Oscillator with Random Mass

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 random 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.


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, where the force F applied 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 t 0 .
The quintessential example of a force is the one-dimensional harmonic oscillator, for which the force F = -kx attempts to return the oscillator to its equilibrium position.Inserting this force into Newton's equation gives 2 2 which is easily solved to yield where 2 = k m  is the angular frequency of osciare determined by the initial position and initial velocity of the oscillator.
Textbooks often present a generalized version of the harmonic oscillator by including a velocity-dependent frictional force, with friction constant , which also can be easily solved to yield where the oscillator is seen to be damped exponentially by the frictional force and the angular frequency ( 2 ) is somewhat reduced.orld were to consist sole If the whole w ly 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 The random force appearing on the right-hand side of the oscillator eq n 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  describing different aspects and many applications of Brownian motion [2].External fluctuations have a different origin, connected with random changes of the oscillator parameters manifested as random frequency and random damping.The former is described by the following equation (with The many appli of this model include different fie 2 d x x 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 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 system [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], chemical 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][16][17][18][19], which is described by the following equation There are ma in chemical and biological so ny situations lutions in which the surrounding medium contains molecules which are capable not only of colliding with the Brownian particle, but also adhering to it.A multiplicative random force arises from the adhesion of surrounding molecules which stick to the Brownian particle for some (random) time, thereby changing its mass.Modern applications of such a model include a nano-mechanical 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 Equation ( 9) describes Brownian motion wi sion.There are many other applications of an wi In the following we will consider noise th adheoscillator th a random mass [23], such as ion-ion reactions [24][25][26], electrodeposition [27], granual flow [28][29][30], cosmology [31][32][33][34], film deposition [35], traffic jams [31,32], and the stock market [38,39].

White and Colored
Two integra tuations: the strength of the noise D and the correlation time ,

White Noise
Traditionally one considers two different forms of noise, .For white noise, the function white and colored noise e "white" noise comes from the The nam fact that the Fourier transform of ( 13) is "white", being c out onstant withany characteristic frequency.Equation (13)  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.

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   (symmetric 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 White noise ( 13) is defined by its strength Ornstein-Uhlenbeck noise is characterized ram D while the by two paeters, 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 Since the oscillator mass is positive, the condition should be satisfied in studies of Equatio The quadratic noise  can be written as Multiplying Equation (19) where In the next two Sections we calculate two first ts, momen x and 2 x f uations (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), and or

26) t
Based on linear response theory, the output   x t of the system to the input x t from Eq ation (27), inserting it into Equation ( 26) and using the well-known formula for splitting the correlations, u- one obtains for white noise, tion of oscillator's mass.

Symmetric Dichotomous Noise
which means the renormaliza Averaging Equation ( 15) over an ensemble of rando functions   t  and using Equation ( 22) with = g x leads to where we assume white-noise correlations of noises The ne tion w func x  quatio enters Equation (30).One can obtain a second e n 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

31
The use of dichotomous noise offers a major advantage over other types of colored noise by terminating an infinite set of higher-order correlations, using the that fact (30) and ( 31), one obtains the following cumbersome eq uations uation for the first moment

Asymmetric Dichotomous Noise
Starting from Equation ( 19) with = 0  , using Equation ( 23) with and averaging ov se yields A second equation for the two functions x and x  can be obtained from Equation ( 20) by using Equation ( 23) (with and from Equations ( 33) and (34), one obtains a follo fourth-order differential equation for

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 - Averaging this equation by using (25) yields In deriving (38), we assumed that   t  is white with the correlator noise Analogously, multiplying Equations ( 36) by y and x, respectively, summing and averaging the sum leads to which yields after averaging Equations ( 38) and ( 41) contain new correlators


by multiply by 2 , 2   ing Equations ( 36) and (40) x y   and ,  respectively, and averaging, In the case of dichotomous noise, we spli order correlators into lower-order correlators by using t the higher- This result coincid with result for pure Br ian motion.The independence of the stationary 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 stationary second moments for the r (7)), which are, re- showing the "energetic" instability [41].It turns out that, in the presence of dichotomous oscillator mass fluctuations, the stationary second moment 2 , x in contrast to its white noise form (45), may lead to instability, 2 < 0. x

Correlation Function
The correlation function can be found along the lines as was done for the second moment by multiplying same Equations (36) by   and averaging the resulting equations, which gives The new correlator  can be found by using Equation ( 28ding to ) lea nd (48), one can e correlation function From Equations ( 47) a find the fourthorder differential equation for th 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 .

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 where   t  is white noise For asymmetric dichotomous noise where It is easy to check that for Equation ( 51) redu the by = 2 k ces to Equation (18), after multiplying latter equation   g t and averaging.
ogous to the previous analysis, Equation (49) written as order differential equations, Anal can be re two first Multiplying the first equation by 2x and the second by 2 y an eraging one gets after using Equations ( 51) and ( 23) Equation ( 54 last equation in Equation ( 55), or, by inserting (51) with k replaced by iplying Equations ( 53) by y and x, respectively, and summing these equations one gets by using Equation (23), using Equation ( 51), which yields, after averaging and Multiplying Equation ( 57) by  and averaging, one obtains In this way we obtain six equations, ( 54)-( 56) and ( 58)-( 59), for the six variables where ; As one can see from Equation ( 60), the polynomial the oscilla dichotomous fluctuations of tor mass can lead to instability, 2 < 0, x for som paras e values of the meter .

Resonance Phenomena
The simplest example of mechanical resonance is a harmonic oscillator subject to a periodic force, where the steady-state amplitude of the oscillator approaches infinity 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 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 Communication Building in New York where the winds the top floors and pried windows loose from their casements.
One of greatest achievements of twentieth-century physics was establishing a deep relationship between determ sound c istic and half-random terms.However, this impression is faulty due to the close connection between deugh apparently different twisted inistic and random phenomena.The widely studied phenomena of "deterministic chaos" and "stochastic resonance" might ontradictory, consisting of halfdetermin terminism and randomness, altho forms of behavior [43].
Here we consider a new manifestation of the resonance of an oscillator.The dynamic equation of motion of a bistable underdamped one-dimensional oscillator driven by a multiplicative random force  , The dynamic resonance mentioned above corresponds to

  
Let us consider some other limiting cases of Equation ( 63).
1) Brownian motion ) 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].


) and small damping, ,    shows two or three peaks in the power spectrum (Fourier component of the correlation function) descriptive of fluctuat mped ion transitions between the two stable points of the potential, small intra-well vibrations and the over-thebarrier vibrations [44].
3) Stochastic resonance (SR) in overda  m the nonlinear case, allows an exact solution [45,46].However, this effect occurs only when the multiplicative noise   t fro  is colored and not white.

5) Vibrational resonance  
which occurs in a deterministic system, manifests itself in the enhancement of a weak periodic signal through a high-frequency pe ochastic reson riodic field, instead of through noise, as in the case of st ance.6) "Erratic" behavior shows up as a "random-like" phenomenon in a simple system

d become c
ation (19) with

Stochastic Resonance
Noise, which always plays a distractive role, appears as a constructive force, increasing the output signal as a function 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 appea has 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 replaced by nonwhite multiplicative noise [46].Deterministic 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, Repeating the procedure leading to Equation (35), one obtains a fourth-order differential equation for , Copyright © 2012 SciRes.

WJM
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, we shall not write down these cumbersome equation ti Analogous to the cases of random frequency and random damping [47], we seek the solution of Equa on (65) in the form One easily finds with One can compare Equations ( 66)-( 68) with the equations for the first moment  (68) x , obtained [52] for the cases of random frequency and random damping, respectively, 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 para

Vibrational Resonance
stochastic resonance, vibrational resonance manitself 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 deterministic equation of motion then has the following form,  of the external ld but with a sligh meters of the noise.
Equation ( 69) describes an oscillator moving in a symmetric double-well potential   with a maximum at and two minima = 0 x  x   with the depth d of the wells, 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 ("vibrational 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 highfrequency signal.The latter plays a role similar to that of noise in SR.If  nce of the outt 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-frequency signal in vibrational resonance change the parameters 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 One can say that Equation (72) is the "coarse-grained" version (with respect to time) of Equation (69).For 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 and for with the proviso that 2 2 > .
b  lts have be All the above resu en obtained for an underoscillator.It turns out [59,60] that a similar effect s place for an overda ped oscillator damped also take m ( 2 2 d d = 0 x t ditional addi-in Equation ( 69)).The influence of the ad tive noise on the vibrational resonance tages of the vibrational resonance compared to the stoch been stud d [61].
to E (Equat subsequent analysis of an oscillator equation with one periodic force is quite analogous to analysis of Equation ( 64), wh stochastic resonance phenomenon.
Equation (69) describes an oscillator moving in symouble-well potential.The vibrational resonance in the quintic oscillator with the potential of the form , and the advanastic resonance in the detection of weak signals have ie For an oscillator with random mass one has to add two periodic fields quations ( 15), (19), and perform the preceding analysis of Equation ( 69), based on dividing its solution in the two time scales ion (71)) followed by the linearization of Equation ( 72 and an appearance of chaos in the Van der Pol oscillator were investigated in [64].Because of the many applications in physics, chemistry, biology and engi brational resonance still attracts great interest, and new applications will surely be found in the future.

"Erratic" Behavior
One of the great achievements of twentieth-century physics was the prediction of deterministic chaos which appears 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.Deterministic chaos occurs if the diffe no inistic 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 oscillator subject to two periodic fields,       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, The Fourier spectrum of the correlation function (88) depends on the ratio  (88) , the spectrum becomes broadband, what is ty terministic chaos.However, this "erratic" be ises from a simple "integrable" Equation (84), w stinguishes it from deterministic chaos.

Conclusion
We considered a new type of sto oscillator which has a random mass.An exam rownian motion with adhesion, where the surro olecules not only collide with the Brownian pa cing a zigzag motion, but also adhere to it for a period of time, thereby increasing the mass of rownian particle.The first two moments are fo dichotomous random noise.An analysis was pe d of the "err m phenomena are complimentary 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 momany plications in the future.
Equation (8) can be rewritten in the following form(25) and One can calculate these and the analogous correlator 2 x 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 stochast ing and counterintuitive phee-scale of ic time-scale, determined by the Kramer transition rate over the barrier.4) Stochastic resonance in a linear overdamped oscillator ( 2 2 = 0 C ), as distinct d d = = = x t b


the phenomenon of dynamic stabilization[57] occurs, namely, the high-frequency external field transforms the previously unstable position = 0  into a stable position.Seeking the solution of Equation (72) of the form substituting in leads to t e following relations between th and frequencies of the two driving fields w the resonant behavior, 80) tion (80) has real solutions for C only if 2 2 > .b Equa  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 and not the potential depth.Then, one obtains, f 0

1 2
  .If this ratio is a rational number, this spectrum will contain a finite number of peaks.However, for irrational1 2