quite recent architectural flaws
M. GITTERMAN 119
uncovered in the US. The first was the Takoma bridge
which was destroyed by the wind force at the resonance
frequency, and the second was the Paramount Commu-
nication Building in New York where the winds
the top floors and pried windows loose from their case-
ments.
One of greatest achievements of twentieth-century phy-
sics was establishing a deep relationship between deter-
m
sound c
istic and half-random terms. However, this im-
pression is faulty due to the close connection between de-
ugh apparently different
twisted
inistic and random phenomena. The widely studied
phenomena of “deterministic chaos” and “stochastic re-
sonance” mightontradictory, consisting of half-
determin
terminism and randomness, altho
forms of behavior [43].
Here we consider a new manifestation of the reso-
nance of an oscillator. The dynamic equation of motion
of a bistable underdamped one-dimensional oscillator
driven by a multiplicative random force

,t

an addi-
tive random force

,t

and two periodic forces,

sin
A
t
and

sin ,Ct has the following form

 
2
23
0
2
dd
d
d
=sinsin
xx
x
tx bx
t
t
tA tCt


 

(63)
The dynamic resonance mentioned above corresponds
to == == =0bC
and 0.
Let us consi-
der some other limiting cases of Equation (63).
1) Brownian motion (0= = ===0bAC
) has
been studied most widely with many applications. The
equilibrium distribution comes from the balance of two
contrary processes: the random force which tends to
increase the velocity of the Brownian particle and the
damped force which tries to stop the particle [1].
2) The double-well oscillator with additive noise
(===0AC
) and small damping, ,
shows
two or three peaks in the power spectrum (Fourier com-
ponent of the correlation function) descriptive of fluctu-
at
mped
ion transitions between the two stable points of the
potential, small intra-well vibrations and the over-the-
barrier vibrations [44].
3) Stochastic resonance (SR) in overda
(22
dd==xt C
nd underdamped (==C
=0
) a0
)
oscillators is a very interest
nomenon, where the noise increases a weak input signal.
SR occurs in the case that a deterministic tim
the external periodic field is synchronized with a stocha-
st
ing and counterintuitive phe-
e-scale of
ic time-scale, determined by the Kramer transition rate
over the barrier.
4) Stochastic resonance in a linear overdamped osci-
llator (22 =0C), as distinct
d d===xt b
m the
nonlinear case, allows an exact solution [45,46]. How-
ever, this effect occurs only when the multiplicative
noise

t
fro
is colored and not white.
5) Vibrational resonance
==0,

which occurs
in a deterministic system, manifests itself in the enhance-
ment of a weak periodic signal through a high-frequency
pe
ochastic reson
riodic field, instead of through noise, as in the case of
stance.
6) “Erratic” behavior shows up as a “random-like”
phenomenon in a simple system
22
dd= b===0xt

with two incommensurate ex-
ternal frequencies,
and .
5.
d
become c
ation (19) with
1. Stochastic Resonance
Noise, which always plays a distractive role, appears as a
constructive force, increasing the output signal as a fun-
ction of noise intensity. This phenomenon was proposed
as the explanation of the periodicity of the ice ages [47,
48] and has found many applications [49].
The standard definition of stochastic resonance (SR) is
the non-monotonic dependence of an output signal, or
some function of it, as a function of some characteristic
of the noise or of the periodic signal [49]. At first glance,
it appears that all three ingredients, nonlinearity, periodic
forcing an random forcing, are necessary for the app-
eahas rance of SR. However, it lear that SR is
generated not only in a typical two-well system, but also
in a periodic structure [50]. Moreover, SR occurs even
when each of these ingredients is absent. Indeed, SR
exists in linear systems when the additive noise is re-
placed by nonwhite multiplicative noise [46]. Determi-
nistic chaos may induce the onset of SR instead of a
random force [49]. Finally, the periodic signal may be
replaced by a constant force in underdamped systems
[51].
Consider the linearized Equ=0
of
ss subject to an oscillator with random ma an external
periodic field,
 
2
22
2
dd
12=sin
d
d
xx
txat
t
t
 

 
 (64)
Repeating the procedure leading to Equation (35), one
obtains a fourth-order differential equation for ,
x




 


 



4
2
22
4
3
222
3
2
222222
2
22 2
22222
222 22
22
d
1d
d
22121d
d
12 3d
d
12
d
1
1sin
21 cos
x
t
x
t
x
t
x
t
x
at
at


 
 







 


 











  

(65)
Copyright © 2012 SciRes. WJM
M. GITTERMAN
120
In a similar way, one can obtain the equation for the
second moment 2,
x
associated with Equation (9),
which is transformed into six equations for six variables,
2,
x
2,
y
,
x
y, 2,
x
2
y
and ,
x
y
but
s. we shall not write down these cumbersome equation
ti
Analogous to the cases of random frequency and ran-
dom damping [47], we seek the solution of Equaon (65)
in the form

=sinAt
x
 (66)
One easily finds
1/2
22
56 5768
1
22
58 67
78
=; =
tan
f
fff
Aff
f
fff
ff
 

 

 (67)
with



2
54263
322
7213 4
222222
84 2143
422
12
222
4
=;=;
=2 22
=2
=
fffaf fa
2
3

12
2
32 2
=1 ;=1;
=2;
f
ffff f
ff ffff
ff
f

 
 



ff
ff f

 



One can compare Equations (66)-(68) with the equa-
tions for the first moment
(68)
x
, obtained [52] for the
cases of random frequency and random damping, re-
spectively, subject to symmetric dichotomous noise, and
extended afterwards [53,54] to the case of asymmetric
noise. All these equation are of fourth order with the
same dependence on the frequency
fie tly different dependence on the pa-
ra
5.2. Vibrational Resonance
stochastic resonance, vibrational resonance mani-
tself in the enhancement of a weak periodic signal
through a high-frequency periodic field, instead of through
noise as in the case of stochastic resonance. The deter-
ministic equation of motion then has the following form,
of the external
ld but with a sligh
meters of the noise.
Like
fests i
 
2
23
0
2
dd =sin sin
d
d
xx
x
bxAt Ct
t
t
 
 (69)
Equation (69) describes an oscillator moving in a sym-
metric double-well potential

22
0
=2Vx x
44bx
with a maximum at and two minima
=0x
x
with
the depth d of the wells,
24
00
==
4
xd
bb
(70)
The amplitude of the output signal as a function of the
amplitude C of the high-frequency field has a bell shape,
showing the phenomenon of vibrational resonance. For
close to the frequency 0
of the free oscillations,
there are two resonance peaks, whereas for smaller
,
there is only one resonance peak. These different results
correspond to two different oscillatory processes, jumps
tions inside one well. between the two wells and oscilla
Assuming that ,
resonance-like behavior (“vi-
brational resonance” [55]) manifests itself in the response
of the system at the low-frequency
, which depends
on the amplitude C and the frequency of the high-
frequency signal. The latter plays a role similar to that of
noise in SR. If the amplitude C is larger than the barrier
height d, theperiod field during each half π
other. transfers the syste one to the
M
m frompotential well
oreover, the two frequencies
and are similar
to the frequencies of the period
rate of jumps betweetwo minima of the under-
damped osllator. Therefore by choosing an appropriate
relation beteen the input si
ithe Kramers
e
ci ,
wgnal
c signal a
sin
nd

n th
A
t
and the am-
plitude C of thor t
one can obtain a non-monotonic depende
pu
e large signal (he strength of the noise)
nce of the out-
t signal on the amplitude C (vibration resonance) or on
the noise strength (stochastic resonance). To put this
another way [56], both noise in SR and the high-fre-
quency signal in vibrational resonance change the para-
meters of the system response to a low-frequency signal.
Let us now pass to an approximate analytical solution
of Equation (69). In accordance with the two times scales
in this equation, we seek a solution of Equation (69) in
the form
 
2
sin
=Ct
xt yt
(71)
where the first term varies significantly only over times
t, while the second term varies much more rapidly. On
substituting Equation (71) into (69), one can average
over a single cycle of

sin .t Then, odd powers of
sint
vanish upon averaging, while the
2
sin t
term gives 12. In this way, one obtains the following
equation for
,
y
t
22
23
0
24
dd 3n
d
d2
yy bC
y t
t
t=siby A


 

 (72)
with
224
0
0
224
0
32
=0; =;
32
=4
bC
yy b
bC
db





(73)
One can say that Equation (72) is the “coarse-grained”
version (with respect to time) of Equation (69). For
242
32>,C0
the phenomenon of dynamic stabili-
zation [57] occurs, namely, the high-frequency external
Copyright © 2012 SciRes. WJM
M. GITTERMAN 121
field transforms the previously unstable position =0
into a stable position.
Seeking the solution of Equation (72) of the form
 
sinyt yt
  (74)
and linearizing Equation (72) gives

in
2
22 22
1
=A


(75)
where

22
22
10
4
3
=3
2
bC by


A resonance in the linearized Equation (72)
when
(76)
occurs
1=,
h
which, after substituting in
leads to te following relations between th
and frequencies of the two driving fields w
the resonant behavior,
Equation (72),
e amplitudes
hich produce
22
22
0
422
33
=24
bC bA


(77)
In addition to the resonance phenomenon, one can
study [58] the influence of the positions and depths of the
potential on the vibrational resonance. Assuming that
2=,b
which means
0, according to Equation (73),
sitions of minima remain fixed, let us raise the q
that
ue-
maximal.
po
stion for
of the out
which value of a control parame
put signal to the input signal A is
ocn
2
ter C the ratio
According to Equation (75), this curs whe

2
22 2
1
=S
 is minimal, which
by the condition
is determined
dd=0,SC which, using (76)
, results in
with
2=b
0


2
22 1
1
22
2
44
=3
=0
2bby




or, for 0=0,y
d
2d
33
C
bC bC


(78)

4
22
2
Cb=3b
and for
(79)
22
0
4
3
=,
2
C
yb


4
22
=2
3
Cb
b
(80)
tion (80) has real solutions for C only if
2
2> .bEqua
Thus far, we considered equal values of two control
parameters, 2
0=b
hanging the depths of potential
and keeping the positions of minima
c
x
unaltere
logously, one can assu
the distance between m
or
d. Ana-
me that 4
0=b
changing thereby
inima and not the potential depth.
Then, one obtains, f
0=0y,
22
22
=2
3
Cb



(81)
and for
22
0
4
3
=2
C
yb

22
22
=2
3
Cb



(82)
with the proviso that 2
2> .b
lts have beAll the above resuen obtained for an under-
oscillator. It turns out [59,60] that a similar effect
s place for an overdaped oscillator
damped
also takem(22
dd=0xt
ditional addi- in Equation (69)). The influence of the ad
tive noise on the vibrational resonance
tages of the vibrational resonance compared to the sto-
ch
been studd [61].
to E
(Equat
subsequent analysis of an oscil-
lator equation with one periodic force is quite analo-
gous to analysis of Equation (64), wh
stochastic resonance phenomenon.
Equation (69) describes an oscillator moving in sym-
ouble-well potential. The vibrational resonance
in the quintic oscillator with the potential of the form
, and the advan-
astic resonance in the detection of weak signals have
ie
For an oscillator with random mass one has to add two
periodic fieldsquations (15), (19), and perform the
preceding analysis of Equation (69), based on dividing its
solution in the two time scalesion (71)) followed
by the linearization of Equation (72) for the slowly
changing solution. The
ich describes the
metric d

224 6
cx
(83)
0
111
=246
Vxx bx

]. Finally, the vibrational resonance
neering, vi
rential equations are
was studied in [62,63
and an appearance of chaos in the Van der Pol oscillator
were investigated in [64]. Because of the many applica-
tions in physics, chemistry, biology and engi-
brational resonance still attracts great interest, and new
applications will surely be found in the future.
5.3. “Erratic” Behavior
One of the great achievements of twentieth-century phy-
sics was the prediction of deterministic chaos which ap-
pears in the equations without any random force [65].
Deterministic chaos means an exponential increase in
time of the solutions for even the smallest change in the
initial conditions. Therefore, to obtain a “deterministic”
solution, one would have to specify the initial conditions
to an infinite number of digits. Otherwise, the solutions
of deterministic equations show chaotic behavior. Deter-
ministic chaos occurs if the diffe
Copyright © 2012 SciRes. WJM
M. GITTERMAN
122
no
ini-
stic chaos may occur only in the underdamped oscillator.
Here, we present an example of “erratic” be
like deterministic chaos, is drawn midway between deter-
nlinear and contain at least three variables. This points
to the important difference between underdamped and
overdamped equations of an oscillator, since determ
havior, which,
ministic and stochastic behavior.
Consider the simple example of an overdamped oscill-
ator subject to two periodic fields,
 
2
1122
d=cos cos
d
x
x
CtC t
t


(84)
We show that the solution
ratic”, being intermediate bet
s of this equation are “er-
ween deterministic and chao-
tic solutions.
The stationary solutions of Equation (84) have the fo-
llowing form
  
12
12
12
=sin sin
CC
x
ttt

(85)
Replace the continuous time in Equation (84) by dis-
crete times 2
2πn
[66]. The solution of this equation
then becomes

11
212
2π=0sin2π
C
xn xn

 
 
 
(86)
If 12
is an irrational number, the sin factor in (86)
will never vanish and the motion becomes “erratic”. The
properties of “erratic” motion can be understood from the
analysis of the correlation function associated with the
n-th and (n + m)-th points,









12
2πCm

12 12
=0
2
11
1
=2π2π
lim
=
1sin2πsin 2π
lim
N
Nn
N
N
xn x nm
N
C
nnm
 
 
 




ell-known relations between the trigono-
metric functions, one obtains
2
11
0
0
N
x
xC
12 12
=0
1sin2πsin 2π
lim
Nnnn
m
N
 




12 12
=0n
N

(87)
Using the w

2
2
111
2π2π
1
=0 cos
C
Cm xm
21
2
2
 
 


The Fourier spectrum of the correlation function (88)
depends on the ratio
(88)
12
. If this ratio is a rational
number, this spectrum will contain a finite number of
peaks. However, for irrational 12
pical of de
havior ar
hich di
chastic
ple is B
unding m
rticle indu
random
the B
und for
rforme
, the spectrum be-
comes broadband, what is tyterministic chaos.
However, this “erratic” beises from a simple
“integrable” Equation (84), wstinguishes it from
deterministic chaos.
6. Conclusion
We considered a new type of stooscillator which
has a random mass. An examrownian motion
with adhesion, where the surroolecules not only
collide with the Brownian pacing a zigzag mo-
tion, but also adhere to it for a period of time,
thereby increasing the mass ofrownian particle.
The first two moments are fodichotomous ran-
dom noise. An analysis was ped of the “err
m
phenomena are compli-
mentary and not contradictory. Due to many applications
in physics, chemistry, biology and engineering, t
del of an oscillator with random mass will find
ap
Oxford Science Publication, Oxford,
Wave Propagation and Scattering in Ran-
atic”
otion, stochastic and vibration resonances, which shows
that deterministic and random
he mo-
many
plications in the future.
REFERENCES
[1] M. Gitterman, “The Noisy Oscillator: The First Hundred
Years, from Einstein until Now,” World Scientific, Sin-
gapore, 2005.
[2] R. Mazo, “Brownian Motion: Fluctuations, Dynamics,
and Applications,”
2002.
[3] A. Ishimaru, “
dom Media,” John Wiley and Sons, New York, 1999.
[4] R. Kubo “A Stochastic Theory in Line Shape,” In: K. E.
Shuler, Ed., Stochastic Processes in Chemical Physics,
Wiley, New York, 1969, pp. 101-128.
[5] O. M. Phillips, “The Dynamics of the Upper Ocean,”
Cambridge University Press, Cambridge, 1977.
[6] M. Turelli, “Theoretical Population Biology,” Academic,
New York, 1977.
[7] H. Takayasu, A.-H. Sato and M. Takayasu, “Stable Infi-
nite Variance Fluctuations in Randomly Amplified Lan-
gevin Systems,” Physical Review Letters, Vol. 79, No. 6,
1997, pp. 966-969. doi:10.1103/PhysRevLett.79.966
[8] B. West and V. Seshadri, “Model of Gravity Wave
Growth Due to Fluctuations in the Air-Sea Coupling Pa-
rameter,” Journal of Geophysical Research, Vol. 86, No.
5, 1981, pp. 4293-4296. doi:10.1029/JC086iC05p04293
[9] M. GittermanMoving Systems,” , “Phase Transitions in
Physical Review E, Vol. 70, No. 3, 2003, Article ID:
036116. doi:10.1103/PhysRevE.70.036116
[10] A. Onuki, “Phase Transitions of Fluids in Shear Flow,”
Journal of Physics: Condensed Matter, Vol. 9, No. 29,
1997, pp. 6119-6159. doi:10.1088/0953-8984/9/29/001
Copyright © 2012 SciRes. WJM
M. GITTERMAN 123
[11] J. M. Chomaz and A. Couairon, “Against the Wind,”
Physics of Fluids, Vol. 11, No. 10, 1999, pp. 2977-2984.
doi:10.1063/1.870157
[12] F. Hestol and A. Libchaber, “Unidirectional Crysta
rtex Matter Driven by Bias Current,”
l
Growth and Crystal Anisotropy,” Physica Scripta, Vol.
1985, No. 9, 1985, pp. 126-129.
[13] A. Saul and K. Showalter, “Propagating Reaction Diffu-
sion Fronts,” In: R. J. Field and M. Burger, Eds., Oscilla-
tions and Traveling Waves in Chemical Systems, Wiley,
New York, 1985, pp. 419-439.
[14] M. Gitterrman, B. Ya. Shapiro and I. Shapiro, “Phase
Transitions in Vo
Physical Review B, Vol. 65, No. 17, 2002, Article ID:
174510. doi:10.1103/PhysRevB.65.174510
[15] M. Gitterman, “New Stochastic Equation for a Harmonic
Oscillator: Brownian Motion with Adhesion,” Journal o
.
, Article ID:
f
Physics C: Conference Series, Vol. 248, No. 1, 2010, Ar-
ticle ID: 012049.
[16] M. Gitterman “New Type of Brownian Motion,” Journal
of Statistical Physics, Vol. 146, No. 1, 2010, pp. 239-243
[17] M. Gitterman and V. I. Kljatskin, “Brownian Motion with
Adhesion: Harmonic Oscillator with Fluctuating Mass,”
Physical Review E, Vol. 81, No. 5, 2010
051139.
[18] M. Gitterman and I. Shapiro, “Stochastic Resonance in a
Harmonic Oscillator with Random Mass Subject to
Asymmetric Dichotomous Noise” Journal of Statistical
Physics, Vol. 144, No. 1, 2011, pp. 139-149.
[19] M. Gitterman “Harmonic Oscillator with Fluctuating
Mass,” Journal of Modern Physics, Vol. 2, 2010, pp.
1136-1140.
[20] J. Portman, M. Khasin, S. W. Shaw and M. I. Dykman,
“The Spectrum of an Oscillator with Fluctuating Mass
and Nanomechanical Mass Sensing,” Bulletin of the
American Physical Society March Meeting, Portland,
15-19 March 2010.
[21] J. Luczka, P. Hanggi and A. Gadomski, “Diffusion of
Clusters with Randomly Growing Masses,” Physical Re-
view E, Vol. 51, No. 6, 1995, pp. 5762-5769.
doi:10.1103/PhysRevE.51.5762
[22] M. S. Abdalla, “Time-Dependent Harmonic Oscillator
with Variable Mass under the Action of a Driving Force,”
Physical Review A, Vol. 34, No. 6, 1986, pp. 4598-4605.
doi:10.1103/PhysRevA.34.4598
[23] R. Lambiotte and M. Ausloos, “Brownian Particle Having
a Fluctuating Mass,” Physical Review E, Vol. 73, No. 1,
2005, Article ID: 011105.
[24] A. Gadomski and J. Siódmiak, “A Kinetic Model of Pro-
tein Crystal Growth in Mass Convection Regime” Crystal
Research and Technology, Vol. 37, No. 2-3, 2002, pp.
281-291.
doi:10.1002/1521-4079(200202)37:2/3<281::AID-CRAT
281>3.3.CO;2-4
[25] M. Rub and A. Gadomski, “Nonequilibrium Thermody-
namics versus Model Grain Growth: Derivation and
Some Physical Implication,” Physica A, Vol. 326, No. 3-4,
2003, pp. 333-343. doi:10.1016/S0378-4371(03)00282-6
[26] A. Gadomski, J. Siódmiak, I. Santamara-Holek, J. M
lle and C. Soria, “Modeling the Elec-
.
Rub and M. Ausloos, “Kinetics of Growth Process Con-
trolled by Mass-Convective Fluctuations and Finite-Size
Curvature Effects,” Acta Physica Polonica B, Vol. 36, No.
5, 2005, pp. 1537-1559.
[27] A. T. Pérez, D. Savi
trophoretic Deposition of Colloidal Particles,” Europhy-
sics Letters, Vol. 55, No. 3, 2001, pp. 425-431.
doi:10.1209/epl/i2001-00431-5
[28] I. Goldhirsch and G. Zanetti, “Clustering Instability in
Dissipative Gases,” Physical Review Letters, Vol. 70, No.
11, 1993, pp. 1619-1622.
doi:10.1103/PhysRevLett.70.1619
[29] S. Luding and H. J. Herrmann, “Cluster-Growth in Freely
Cooling Granular Media,” Chaos, Vol. 9, No. 3, 1999, pp.
673-682. doi:10.1063/1.166441
I. Temizer, “A Computatio[30] nal Model for Aggregation in a
Class of Granual Materials,” Master Thesis, University of
California, Berkeley, 2001.
[31] W. Benz, “From Dust to Planets,” Spatium, Vol. 6, 2000,
pp. 3-15.
[32] J. Blum, et al., “Growth and Form of Planetary Seedlings:
Results from a Microgravity Aggregation Experiment,”
Physical Review Letters, Vol. 85, No. 12, 2000, pp. 2426-
2429.
[33] J. Blum and G. Wurm, “Experiments on Sticking, Re-
structuring and Fragmentation of Preplanetary Dust Ag-
gregates,” Icarus, Vol. 143, No. 1, 2000, pp. 138-146.
doi:10.1006/icar.1999.6234
[34] S. J. Weidenschilling, D. Spaute, D. R. Davis, F. Marzari
and K. Ohtsuki, “Accretional Evolution of a Planetesimal
Swarm,” Icarus, Vol. 128, No. 2, 1997, pp. 429-455.
doi:10.1006/icar.1997.5747
[35] N. Kaiser, “Review of the Fundamentals of T
Growth,” Applied Optics, Vol.
hin-Film
41, No. 16, 2002, pp.
3053-3060. doi:10.1364/AO.41.003053
[36] T. Nagatani, “Kinetics of Clustering and Acceleration in
1D Traffic Flow,” Journal o
65, 1996, pp. 3386-3389.
f Physical Society Japan, Vol.
3/JPSJ.65.3386doi:10.114
[37] E. Ben-Naim, P. L. Krapivsky and S. Redner, “Kinetics
of Clustering in Traffic Flows,” Physical Review E, Vol.
50, No. 2, 1994, pp. 822-829.
doi:10.1103/PhysRevE.50.822
[38] M. Ausloos and K. Ivanova, “Mechanistic Approach to
Generalized Technical Analysis of Share Prices and Stock
2, pp. 177-187.
Market Indices,” European Journal of Physics B, Vol. 27,
No. 4, 200
doi:10.1007/s10051-002-9018-9
[39] M. Ausloos and K. Ivanova, “Generalized Technical
Analysis. Effects of Transaction Volume and Risk,” In: H.
Takayasu, Ed., The Applications of Econophysics, Sprin-
,”
. 563-574.
ger Verlag, Berlin, 2004, pp. 117-124.
[40] V. E. Shapiro and V. M. Loginov, “Formulae of Differen-
tiation” and Their Use for Solving Stochastic Equations
Physica A, Vol. 91, 1978, pp
[41] M. Gitterman, “Classical Harmonic Oscillator with Mul-
tiplicative Noise,” Physica A, Vol. 352, No. 2-4, 2005, pp.
309-334. doi:10.1016/j.physa.2005.01.008
Copyright © 2012 SciRes. WJM
M. GITTERMAN
Copyright © 2012 SciRes. WJM
124
Peng and M. K. Luo, “Sto-[42] L. Zhang, S. C. Zhong, H.
chastic Multi-Resonance in a Linear System Driven by
Multiplicative Polynomial Dichotomous Noise,” Chinese
Physics Letters, Vol. 28, No. 9. 2011, Article ID: 090505.
[43] M. Gitterman “Order and Chaos: Are They Contradictory
or Complementary?” European Journal of Physics, Vol.
23, No. 2, 2002, pp. 119-122.
doi:10.1088/0143-0807/23/2/304
[44] M. I. Dykman, R. Mannela, P. V. E. McClintock, F. Moss,
and M. Soskin, “Spectral Density of Fluctuations of a
Double-Well Duffing Oscillator
Physical Review A, Vol. 37, No.
Driven by White Noise,”
4, 1988, pp. 1303-1313.
doi:10.1103/PhysRevA.37.1303
[45] A. Fulinski, “Relaxation, Noise-Induced Transitions, and
Stochastic Resonance Driven by Non-Markovian Di-
chotomic Noise,” Physical Revi
1995, pp. 4523-4526.
ew E, Vol. 52, No. 4,
ysRevE.52.4523
doi:10.1103/Ph
[46] V. Berdichevsky and M. Gitterman, “Multiplicative Sto-
chastic Resonance in Linear Systems: Analytical Solu-
tion,” Europhysical Letters, Vol. 36, No. 3, 1996, pp.
161-166. doi:10.1209/epl/i1996-00203-9
-4470/14/11/006
[47] R. Benzi, S. Sutera and A. Vulpani, “The Mechanism of
Stochastic Resonance,” Journal of Physics A, Vol. 14, No.
11, 1981, pp. 453-458. doi:10.1088/0305
[48] G. Nicolis, “Stochastic Aspects of Climatic Transitions—
Response to a Periodic Forcing,” Tellus, Vol. 34, No. 1,
1982, pp. 1-9. doi:10.1111/j.2153-3490.1982.tb01786.x
[49] L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni,
“Stochastic Resonance,” Review of Modern Physics, Vol.
70, No. 1, 1998, pp. 223-287.
doi:10.1103/RevModPhys.70.223
[50] N. G. Stokes, N. D. Stein and V. P. E. McClintock, “Sto-
chastic Resonance in Monostable Systems,” Journal of
Physics A, Vol. 26, No. 7, 1993, pp. 385-390.
doi:10.1088/0305-4470/26/7/007
[51] F. Marchesoni, “Comment on Stochastic Resonance in
Washboard Potentials,” Physics Letters A, Vol. 352, No.
1-2, 1997, pp. 61-64.
doi:10.1016/S0375-9601(97)00232-6
[52] M. Gitterman, “Classical Harmonic Oscillator with Multi-
plicative Noise,” Physica A, Vol. 352, No. 2-4, 2005, pp.
309-334. doi:10.1016/j.physa.2005.01.008
[53] S.-Q. Jiang, B. Wu and T.-X. Gu, “Stochastic Resonance
in a Harmonic Oscillator Fluctuating Intrinsic Frequency
by Asymmetric Dichotomous Noise,” Journal of Elec-
tronic Science and Technology, Vol. 5, No. 4, 2007, pp.
344-347.
[54] S. Jiang, F. Guo, Y. Zhow and T. Gu, “Stochastic Reso-
nance in a Harmonic Oscillator with Randomizing Dam-
ping by Asymmetric Dichotomous Noise,” International
Conference on Communications, Circuits and Systems,
Kokura, 11-13 July 2007, pp. 1044-1047.
[55] P. S. Landa and P. V. E. McClintock, “Vibration
nance,” Journal of Physics A, Vol
al Reso-
. 33, No. 45, 2000, pp.
433-438. doi:10.1088/0305-4470/33/45/103
[56] Y. Braiman and I. Goldhirsch, “Taming Chaotic Dynam-
ics With Weak Periodic Perturbations,”
Letters, Vol. 66, No. 20, 1991, pp. 254
Physical Review
5-2548.
doi:10.1103/PhysRevLett.66.2545
[57] Y. Kim, S. Y. Lee and S. Y. Kim, “Experimental Obser-
vation of Dynamic Stabilization in a Double-Well
Duffing Oscillator,” Physics Letters A, Vol. 275, No. 4,
2000, pp. 254-259. doi:10.1016/S0375-9601(00)00572-7
[58] S. Rajasekar, S. Jeyakumari and M. A. F. Sanjuan, “Role
of Depth and Location of Minima of a Double-Well Po-
tential on Vibrational Resonance,” Journal of Physics A,
.
Vol. 43, No. 46, 2010, Article ID: 465101.
[59] I. Blekhmam and P. S. Landa, “Conjugate Resonances
and Bifurcations in Nonlinear Systems under Biharmoni-
cal Excitation,” International Journal of Non-Linear Me-
chanics, Vol. 39, No. 3, 2004, pp. 421-426
doi:10.1016/S0020-7462(02)00201-9
[60] J. P. Baltanas, et al., “Experimental Evidence, Numerics,
and Theory of Vibrational Resonance in Bistable Sys-
ement of
Optical System: Com-
oafo, “Behavior of
tems,” Physical Review E, Vol. 67, No. 6, 2003, Article
ID: 066119.
[61] V. N. Chizhevsky and G. Giacomelli, “Improv
Signal-to-Noise Ratio in a Bistable
parison between Vibrational and Stochastic Resonance,”
Physical Review A, Vol. 71, No. 1, 2005, Article ID:
011801.
[62] S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A.
F. Sanjuan, “Single and Multiple Vibrational Resonance
in a Quintic Oscillator with Monostable Potentials,”
Physical Review E, Vol. 80, No. 4, 2009, Article ID:
046608.
[63] S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A.
F. Sanjuan, “Analysis of Vibrational Resonance in a
Quintic Oscillator,” Chaos, Vol. 19, No. 4, 2009, Article
ID: 043128.
[64] J. C. Chedjou, H. B. Fotsin and P. W
the Van der Pol Oscillator with Two External Periodic
Forces,” Physica Scripta, Vol. 55, No. 4, 1997, pp. 390-
393. doi:10.1088/0031-8949/55/4/002
[65] H. G. Schuster and W. Just, “Deterministic Chaos: An
Introduction,” Wiley, New York, 2005.
doi:10.1002/3527604804
[66] E. Ott, “Chaos in Dynamical Systems,” Cambridge Uni-
versity Press, Cambridge, 2002.