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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