World Journal of Mechanics, 2012, 2, 113-124
doi:10.4236/wjm.2012.22013 Published Online April 2012 (http://www.SciRP.org/journal/wjm)
Oscillator with Random Mass
Moshe Gitterman
Department of Physics, Bar Ilan University, Tel Aviv, Israel
Email: gittem2001@yahoo.com
Received January 12, 2012; revised February 20, 2012; accepted March 1, 2012
ABSTRACT
We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscilla-
tor for some random time after the collision (Brownian motion with adhesion). This is another form of a stochastic os-
cillator, different from oscillator usually studied that is subject to a random force or having random frequency or ran-
dom damping. We calculated first two moments for different form of a random force, and studied different resonance
phenomena (stochastic resonance, vibration resonance and “erratic” behavior) interposed between order and chaos.
Keywords: Harmonic Oscillator; Fluctuating Mass; Stability Conditions; Resonance Phenomena
1. Introduction
The simplest, most general and the most widely used
model in physics is the harmonic oscillator. This model
has been applied everywhere, from quarks to cosmology.
Moreover, a person who is worried by oscillations in the
stock market can relax to classical music produced by the
oscillations of string instruments. The ancient Greeks
already had a general idea of oscillations and used them
in musical instruments. Regarding practical applications,
we note the Galilean discovery of the universality of the
period for small oscillations, which was used in 1602 for
measuring the human pulse. Many other applications
have been found in the last 400 years. Our interest here is
centered on the influence of noise on the harmonic
oscillator [1].
All of Newtonian mechanics is encapsulated in the
basic equation, 22
=d d,
F
mxt where the force F ap-
plied to a particle of mass m, situated at position x(t) at
time t, causes the particle to accelerate. The goal of
mechanics is to find x(t), the position of the particle at
any future time, given its position and velocity at some
initial time t0.
The quintessential example of a force is the one-di-
mensional harmonic oscillator, for which the force F =
kx attempts to return the oscillator to its equilibrium po-
sition. Inserting this force into Newton’s equation gives
22
dd=mxt kx (1)
which is easily solved to yield
=sinxC
twt
(2)
where 2=km
is the angular frequency of osci-
are determined
by the initial position and initial velocity of the oscillator.
Textbooks often present a generalized version of the har-
monic oscillator by including a velocity-dependent fri-
ctional force, with friction constant ,
2
dd
2=d
d
x
x
mkx
t
t
 (3)
which also can be easily solved to yield


1
=exp cos
2
t
xC t
tm

 (4)
where the oscillator is seen to be damped exponentially

by the frictional force and the angular frequency
(222
1=4

) is somewhat reduced.
orld were to consist soleIf the whole wly of uncoupled
ha
n
th
rmonic oscillators, the subject of mechanics could end
right here. However, most mechanical systems are much
more complicated. Among the generalizations of the
simple harmonic oscillator that have been considered in
recent years is the stochastic oscillator, which is an
oscillator that is subject to random external influences.
There are different ways of including fluctuations i
e oscillator model. These may arise from internal
fluctuations (thermal noise) described by

22
ddxx
2=
d
d
x
t
mt m
t
 (5)
The random force

t
uatio
appearing on the right-hand
side of the oscillator eqn describes Brownian motion.
An additive random force, originating from the random
number of molecules of the surrounding medium that
collide with the Brownian particle from opposite sides,
results in random zigzag motion. There are many books
llation and the two constants C and
Copyright © 2012 SciRes. WJM
M. GITTERMAN
114
describing different aspects and many applications of
Brownian motion [2]. External fluctuations have a dif-
ferent origin, connected with random changes of the os-
cillator parameters manifested as random frequency and
random damping. The former is described by the follow-
ing equation (with =1m),

2
d2
d1=0
dtx
t

 


(6)
The many appliof this model include different
fie
2d
xx
t
cations
lds in physics, such as wave propagation in a random
medium [3], spin precession in a random external field
[4], turbulent flow on the ocean surface [5], and as well
as in biology (population dynamics [6]), in economics
(stock market prices [7]) and so on. The case of random
damping is described by the equation

2
ddxx
2
1=
0
d
dtx
t
 
 
 (7)
This equation was fi used [8] to analyze water waves
in
ssibility for
in
2
t
rst

fluenced by a turbulent wind field. However, this
equation, with the coordinate x and time t replaced by the
order parameter and coordinate, respectively, transforms
into the Ginzburg-Landau equation with a convective
term which describes phase transitions in a moving sy-
stem [9]. There are many problems in which the particle
advected by the mean flow passes through the region
under study, including phase transitions under shear [10],
open flows of liquids [11], dendritic growth [12], chemi-
cal waves [13], motion of vortices [14], etc.
In this article, we discuss still another po
troducing randomness in the oscillator equation, namely,
by introducing a random mass [15-19], which is des-
cribed by the following equation
2
dx2
2
d
1=0
d
d
x
tx
t
t


(8)
There are ma in chemical and biological
so


ny situations
lutions in which the surrounding medium contains mo-
lecules which are capable not only of colliding with the
Brownian particle, but also adhering to it. A multipli-
cative random force arises from the adhesion of surr-
ounding molecules which stick to the Brownian particle
for some (random) time, thereby changing its mass. Mo-
dern applications of such a model include a nano-me-
chanical resonator which randomly absorbs and desorbs
molecules [20]. The diffusion of clusters with randomly
growing masses has also been considered [21]. There are
some applications of a variable-mass oscillator [22]. The
oscillator equation may contain both a multiplicative
random force

t
, as in Equation (8), and an additive
random force

t
 
2
2
2
dd
1
d
d
xx
tx
t
t


 =
t
(9)
Equation (9) describes Brownian motion wi
sion. There are many other applications of an
wi
In the following we will consider noise
th adhe-
oscillator
th a random mass [23], such as ion-ion reactions [24-
26], electrodeposition [27], granual flow [28-30], cos-
mology [31-34], film deposition [35], traffic jams [31,32],
and the stock market [38,39].
2. White and Colored Noise

t
with
=0t
having the correlator

12 12
=tt rtt

ls characterize the fluc
(10)
Two integratuations: the strength
of the noise D
 
0
=dDzttz

(11)
and the correlation time ,
 
0
1
=dzz
ttz
D

(12)
2.1. White Noise
Traditionally one considers two different forms of noise,
. For white noise, the function white and colored noise
12
rt t
has the form of a delta-function,

12 12
=tt Dtt
 
(13)
e “white” noise comes from the The namfact that the
Fourier transform of (13) is “white”, being c
out
onstant with-
any characteristic frequency. Equation (13) means
that noises
1
t
and
2
t
are statistically indepen-
dent, no matter how close 1
t and 2
t are. This extreme
assumption, which leads to a non-physical infinite value
of
2t
in (13), means that the correlation time
is not zero, as assumed in (13), but smaller than all other
characteristic times in the problem. It is clear that for our
problem, the noise in (8) cannot be white since a large
negative noise,
0,t
implies a negative mass of the
oscillator.
2.2. Colored Noise
All non-white sources of noise are called colored noise.
type of noise, the so-called dicho- We consider a special
tomous noise (random telegraph signal), which randomly
jumps between two different values, either
(sym-
metric dichotomous noise) or
A
and B (asymmetric
di- chotomous noise), which are characterized by the
Orn- stein-Uhlenbeck correlation function. For the
symmetric noise, one has the following form

2
12 12
=exptt tt
 
 

(14)
Copyright © 2012 SciRes. WJM
M. GITTERMAN 115
White noise (13) is defined by its strength
Ornstein-Uhlenbeck noise is characterized
ram
D while the
by two pa-
eters, 2
and .
The transition from the Ornstein-
Uhlenbeck noise (14) to white noise (13) occurs in the
limit 2
 and ,
 with 2=D

in (14).
Returning to our Equation (8) and multiplying it by

1,t
one obtains



2
22
2
dd
11 =0
d
xtx
t


 

(15)
dt

Since the oscillator mass is positive, the condition
should be satisfied in studies of Equatio
2<1
For the asym
n (15).
ptotic case of small oscillations of the mass,
21, Equation (15) can be rewritten as

2
22
2
dd d
=
dd
d
x
x
tx
tt
t


 

(16)
Therefore, the small dichotomous fluctuations
are equivalent to simultaneous fluctuations o
que
of mass
f the fre-
ncy and the damping coefficient.
For asymmetric dichotomous noise, it is convenient to
replace

t
in Equation (8) by a positive random force

2,t
which corresponds to the fact that the mass of
the Brownian particle can only increase due to the adhe-
sion of the molecules of the surrounding medium,
 
2
22
2
dd
1=
d
d
xx
txt
t
t



 (17)
The quadratic noise
2t
can be written as

22
=t
 (18)
with 2=
A
B
and .=
A
B Indeed, for =
A
,
=
one obtains

=
22
A
BABA A
, and for =,B
22
B =
Then, Equation (17) takesllowing form,
the fo
 
2
22
ddxx
 
  (19)
Multiplying Equation (19) with
2=
d
d
tx
t
t
t

1
=0
by
one obtains

2
1,t



2
22
dd d
1=
d
xx x
Rx
t


 

2
2d
dx
t
t
 



(20)
where 2

2
22
=1 .R


In the next two Sections we calculate two first
ts, momen
x
and 2
x
fuations (15) and (19) or Eq
To split the correlations, we use the well-known Shapiro-
Loginov procedure [40] which yields for exponentially
correlated noise (14),
dd
=
d
g
dd
=
dd
g
g
tt




(22)
and
dd
=d
d
n
n
n
g
g
t
t


 (23)
d
g
g
tt

(21)
or
If dd= ,gt A
Equation (21) becomes
2
d=
d
g
g
t

anr white noise (
(24)
d fo2
 and
 with
2=D

), one gets
=
g
D
First M
3.1. White Noise
Equation (8) can be rewritten in the following form
(25)
3. oments
2
d2
2
2
d d
=
dd
2
d
x
xx
x
tt
 
 (26)
t
Based on linear response theory, the output
x
t of
the system to the input 22
dd
x
t
is
 
2
0
1
d
1
=exp sin
2
tx
tu u u

 



12d
d
xt
u
tt

(27)
where 22
1=4

.
Finding 22
dd
x
t from Eq
ation (27), inserting it into Equation (26) and using the
well-known formula for splitting the correlations,
u-
 
2
dd
=
xx
tt t
2
11 1 1
dd
tt t
tt
 
(28)
one obtains for white noise,

2
2
2
dd
1=
d
d
Dxx x
t
t


0
tion of oscillator’s mass.
3.2. Symmetric Dichotomous Noise
m
(29)
which means the renormaliza
Averaging Equation (15) over an ensemble of rando
functions
t
and using Equation (22) with =
g
x
leads to

2
22
2
dd
1d
d
x
2
d=
d
x
t
t





)
x
t

 


 


(30
where we assume white-noise correlations of noises
Copyright © 2012 SciRes. WJM
M. GITTERMAN
116

t
and that is,

,t
 
11
=.tt tt
 
The netion w func
x
quatio
enters Equation (30). One
can obtain a second en for the two functions
x
and
x
by multiplying Equation (15) by
t
and
averaging, using again Equation (22) with =
g
x and
=d d
g
xt,

2
d
22
22
d
1dd
d=0
d
x
tt
x
t
 
 

 









()

31
The use of dichotomous noise offers a major advan-
tage over other types of colored noise by terminating an
infinite set of higher-order correlations, using the
that
fact

22
=.t
Eliminating
x
from Eq
(30) and (31), one obtains the following cumbersome
eq
uations
uation for the first moment

x
t
 





22
22
22 2
23
d
=1
43
22
43
dd
1221
dd
2
22d
12
2
d
d
2
x
x
tt




x

 t
 

 x
t
x

 
 








(32)
3.3. Asymmetric Dichotomous Noise
Starting from Equation (19) with =0
, using Equation
(23) with and averaging ovse yields
=2n,er noi

2
22
2
2
d
dd
1d
d
=0
d
x
t
t




(33)
A second equation for the two functions
x
t






x
and
x
can be obtained from Equation (20) by using
Equation (23) (with and
=2n=1n),

2
22
2
2
32
dd =0
dd x
tt



 





)
dd
1d
d
Rx x
t
t



 




(34
Excluding the correlator
x
wing




 

from Equations (33)
and (34), one obtains a follo fourth-order diffe-
rential equation for
x

4
2
22
4
3
222
3
2
222222
22222
d
1d
d
22121d
d
3
d
1=0
x
t
x
t
2
22 2
12 d
d
12
x
t
x
 




t
x
  
  
 





 


 



(35)
4. Second Moments
We restrict ourselves to the case of symmetric dichoto
mous noise. One can rewrite the second-order differential
Equation (9) as an equivalent system of two first-order
differential equations
-
2
ddd
=; =
ddd
xyy
yy
ttt
x


(36)
Equation (36) by Multiplying the first of2
x
and the
second by 2
y
yields
22 22
ddd
=2 ;222=2
ddd
y
x
xy yyyxyy
ttt
 
 (37)
Averaging this equation by using (25) yields
2
2
222
d
dd
2
dd
x
tt





(38)
1
d=2 ;
d
=2
d
xxy
t
yy D
t
In deriving (38), we assumed that

t
is white
with the correlator
noise

=tt Dtt
 
12121 (39)
Analogously, multiplying Equations (36) by y and x,
respectively, summing and averaging the sum leads to
22
dd
=
dd
y2
x
yyxxyx x
tt
 
  (40)
which yields after averaging
2222
d
=d
d
dxy
t
y
xy yxyx
t
 

  


(41)
Equations (38) and (41) contain new correlators 2
y
.
x
y
and One can calculate these and the analogous
correlator 2
x
by multiply
by 2,2
ing Equations (36) and (40)
x
y
and ,
respectively, and averaging,
Copyright © 2012 SciRes. WJM
M. GITTERMAN 117
2
d=2
d
x
xy
t
 


 (42)
22 22
dd
22
dd
yyxy
tt
 

 

 =0
(43)
22 2222
d
dxy
t
xyyx
 




(44)
d
=d
y
t
In the case of dichotomous noise, we spli
order correlators into lower-order correlators by using
t the higher-
22
=
, so that, for example, 22
y
22
=.y
By
(44), this m
for the six
eans we obtain six equations, (38) and (40) -
variables 2,
x
2,y ,
x
y 2,
x
2,y
and .
x
y
namic equat
We will not write down the cumbersome
dyions for the second moments, w
be easily obtained from this system of differential e
tions, but shall restrict our attention to the limiting case
of
hich can
qua-
white noise which gives
21
=D
x2
es the well-known
own
andom frequency (Equation
(6)) and random damping (Equation
spectively,
(45)
This result coincidwithresult for
pure Brian motion. The independence of the stati-
onary results on the mass fluctuation is due to the fact
that the multiplicative random force appears in Equation
(8) in front of the higher derivative. It is remarkable that
these results are significantly different from the station-
ary second moments for the r
(7)), which are, re-


22
11
2
22
=:=
12
2
DD
xx
D
D

 
(46)
showing the “energetic” instability [41].
It turns out that, in the presence of dichotomous os-
cillator mass fluctuations, the stationary second moment
2,
x
in contrast to its white noise form (45), may lead
to instability, 2<0.x
4.1. Correlation Function
The correlation function can be found along the
lines as was done for the second moment by multiplying
same
Equations (36) by

1
x
t and averaging the resulting
equations, which gives
 
11
d=
d
x
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.
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