Research Article The Stochastic Resonance Behaviors of a ...downloads.hindawi.com/journals/amp/2016/5164575.pdf · Harmonic Oscillator Subject to Multiplicative and Periodically Modulated

Research ArticleThe Stochastic Resonance Behaviors of a GeneralizedHarmonic Oscillator Subject to Multiplicative and PeriodicallyModulated Noises

Suchuan Zhong1 Kun Wei2 Lu Zhang3 Hong Ma3 and Maokang Luo3

1College of Aeronautics and Astronautics Sichuan University Chengdu 610065 China2School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu 611731 China3College of Mathematics Sichuan University Chengdu 610065 China

Correspondence should be addressed to Maokang Luo makaluoscueducn

Received 21 September 2016 Revised 9 November 2016 Accepted 20 November 2016

Academic Editor Maria L Gandarias

Copyright copy 2016 Suchuan Zhong et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The stochastic resonance (SR) characteristics of a generalized Langevin linear system driven by a multiplicative noise and aperiodically modulated noise are studied (the two noises are correlated) In this paper we consider a generalized Langevin equation(GLE) driven by an internal noise with long-memory and long-range dependence such as fractional Gaussian noise (fGn) andMittag-Leffler noise (M-Ln) Such a model is appropriate to characterize the chemical and biological solutions as well as to somenanotechnological devices An exact analytic expression of the output amplitude is obtained Based on it some characteristicfeatures of stochastic resonance phenomenon are revealed On the other hand by the use of the exact expression we obtain thephase diagram for the resonant behaviors of the output amplitude versus noise intensity under different values of systemparametersThese useful results presented in this paper can give the theoretical basis for practical use and control of the SR phenomenon ofthis mathematical model in future works

1 Introduction

The phenomenon of stochastic resonance (SR) characterizedthe cooperative effect between weak signal and noise ina nonlinear systems which was originally perceived forexplaining the periodicity of ice ages in early 1980s [1 2] Inthe past three decades the SR phenomenon attracted greatattentions and has been documented in a large number ofliteratures in biology physics chemistry and engineering [3ndash6] such as SR in magnetic systems [7] and tunnel diode [8]and in cancer growth models [9 10]

Nowadays there have been considerable developments inSR and the original understanding of SR is extended Firstlyin the initial stage of investigation of SR the nonlinearitysystem noise and periodic signal were thought of as threeessential ingredients for the presence of SR However in 1996Berdichevsky and Gitterman [11] found that SR phenomenoncan occur in the linear system with multiplicative colorednoise [12 13] It is because the multiplicative noise breaksthe symmetry of the potential of the linear system and this

gives rise to the SR phenomenon which has been explainedbyValenti et al in literatures [14 15] Valenti et al investigatedthe SR phenomenon in population dynamics where themul-tiplicative noise source is Gaussian or Levy type Moreoverthe appearance of SR in a trapping overdamped monostablesystem was also investigated in literature [16] recently Sec-ondly the conventional definition about SR phenomenonis the signal-to-noise ratio (SNR) versus the noise intensityexhibits a peak [17 18] whereas the generalized SR [19]implies the nonmonotonic behaviors of a certain function ofthe output signal (such as the first and second moments theautocorrelation function) on the system parameters Thirdlyanother interesting extension of SR lies in the fact that outputamplitude 119860 attains a maximum value by increasing thedriving frequency Ω Such phenomenon indicates the bonafide SR which was introduced by Gammaitoni et al appearsat some value ofΩ [20 21]

The SR phenomenon driven by Gaussian noise has beeninvestigated both theoretically and experimentally However

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2016 Article ID 5164575 12 pageshttpdxdoiorg10115520165164575

2 Advances in Mathematical Physics

the Gaussian noise is just an ideal model for actual fluctua-tions and not always appropriate to describe the real noisyenvironment For instance in the nonequilibrium situationthe stochastic processes describing the interactions of a testparticle with the environment exhibit a heavy tailed non-Gaussian distribution Nowadays Dybiec et al investigatedthe resonant behaviors of a stochastic dynamics systemwith Levy stable noise and an isotropic 120572-stable noise withheavy tails and jumps in literatures [22 23] Dubkov et alinvestigated the Levy flight superdiffusion as a self-similarLevy process and derive the fractional Fokker-Planck equa-tion (FFPE) for probability distribution from the Langevinequation with Levy stable noise [24ndash26]

Recently more and more scholars began to pay atten-tion to another important class of non-Gaussian noise thebounded noise It should be emphasized is that the boundednoise is a more realistic and versatile mathematical model ofstochastic fluctuations in applications and it is widely appliedin the domains of statistical physics biology and engineeringin the last 20 years Furthermore the well-known telegraphnoise such as dichotomous noise (DN) and trichotomousnoise that are widely used in the studied of SR phenomenais a special case of bounded noise The deepening anddevelopment of theoretical studies on bounded noise led tothe fact that lots of scholars investigated the effect of boundednoise on the stochastic resonant behaviors in the specialmodel in physics biology and engineering [27]

Since Richardsonrsquos work in literature [28] a large numberof observations related to anomalous diffusion [29ndash33] havebeen reported in several scientific fields for example brainstudies [34 35] social systems [36] biological cells [37 38]animal foraging behavior [39] nanoscience [40 41] andgeophysical systems [42] One of the main aspects of thesesituations is the correlation functions of the above anomalousdiffusion phenomena which may be related to the non-Markovian characteristic of the stochastic process [43] Thetypical characteristic of anomalous diffusion lies in themean-square displacement (MSD) which satisfies ⟨119909(119905)2⟩ sim 119905120572where the diffusion exponents 0 lt 120572 lt 1 and 120572 gt 1 indicatesubdiffusion and superdiffusion respectively When 120572 = 1the normal diffusion is recovered [44]

It is well-known that the normal diffusion can bemodeledby a Langevin equation where a Brownian particle subjectedto a viscous drag from the surrounding medium is charac-terized by a friction force and it also subjected to a stochasticforce that arises from the surrounding environmentThe fric-tion constant determines how quickly the system exchangesenergy with the surrounding environment For a realisticdescription of the surrounding environment it is difficult tochoose a universal value of the friction constant Indeed inorder to depict the real situation more effectively a differentvalue of the friction constant should be adopted Hence ageneralization of the Langevin equation is needed leading tothe so-called generalized Langevin equation (GLE) [45] TheGLE is an equation ofmotion for a non-Markovian stochasticprocess where the particle has a memory effect to its velocity

Nowadays the GLE driven by a fractional Gaussian noise(fGn) [46]with a power-law friction kernel is extensively used

for modeling anomalous diffusion processes For instance inthe study on dynamics of single-molecule when the electrontransfer (ET) was used to probe the conformational fluctu-ations of single-molecule enzyme the distance between theET donor and acceptor can be modeled well through a GLEdriven by an fGn [47ndash54] Besides Vinales and Despositohave introduced a novel noise whose correlation function isproportional to a Mittag-Leffler function which is called theMittag-Leffler noise (M-Ln) [55ndash57] The correlation func-tion behaves as a power-law for large times but is nonsingularat the origin due to the inclusion of a characteristic time

Theoverwhelmingmajority of previous studies of SRhaverelated to the case where the external noise and the weakperiodic force are introduced additively However Dykmanet al [58] studied the case where the signal is multiplied tonoise namely the noise is modulated by a signal They foundthatwhen an asymmetric bistable system is driven by a signal-modulated noise stochastic resonance appeared in contrastto the additive noise new characteristics emerge and theirresults were in agreement with experiments FurthermoreCao and Wu [59] studied the SR characteristics of a linearsystem driven by a signal-modulated noise and an additivenoise It seems that a periodically modulated noise is notuncommon and arises for example at the output of anyamplifier (optics or radio astronomy) whose amplificationfactor varies periodically with time

Due to the synergy of generalized friction kernel of a GLEand the periodically modulated noise the stochastic resonantbehaviors of a GLE can be influenced In contrast to thecase that has been investigated before new dynamic char-acteristics emerge Motivated by the above discussions wewould like to explore the stochastic resonance phenomenonin a generalized harmonic oscillator with multiplicative andperiodically modulated noises Moreover we consider theGLE is driven by a fractional Gaussian noise and a Mittag-Leffler noise respectively in this paper We focus on thevarious nonmonotonic behaviors of the output amplitude 119860with the systemparameters and the parameters of the internaldriven noise

The physical motivations of this paper are as follows (1)in view of the importance of stochastic generalized harmonicoscillator (linear oscillator) with memory in physics chem-istry and biology and due to the periodicallymodulated noisearising at the output of the amplifier of the optics deviceand radio astronomy device to establish a physical model inwhich the SR can contain the effects of the two factors thelinearity of the system and the periodical modulation of thenoise (2) The second one is to give a theoretical foundationfor the study of SR characteristic features of a generalizedharmonic oscillator subject to multiplicative periodicallymodulated noises and external periodic force Our studyshows that such a model leads to stochastic resonancephenomenon Meanwhile an exact analytic expression of theoutput amplitude is obtained Based on it some characteristicfeatures of SR are revealed

The paper is organized as follows Section 2 gives theintroductions of the generalized Langevin equation the frac-tional Gaussian noise and the Mittag-Leffler noise Section 3gives analytical expression of the output amplitude of the

Advances in Mathematical Physics 3

systemrsquos steady response Section 4 presents the simulationresults and gives the discussions Section 5 gives conclusion

2 System Model

21 The Generalized Langevin Equation The generalizedLangevin equation (GLE) is an equation of motion for thenon-Markovian stochastic process where the particle has amemory effect to its velocity Anomalous diffusion in physicaland biological systems can be formulated in the framework ofaGLE that reads asNewtonrsquos law for a particle of the unitmass(119898 = 1) [11 47 60ndash66]

(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)

where 119909(119905) is the displacement of the Brownian particle attime 119905 120574 gt 0 is the friction constant 119870(119905) represents thememory kernel of the frictional force and 119889119880(119909)119889119909 is theexternal force under the potential 119880(119909) The random force120578(119905) is zero-centered and stationary Gaussian that obeys thegeneralized second fluctuation-dissipation theorem [67]

⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)

where 119896119861 is the Boltzmann constant and 119879 is the absolutetemperature of the environment

22TheFractional GaussianNoise FractionalGaussian noise(fGn) and fractional Brownian motion (fBm) were originallyintroduced by Mandelbrot and Van Ness [46] for modelingstochastic fractal processes The fGn 120576119867(119905) = 120576119867(119905) 119905 gt 0with a constant Hurst parameter 12 lt 119867 lt 1 can beused to more accurately characterize the long-range depen-dent process [the autocorrelation function 119903(119896) satisfiessuminfin119896=minusinfin 119903(119896) = infin] than traditional short-range dependentstochastic process [the autocorrelation function 119903(119896) satisfiessuminfin119896=minusinfin 119903(119896) lt infin] The short-range dependent stochasticprocess is for example Markov Poisson or autoregressivemoving average (ARMA) process

Now consider the one-side normalized fBm which is aGaussian process 119861119867(119905) = 119861119867(119905) 119905 gt 0 which shows theproperties below [68]

(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)

The fGn 120576119867(119905) = 120576119867(119905) 119905 gt 0 given by [47ndash49]

120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)

is a stationary Gaussian process with ⟨120576119867(119905)⟩ = 0 Thereforeaccording to (3) and (4) and the LrsquoHospitalrsquos rule theautocorrelation function 119862(119905) of fGn can be derived as

119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+

⟨[119861119867 (0 + 119904) minus 119861119867 (0)]119904 [119861119867 (119905 + 119904) minus 119861119867 (119905)]119904 ⟩= 2119896119861119879 sdot lim

119904rarr0+

|119905 + 119904|2119867 + |119905 minus 119904|2119867 minus 2 |119905|211986721199042 = 2119896119861119879sdot lim119904rarr0+

2119867 |119905 + 119904|2119867minus1 minus 2119867 |119905 minus 119904|2119867minus14119904 = 2119896119861119879sdot lim119904rarr0+

2119867 (2119867 minus 1) |119905 + 119904|2119867minus2 + 2119867 (2119867 minus 1) |119905 minus 119904|2119867minus24 = 2119896119861119879sdot 2119867 (2119867 minus 1) |119905 + 0|2119867minus2 + 2119867 (2119867 minus 1) |119905 minus 0|2119867minus24= 2119896119861119879 sdot 119867 (2119867 minus 1) 1199052119867minus2 119905 gt 0

(5)

23 The Mittag-Leffler Noise It is well-known that the phys-ical origin of anomalous diffusion is related to the long-timetail correlationsThus in order tomodel anomalous diffusionprocess a lot of different power-law correlation functions areemployed in (1) and (2)

Vinales and Desposito have introduced a novel noisewhose correlation function is proportional to aMittag-Lefflerfunction which is called Mittag-Leffler noise [55ndash57] Thecorrelation function of Mittag-Leffler noise behaves as apower-law for large times but is nonsingular at the origindue to the inclusion of a characteristic time The correlationfunction of Mittag-Leffler noise is given by

119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)

where 120591 is called characteristicmemory time and thememoryexponent 120572 can be taken as 0 lt 120572 lt 2 The 119864120572(sdot) denotes theMittag-Leffler function that is defined through the series

119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)

which behaves as a stretched exponential for short times andas inverse power-law in the long-time regime when 120572 = 1Meanwhile when 120572 = 1 the correlation function equation(4) reduces to the exponential form

119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)

which describes a standard Ornstein-Uhlenbeck process


24 The System Model In this paper we consider a periodi-cally driven linear system with multiplicative noise and peri-odically modulated additive noise described by the followinggeneralized Langevin equation

(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)

where 1205960 is the intrinsic frequency of the harmonic oscillator119880(119909) = 1205962011990922 The fluctuations of 12059620 in (9) are modeled asa Markovian dichotomous noise 1205851(119905) [69] which consists ofjumps between two values minus119886 and 119886 119886 gt 0 with stationaryprobabilities 119875119904(minus119886) = 119875119904(119886) = 12 The statistical propertiesof 1205851(119905) are ⟨1205851 (119905)⟩ = 0

⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)

where 1198862 is the noise intensity and V is the correlation ratewith 1205910 = 1V being the correlation time1205852(119905) is a zero mean signal-modulated noise with cou-pling strength119863 with noise 1205851(119905) [70 71] that is⟨1205851 (119905) 1205852 (119904)⟩ = 119863120575 (119905 minus 119904) (11)

In (9) 1198601 and Ω are the amplitude and frequency of theexternal periodic force1198601 sin(Ω119905) Meanwhile1198602 andΩ arethe amplitude and frequency of the periodically modulatedadditive noise 1198602 sin(Ω119905)1205852(119905) respectively

In this paper we assume that the external noise 1205851(119905) 1205852(119905)and the internal noise 120578(119905) satisfy ⟨120578(119905)1205851(119904)⟩ = ⟨120578(119905)1205852(119904)⟩ =0with different origins In the next section we will obtain theexact expression of the first moment of the output signal

3 First Moment

31 The Analytical Expression of the Output Amplitude of aGLE First of all we should transfer the stochastic equation(9) to the deterministic equation for the average value ⟨119909⟩For this purpose we use the well-known Shapiro-Loginov[72] procedure which yields for exponentially correlatednoise (10)

⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)

Equation (9) depicted the motion of 119909(119905) is bounded by anoisy harmonic force field by averaging realization of thetrajectory of the stochastic equation (9) and applying thecharacteristics of the noises 1205851(119905) 1205852(119905) and 120578(119905) we obtainthe equation of the particlersquos average displacement ⟨119909⟩1198892 ⟨119909⟩1198891199052 + 120574int119905

0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)

It can be found that (13) shows the synthetic affections ofthe particlersquos average displacement ⟨119909⟩ and the multiplicative

noise coupling term ⟨1205851119909⟩ In order to deal with the newmultiplicative noise coupling term ⟨1205851119909⟩ we multiply bothsides of (9) with 1205851(119905) and then average to construct a closedequations of ⟨119909⟩ and ⟨1205851119909⟩

⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)

Using the Shapiro-Loginov formula (12) and the charac-teristics of the generalized integration (14) turns to be

[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)

To summarize for the linear generalized Langevin equa-tion (9) to be investigated in this paper it is a stochasticdifferential equation driven by an internal noise 120578(119905) Whenwe want to obtain the particlersquos average displacement ⟨119909⟩from (9) we average (9) and obtain the traditional classicaldifferential equation (13) for ⟨119909⟩ and the new multiplicativenoise coupling term ⟨1205851119909⟩ In order to deal with the newcoupling term ⟨1205851119909⟩ we do some mathematical calculationsand have another ordinary differential equation (15) Finallywe obtain two linear closed equations (13) and (15) for 1199091 =⟨119909⟩ and 1199092 = ⟨1205851119909⟩

In order to solve the closed equations (13) and (15) weuse the Laplace transform technique 119883119894 = LT[119909119894(119905)] ≜int+infin0

119909119894(119905)119890minus119904119905119889119905 119894 = 1 2 [73] under the long time limit 119905 rarrinfin condition we obtain the following equations

[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)

(119904) = LT[119870(119905)]means performing Laplace transformThe solutions of (16) can be represented as

119883(119886119904)1 (119904) = 11 (119904) sdot LT [1198601 sin (Ω119905)] + 21 (119904)sdot LT [1198602119863 sin (Ω119905)]


(17)


where 119896119894(119904) = LT[119867119896119894(119905)] 119896 119894 = 1 2 and 119896119894(119904) can beobtained from (16) Particularly we have

11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)

with (119904) = [12059620 + 1199042 +120574119904 sdot (119904)][12059620 + (119904+ V)2 +120574(119904+ V) sdot (119904 +V)] minus 1198862

Applying the inverse Laplace transform technique by thetheory of ldquosignals and systemsrdquo the product of the Laplacedomain functions corresponding to the convolution of thetime domain functions we can obtain the solutions 119909(119886119904)1 (119905) =⟨119909(119905)⟩119886119904 ≜ lim119905rarrinfin⟨119909(119905)⟩ and 119909(119886119904)2 (119905) = ⟨1205851(119905)119909(119905)⟩119886119904 ≜lim119905rarrinfin⟨1205851(119905)119909(119905)⟩ through (17)

119909(119886119904)1 (119905) = 11986711 (119905) lowast [1198601 sin (Ω119905)] + 11986721 (119905)lowast [1198602119863 sin (Ω119905)]


(19)

where lowast is the convolution operatorEquations (13) and (15) with 1199091 = ⟨119909⟩ and 1199092 = ⟨1205851119909⟩ can

be regarded as a linear system and the forces 1198601 sin(Ω119905) and1198602119863 sin(Ω119905) can be regarded as the input periodic signalsBy the theory of ldquosignals and systemsrdquo when we put periodicsignals 1198601 sin(Ω119905) and 1198602119863 sin(Ω119905) into a linear systemwith system functions 11986711(119905) and 11986721(119905) the output signalsare still periodic signals meanwhile the frequency of theoutput signals are the same as the frequency of the inputsignals We denote the output signals as 11986011 sin(Ω119905 + 12059311)and 11986021 sin(Ω119905 + 12059321) respectively which can be shown inFigure 1

Moreover the amplitudes 11986011 and 11986021 of the outputsignals are proportion to the amplitudes 1198601 and 1198602119863 of theinput signals and the proportion constants are the amplitudesof the frequency response functions 11986711(119890119895Ω) and 11986721(119890119895Ω)that is 11986011 = 1198601|11986711(119890119895Ω)| and 11986021 = 1198602119863|11986721(119890119895Ω)|

Thus from (19) we obtain the following expression ofparticlersquos average displacement 119909(119886119904)1 (119905)119909(119886119904)1 (119905) = 11986711 (119905) lowast [1198601 sin (Ω119905)] + 11986721 (119905) lowast [1198602119863

sdot sin (Ω119905)] = 11986011 sin (Ω119905 + 12059311) + 11986021 sin (Ω119905+ 12059321) = radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321)sdot sin[[[

Ω119905

+ arcsin (11986011 sin12059311 + 11986021 sin12059321)radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321)]]]

(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)

Figure 1 The relationships of the input periodic signals and theoutput signals by the theory of ldquosignals and systemsrdquo

where11986011 11986021 and 12059311 12059321 are the amplitude and phase shiftof the long-time behaviors of the output signals respectively

In addition we can obtain the expressions of frequencyresponse functions 11986711(119890119895Ω) and 11986721(119890119895Ω) through (18)Through the derivation we can suppose that the expressionsof11986711(119890119895Ω) and11986721(119890119895Ω) can be simplified as follows

11986711 (119890119895Ω) ≜ 1198601 + 1198611 sdot 1198951198621 + 1198631 sdot 119895 11986721 (119890119895Ω) ≜ 1198602 + 1198612 sdot 1198951198622 + 1198632 sdot 119895

(21)

where 119860 119894 119861119894 119862119894 119863119894 119894 = 1 2 are real numbersThen the amplitudes of frequency response functions11986711(119890119895Ω) and11986721(119890119895Ω) are1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = radic11986711 (119890119895Ω)119867lowast11 (119890119895Ω) = radic11986021 + 1198612111986221 + 11986321 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = radic11986721 (119890119895Ω)119867lowast21 (119890119895Ω) = radic11986022 + 1198612211986222 + 11986322

(22)

where 119867lowast1198941(119890119895Ω) 119894 = 1 2 is the conjugation of 1198671198941(119890119895Ω) 119894 =1 2Meanwhile the amplitudes of the output signals are

11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)

In this paper with the expression of the particlersquos averagedisplacement 119909(119886119904)1 (119905) in (20) we mainly discuss the resonantbehaviors of the output amplitude 119860 which is defined as

119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)

It should be emphasized that the following inequalitymust hold for the sake of the stability of solutions [69]

0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)

In this paper we assume the stability condition (25) issatisfied


32 The Output Amplitude of a GLE with Fractional Gaus-sian Noise When the internal noise 120578(119905) in the generalizedLangevin equation (9) is fractional Gaussian noise withcorrelation function (5) from the fluctuation-dissipationtheorem (2) we derive the power-law memory kernel 119870(119905)presented by Hurst exponent119867 0 lt 119867 lt 1

119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)

Performing the Laplace transform we obtain the related(119904) = LT[119870(119905)] (119904) = Γ (2119867 + 1) sdot 1199041minus2119867 0 lt 119867 lt 1 (27)

From (18) (20) (24) and (27) we get the output amplitude119860expressed by (24) with

11986011 = 1198601radic11989112 + 1198912211989132 + 11989142 12059311 = arctan(11989121198913 minus 1198911119891411989111198913 + 11989121198914) 11986021 = 1198602119863radic 111989132 + 11989142

(28)

12059321 = arctan(minus11989141198913) (29)

where

1198911 = 12059620 + 1198872 cos (2120579) + 1205741198872minus2119867sdot Γ (2119867 + 1) cos [(2 minus 2119867) 120579] 1198912 = 1198872 sin (2120579) + 1205741198872minus2119867sdot Γ (2119867 + 1) sin [(2 minus 2119867) 120579] 1198913 = 1198720 +1198721 cos (2120579) + 1198722 cos [(2 minus 2119867) 120579]

+1198723 cos [(2 minus 2119867) 1205872 + 2120579]+1198724 cos [(1205872 + 120579) (2 minus 2119867)]+1198725 cos [(2 minus 2119867) 1205872 ]

1198914 = 1198721 sin (2120579) + 1198722 sin [(2 minus 2119867) 120579]+1198723 sin [(2 minus 2119867) 1205872 + 2120579]+1198724 sin [(1205872 + 120579) (2 minus 2119867)]+1198725 sin [(2 minus 2119867) 1205872 ]

119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059640 minus 1198862 minus Ω212059620 1198721 = (12059620 minus Ω2) 11988721198722 = 120574 (12059620 minus Ω2) 1198872minus2119867Γ (2119867 + 1) 1198723 = 120574Ω2minus21198671198872Γ (2119867 + 1) 1198724 = 1205742Ω2minus21198671198872minus2119867Γ2 (2119867 + 1) 1198725 = 12057412059620Ω2minus2119867Γ (2119867 + 1)

(30)

33The Output Amplitude of a GLE withMittag-Leffler NoiseWhen the internal noise 120578(119905) in the generalized Langevinequation (9) is Mittag-Leffler noise with correlation function(6) from the fluctuation-dissipation theorem (2) we derivetheMittag-Lefflermemory kernel119870(119905) presented bymemorytime 120591 and memory exponent 120572

119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)

Performing the Laplace transform we obtain the related(119904) = LT[119870(119905)] (119904) = 119904120572minus11 + (120591119904)120572 0 lt 120572 lt 2 120591 gt 0 (32)

From (18) (20) (24) and (32) we obtain the output ampli-tude 119860 expressed by (24) with

11986011 = 1198601radic11989212 + 1198922211989232 + 11989242 12059311 = arctan(11989221198923 minus 1198921119892411989211198923 + 11989221198924) 11986021 = 1198602119863radic ℎ12 + ℎ2211989232 + 11989242 12059321 = arctan(ℎ21198923 minus ℎ11198924ℎ11198923 + ℎ21198924)

(33)

where

1198921 = 12059620 +1198727 cos (120572120579) +1198722 cos (2120579)+ 11987221198725 cos [(2 + 120572) 120579]+ 119872412059620 cos(1205872 120572)+11987241198727 cos [(1205872 + 120579) 120572]


+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]

1198922 = 1198727 sin (120572120579) + 1198722 sin (2120579)+ 11987221198725 sin [(2 + 120572) 120579] + 119872412059620 sin(1205872 120572)+11987241198727 sin [(1205872 + 120579) 120572]+11987221198724 sin(2120579 + 1205872 120572)+119872211987241198725 sin [(2 + 120572) 120579 + 1205872 120572]

ℎ1 = minus1 minus1198724 cos(1205872 120572) minus1198725 cos (120572120579)minus11987241198725 cos [(1205872 + 120579) 120572]

ℎ2 = minus1198724 sin(1205872 120572) minus1198725 sin (120572120579)minus11987241198725 sin [(1205872 + 120579) 120572]

1198923 = 1198728 +11987201198722 cos (2120579) + 1198729 cos (120572120579)+119872011987221198725 cos [(2 + 120572) 120579]+ 11987210 cos(1205872 120572) +11987211 cos(2120579 + 1205872 120572)+11987212 cos [(1205872 + 120579) 120572]+11987213 cos [(2 + 120572) 120579 + 1205872 120572]

1198924 = 11987201198722 sin (2120579) + 1198729 sin (120572120579)+119872011987221198725 sin [(2 + 120572) 120579]+ 11987210 sin(1205872 120572) +11987211 sin(2120579 + 1205872 120572)+11987212 sin [(1205872 + 120579) 120572]+11987213 sin [(2 + 120572) 120579 + 1205872 120572]

119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)

4 Stochastic Resonance Behaviors ofa GLE with Multiplicative and PeriodicallyModulated Noises

In this section we will perform the numerical simulationson the above analytical expression in (24) with the internalnoise 120578(119905) modeled as fractional Gaussian noise and Mittag-Leffler noise in Sections 41 and 42 respectively It can beseen from the analytical expression in (24) the behaviorsof 119860 are fully determined by the combination of the systemparameters 120574 12059620 1198862 V 1198601 1198602 119863 and Ω and the parametersof 120578(119905) Based on it some characteristic features of stochasticresonance behaviors are revealed

41 The Stochastic Resonance Behaviors of GLE with a Frac-tional Gaussian Noise From (25) and (27) we obtain thestability condition ofGLEwith fractional Gaussian noise 120578(119905)

0 lt 1198862 le 1198862crfGn= 12059620 [12059620 + V2 + 120574 sdot Γ (2119867 + 1) sdot V2minus2119867] (35)

with 0 lt 119867 lt 1It can be seen from the stability condition (35) that the

critical noise intensity 1198862crfGn is determined by 12059620 V 120574 and119867 Besides from (24) the behaviors of output amplitude 119860are fully determined by the combination of the parameters120574 12059620 1198862 V 1198601 1198602 119863Ω and119867 Thus in Figure 2 we depictthe phase diagram in the 119863 minus Ω plane for the emergence ofthe stochastic resonance behaviors of 119860 versus 1198862 at 12059620 = 11198601 = 1119867 = 055 1198602 = 1 120574 = 03 and V = 001

In the unshaded region [see the domain (0) of level= 0 which corresponds to Figure 2] the output amplitude119860(1198862) varies monotonically as the noise intensity 1198862 varieswhich means the SR phenomenon is impossible Meanwhilein the shaded regions it corresponds to the traditional SR


X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1

Figure 2 The phase diagram for the stochastic resonance behaviors of the output amplitude 119860(1198862) at 12059620 = 1 1198601 = 1 119867 = 055 1198602 = 1120574 = 03 and V = 001 and the values of Level in Figure 2 reflect the number of SR peaks for different combination of119863 andΩ

05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)

Figure 3 The output amplitude 119860 versus the noise intensity 1198862 with various (119863Ω) be chose through Figure 2 with parameters 12059620 = 11198601 = 1 1198602 = 1 V = 001119867 = 055 and 120574 = 03 and (a)119863 = 03 and Ω = 095 (b)119863 = 09 and Ω = 02 (c)119863 = 13 and Ω = 06phenomenon taking place and two phases can be discernedin the resonant domain

(1) The light shaded region (i) corresponds to the single-peak SR phenomenon [see the domain of level = 1which corresponds to Figure 2]

(2) The dark shaded region (ii) corresponds to thedouble-peaks SR phenomenon [see the domain oflevel = 2 which corresponds to Figure 2]

From Figure 2 we can find that when the drivingfrequency Ω is large enough [Ω gt 095] or small enough[Ω lt 001] it is impossible to induce the SR phenomenon

In Figure 3 we plot the curves given by (24) and (28)in which the dependence of the output amplitude 119860 on thenoise intensity 1198862 for different values of the systemparameters(119863 Ω) can be chosen from Figure 2 to verify the correctnessof the results shown in Figure 2 As shown in Figure 3(a)when 119863 = 03 and Ω = 095 which belongs to the unshadeddomain in Figure 2 the output amplitude 119860(1198862) monotonicbehavior decreased with the increasing of 1198862 which meansthe SR phenomenon does not take place In Figure 3(b) when119863 = 09 and Ω = 02 which corresponds to the lightgrey domain in Figure 2 the curve shows that the outputamplitude 119860(1198862) attains a maximum value at some values of1198862 that is the single-peak SR phenomenon takes place by


X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D

Figure 4The phase diagram for the stochastic resonance behaviors of the output amplitude119860(1198862) at12059620 = 11198601 = 1 120572 = 06 120591 = 021198602 = 1120574 = 03 and V = 005 and the values of level in Figure 3 reflect the number of SR peaks for different combination of 119863 andΩ

increasing 1198862 Furthermore one can see from Figure 3(c) thatthe double-peaks SR phenomenon happens for the reasonthat the parameter combination of 119863 = 13 and Ω = 06belongs to the dark grey domain in Figure 2 It should beemphasized that the double-peaks SR phenomenon happensbecause of the presence of two types of noise external andinternal noise sources The double-peaks SR phenomenoncan take place in biological systems such as neuronal systems[74] in which the internal noise is due to signals coming fromall other neurons and external noise is the environmentalnoise due to its interaction with the neuronal system

The main contribution of this section is as follows withthe help of phase diagram for the SR phenomenon we caneffectively control the SR phenomenon of this generalizedharmonic system in a certain range and further broaden theapplication scope of the SR phenomenon in physics biologyand engineering such as the detection of weak stimuli byspiking neurons in the presence of certain level of noisybackground neural activity [75]

42The Stochastic Resonance Behaviors of GLE with aMittag-Leffler Noise From (25) and (32) we obtain the stabilitycondition of GLE with a Mittag-Leffler noise 120578(119905)

0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)

with 0 lt 120572 lt 2 120591 gt 0It is found from the stability condition (36) that the

critical noise intensity 1198862crMLn is determined by 12059620 V 120574 120591 and120572 In Figure 4 the phase diagram in the 119863 minus Ω plane for theemergence of SR phenomenon of 119860(1198862) at 12059620 = 1 1198601 = 1120572 = 06 120591 = 02 1198602 = 1 120574 = 03 and V = 005 is shownThe same as Figure 2 when parameters (119863Ω) belong tothe unshaded regions (0) the SR phenomenon is impossiblewhen (119863Ω) belong to the light grey regions (i) the single-peak SR phenomenon happens when (119863Ω) belong to the

dark grey regions (ii) the double-peaks SR phenomenontakes place Moreover the sufficiently large driving frequencyΩ [Ω gt 098] or small enough Ω [Ω lt 001] cannot inducethe system to produce SR phenomenon

In Figure 5 we also show the curves given by (24) and (33)in which the dependence of the output amplitude 119860 on thenoise intensity 1198862 for different values of the systemparameters(119863Ω) can be chosen from Figure 4 to verify the correctnessof the results shown in Figure 4

As shown in Figure 5(a) when 119863 = 01 and Ω = 005which belongs to the unshaded domain in Figure 4 theoutput amplitude 119860(1198862) monotonic increase occurs with theincreasing of 1198862 which means the SR phenomenon does nottake place In Figure 5(b) when 119863 = 11 and Ω = 025which corresponds to the light grey domain in Figure 4the curve shows that the output amplitude 119860(1198862) attains amaximum value at some values of 1198862 that is the single-peakSR phenomenon takes place by increasing 1198862 Furthermoreone can see from Figure 5(c) that the double-peaks SRphenomenon happens for the reason that the parametercombination of 119863 = 04 and Ω = 07 belongs to the darkgrey domain in Figure 4

5 Conclusions

To summarize in this paper we explore the SR phenomenonin a generalized Langevin equation with multiplicativeperiodically modulated noises and external periodic forceMoreover the system internal noise is modeled as a frac-tional Gaussian noise and aMittag-Leffler noise respectivelyWithout loss of generality the fluctuations of system intrinsicfrequency aremodeled as amultiplicative dichotomous noiseBy the use of the stochastic averagingmethod and the Laplacetransform technique we obtain the exact expression of theoutput amplitude 119860 given by (24)


0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)

Figure 5The output amplitude119860 versus the noise intensity 1198862 with various (119863Ω) chosen through Figure 4 with parameters12059620 = 11198601 = 1120572 = 06 120591 = 02 1198602 = 1 120574 = 03 and V = 005 and (a)119863 = 01 andΩ = 005 (b)119863 = 11 andΩ = 025 (c)119863 = 04 and Ω = 07

We focus on the various nonmonotonic behaviorsof the output amplitude 119860 with the system parameters120574 12059620 1198862 V 1198601 1198602 119863 and Ω and the parameters of theinternal driven noiseWith the exact expression of the outputamplitude 119860 we find the conventional SR takes place withthe increases of the noise intensity 1198862 for fractional Gaussiannoise and Mittag-Leffler noise respectively Moreover wegive the phase diagram in119863minusΩ plane for the emergence of SRphenomenon of 119860(1198862) and find the single-peak and double-peaks SR phenomena

We believe all the results in this paper not only supplythe theoretical investigations of the generalized harmonicoscillator subject to multiplicative periodically modulatednoises and external periodic force but also can suggest someexperimental anomalous diffusion results in physical andbiological applications in the future [76]

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the Key Program of NationalNatural Science Foundation of China (Grant nos 11171238and 11601066) and Natural Science Foundation for the Youth(Grant nos 11501386 and 11401405)

References

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[2] R Benzi G Parisi and A Vulpiani ldquoStochastic resonance inclimatic changerdquo Tellus vol 34 article 10 1982

[3] R Benzi ldquoStochastic resonance from climate to biologyrdquoNonlinear Processes in Geophysics vol 17 no 5 pp 431ndash4412010

[4] M D McDonnell and D Abbott ldquoWhat is stochastic reso-nance Definitions misconceptions debates and its relevanceto biologyrdquo PLoS Computational Biology vol 5 no 5 Article IDe1000348 2009

[5] L Gammaitoni P Hanggi P Jung and FMarchesoni ldquoStochas-tic resonance a remarkable idea that changed our perception ofnoiserdquo European Physical Journal B vol 69 no 1 pp 1ndash3 2009

[6] T Wellens V Shatokhin and A Buchleitner ldquoStochastic reso-nancerdquo Reports on Progress in Physics vol 67 no 1 pp 45ndash1052004

[7] R N Mantegna B Spagnolo L Testa and M TrapaneseldquoStochastic resonance in magnetic systems described bypreisach hysteresis modelrdquo Journal of Applied Physics vol 97no 10 Article ID 10E519 2005

[8] R N Mantegna and B Spagnolo ldquoStochastic resonance in atunnel diode in the presence of white or coloured noiserdquo IlNuovo Cimento D vol 17 no 7-8 pp 873ndash881 1995

[9] A Behera and S F C OrsquoRourke ldquoThe effect of correlatednoise in a Gompertz tumor growth modelrdquo Brazilian Journalof Physics vol 38 no 2 pp 272ndash278 2008

[10] A Fiasconaro A Ochab-Marcinek B Spagnolo and EGudowska-Nowak ldquoMonitoring noise-resonant effects in can-cer growth influenced by external fluctuations and periodic


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[11] V Berdichevsky and M Gitterman ldquoStochastic resonance inlinear systems subject to multiplicative and additive noiserdquoPhysical Review E vol 60 no 2 pp 1494ndash1499 1999

[12] M Gitterman ldquoHarmonic oscillator with fluctuating dampingparameterrdquo Physical Review E vol 69 no 4 Article ID 0411012004

[13] J-H Li and Y-X Han ldquoPhenomenon of stochastic resonancecaused by multiplicative asymmetric dichotomous noiserdquo Phys-ical Review E vol 74 no 5 Article ID 051115 2006

[14] D Valenti A Fiasconaro and B Spagnolo ldquoStochastic reso-nance andnoise delayed extinction in amodel of two competingspeciesrdquo Physica A Statistical Mechanics and Its Applicationsvol 331 no 3-4 pp 477ndash486 2004

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[16] N V Agudov A V Krichigin D Valenti and B SpagnololdquoStochastic resonance in a trapping overdamped monostablesystemrdquo Physical Review E vol 81 no 5 Article ID 051123 8pages 2010

[17] J K Douglass L Wilkens E Pantazelou and F Moss ldquoNoiseenhancement of information transfer in crayfish mechanore-ceptors by stochastic resonancerdquo Nature vol 365 no 6444 pp337ndash340 1993

[18] K Wiesenfeld and F Moss ldquoStochastic resonance and thebenefits of noise from ice ages to crayfish and SQUIDsrdquoNaturevol 373 no 6509 pp 33ndash36 1995

[19] M Gitterman ldquoClassical harmonic oscillator with multiplica-tive noiserdquo Physica A vol 352 no 2ndash4 pp 309ndash334 2005

[20] L Gammaitoni F Marchesoni and S Santucci ldquoStochasticresonance as a bona fide resonancerdquoPhysical Review Letters vol74 no 7 pp 1052ndash1055 1995

[21] L Gammaitoni P Hanggi P Jung and FMarchesoni ldquoStochas-tic resonancerdquoReviews ofModern Physics vol 70 no 1 pp 223ndash287 1998

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[24] G M Viswanathan E P Raposo and M G E da LuzldquoLevy flights and superdiffusion in the context of biologicalencounters and random searchesrdquo Physics of Life Reviews vol5 no 3 pp 133ndash150 2008

[25] A A Dubkov B Spagnolo and V V Uchaikin ldquoLevy flightsuperdiffusion an introductionrdquo International Journal of Bifur-cation and Chaos in Applied Sciences and Engineering vol 18no 9 pp 2649ndash2672 2008

[26] A Dubkov and B Spagnolo ldquoLangevin approach to Levyflights in fixed potentials exact results for stationary probabilitydistributionsrdquo Acta Physica Polonica B vol 38 no 5 pp 1745ndash1758 2007

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[28] L F Richardson ldquoAtmospheric diffusion shown on a distanceneighbor graphrdquo Proceedings of the Royal Society A vol 110 no765 pp 709ndash737 1926

[29] B OrsquoShaughnessy and I Procaccia ldquoAnalytical solutions fordiffusion on fractal objectsrdquo Physical Review Letters vol 54 no5 pp 455ndash458 1985

[30] R Metzler W G Glockle and T F Nonnenmacher ldquoFractionalmodel equation for anomalous diffusionrdquo Physica A vol 211 no1 pp 13ndash24 1994

[31] L Acedo and S Bravo Yuste ldquoShort-time regime propagator infractalsrdquo Physical Review E - Statistical Physics Plasmas Fluidsand Related Interdisciplinary Topics vol 57 no 5 pp 5160ndash51671998

[32] DCamposVMendez and J Fort ldquoDescription of diffusive andpropagative behavior on fractalsrdquo Physical Review E vol 69 no3 Article ID 031115 2004

[33] D H N Anh P Blaudeck K H Hoffmann J Prehl and STarafdar ldquoAnomalous diffusion on random fractal compositesrdquoJournal of Physics A Mathematical and Theoretical vol 40 no38 pp 11453ndash11465 2007

[34] F Santamaria S Wils E De Schutter and G J AugustineldquoAnomalous diffusion in purkinje cell dendrites caused byspinesrdquo Neuron vol 52 no 4 pp 635ndash648 2006

[35] X J Zhou Q Gao O Abdullah and R L Magin ldquoStudies ofanomalous diffusion in the human brain using fractional ordercalculusrdquo Magnetic Resonance in Medicine vol 63 no 3 pp562ndash569 2010

[36] S Hapca J W Crawford and I M Young ldquoAnomalous dif-fusion of heterogeneous populations characterized by normaldiffusion at the individual levelrdquo Journal of the Royal SocietyInterface vol 6 no 30 pp 111ndash122 2009

[37] D V Nicolau Jr J F Hancock and K Burrage ldquoSources ofanomalous diffusion on cell membranes a Monte Carlo studyrdquoBiophysical Journal vol 92 no 6 pp 1975ndash1987 2007

[38] G Guigas and M Weiss ldquoSampling the cell with anomalousdiffusionmdashthe discovery of slownessrdquo Biophysical Journal vol94 no 1 pp 90ndash94 2008

[39] A M Edwards R A Phillips N W Watkins et al ldquoRevisitingLevy flight search patterns of wandering albatrosses bumble-bees and deerrdquo Nature vol 449 no 7165 pp 1044ndash1048 2007

[40] N Yang G Zhang and B Li ldquoViolation of Fourierrsquos law andanomalous heat diffusion in silicon nanowiresrdquo Nano Todayvol 5 no 2 pp 85ndash90 2010

[41] S Ozturk Y A Hassan and V M Ugaz ldquoInterfacial complexa-tion explains anomalous diffusion in nanofluidsrdquo Nano Lettersvol 10 no 2 pp 665ndash671 2010

[42] V R Voller and C Paola ldquoCan anomalous diffusion describedepositional fluvial profilesrdquo Journal of Geophysical ResearchEarth Surface vol 115 no F2 2010

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[45] H Ness L Stella C D Lorenz and L Kantorovich ldquoApplica-tions of the generalized Langevin equation towards a realisticdescription of the bathsrdquo Physical Review B vol 91 no 1 ArticleID 014301 pp 1ndash15 2015

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moleculerdquo Physical Review Letters vol 93 no 18 Article ID180603 4 pages 2004

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[50] WMin andX S Xie ldquoKramersmodel with a power-law frictionkernel dispersed kinetics and dynamic disorder of biochemicalreactionsrdquo Physical Review E vol 73 no 1 Article ID 010902pp 1ndash4 2006

[51] H Yang G Luo P Karnchanaphanurach et al ldquoProteinconformational dynamics probed by single-molecule electrontransferrdquo Science vol 302 no 5643 pp 262ndash266 2003

[52] S Zhong K Wei S Gao and H Ma ldquoStochastic resonancein a linear fractional Langevin equationrdquo Journal of StatisticalPhysics vol 150 no 5 pp 867ndash880 2013

[53] S Zhong K Wei S Gao and H Ma ldquoTrichotomous noiseinduced resonance behavior for a fractional oscillator withrandom massrdquo Journal of Statistical Physics vol 159 no 1 pp195ndash209 2015

[54] S Zhong H Ma H Peng and L Zhang ldquoStochastic resonancein a harmonic oscillator with fractional-order external andintrinsic dampingsrdquo Nonlinear Dynamics vol 82 no 1-2 pp535ndash545 2015

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[60] D S Grebenkov andM Vahabi ldquoAnalytical solution of the gen-eralized Langevin equation with hydrodynamic interactionssubdiffusion of heavy tracersrdquo Physical Review E vol 89 no 1Article ID 012130 18 pages 2014

[61] S Burov and E Barkai ldquoCritical exponent of the fractionalLangevin equationrdquo Physical Review Letters vol 100 no 7Article ID 070601 2008

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[65] K G Wang and M Tokuyama ldquoNonequilibrium statisticaldescription of anomalous diffusionrdquo Physica A vol 265 no 3pp 341ndash351 1999

[66] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 7 pages 2009

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[70] Z-Q Huang and F Guo ldquoStochastic resonance in a fractionallinear oscillator subject to random viscous damping and signal-modulated noiserdquo Chinese Journal of Physics vol 54 no 1 pp69ndash76 2016

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[72] V E Shapiro and V M Loginov ldquolsquoFormulae of differentiationrsquoand their use for solving stochastic equationsrdquo Physica AStatistical Mechanics and Its Applications vol 91 no 3-4 pp563ndash574 1978

[73] A V Oppenheim A S Willsky and S H Nawab Signals andSystems Prentice Hall Beijing China 2005

[74] J M G Vilar and J M Rubı ldquoStochastic multiresonancerdquoPhysical Review Letters vol 78 no 15 pp 2882ndash2885 1997

[75] J F Mejias and J J Torres ldquoEmergence of resonances in neuralsystems the interplay between adaptive threshold and short-term synaptic plasticityrdquo PLoS ONE vol 6 no 3 Article IDe17255 pp 869ndash880 2011

[76] K Laas R Mankin and A Rekker ldquoConstructive influenceof noise flatness and friction on the resonant behavior of aharmonic oscillatorwith fluctuating frequencyrdquoPhysical ReviewE vol 79 no 5 Article ID 051128 2009

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Stochastic AnalysisInternational Journal of


the Gaussian noise is just an ideal model for actual fluctua-tions and not always appropriate to describe the real noisyenvironment For instance in the nonequilibrium situationthe stochastic processes describing the interactions of a testparticle with the environment exhibit a heavy tailed non-Gaussian distribution Nowadays Dybiec et al investigatedthe resonant behaviors of a stochastic dynamics systemwith Levy stable noise and an isotropic 120572-stable noise withheavy tails and jumps in literatures [22 23] Dubkov et alinvestigated the Levy flight superdiffusion as a self-similarLevy process and derive the fractional Fokker-Planck equa-tion (FFPE) for probability distribution from the Langevinequation with Levy stable noise [24ndash26]

Recently more and more scholars began to pay atten-tion to another important class of non-Gaussian noise thebounded noise It should be emphasized is that the boundednoise is a more realistic and versatile mathematical model ofstochastic fluctuations in applications and it is widely appliedin the domains of statistical physics biology and engineeringin the last 20 years Furthermore the well-known telegraphnoise such as dichotomous noise (DN) and trichotomousnoise that are widely used in the studied of SR phenomenais a special case of bounded noise The deepening anddevelopment of theoretical studies on bounded noise led tothe fact that lots of scholars investigated the effect of boundednoise on the stochastic resonant behaviors in the specialmodel in physics biology and engineering [27]

Since Richardsonrsquos work in literature [28] a large numberof observations related to anomalous diffusion [29ndash33] havebeen reported in several scientific fields for example brainstudies [34 35] social systems [36] biological cells [37 38]animal foraging behavior [39] nanoscience [40 41] andgeophysical systems [42] One of the main aspects of thesesituations is the correlation functions of the above anomalousdiffusion phenomena which may be related to the non-Markovian characteristic of the stochastic process [43] Thetypical characteristic of anomalous diffusion lies in themean-square displacement (MSD) which satisfies ⟨119909(119905)2⟩ sim 119905120572where the diffusion exponents 0 lt 120572 lt 1 and 120572 gt 1 indicatesubdiffusion and superdiffusion respectively When 120572 = 1the normal diffusion is recovered [44]

It is well-known that the normal diffusion can bemodeledby a Langevin equation where a Brownian particle subjectedto a viscous drag from the surrounding medium is charac-terized by a friction force and it also subjected to a stochasticforce that arises from the surrounding environmentThe fric-tion constant determines how quickly the system exchangesenergy with the surrounding environment For a realisticdescription of the surrounding environment it is difficult tochoose a universal value of the friction constant Indeed inorder to depict the real situation more effectively a differentvalue of the friction constant should be adopted Hence ageneralization of the Langevin equation is needed leading tothe so-called generalized Langevin equation (GLE) [45] TheGLE is an equation ofmotion for a non-Markovian stochasticprocess where the particle has a memory effect to its velocity

Nowadays the GLE driven by a fractional Gaussian noise(fGn) [46]with a power-law friction kernel is extensively used

for modeling anomalous diffusion processes For instance inthe study on dynamics of single-molecule when the electrontransfer (ET) was used to probe the conformational fluctu-ations of single-molecule enzyme the distance between theET donor and acceptor can be modeled well through a GLEdriven by an fGn [47ndash54] Besides Vinales and Despositohave introduced a novel noise whose correlation function isproportional to a Mittag-Leffler function which is called theMittag-Leffler noise (M-Ln) [55ndash57] The correlation func-tion behaves as a power-law for large times but is nonsingularat the origin due to the inclusion of a characteristic time

Theoverwhelmingmajority of previous studies of SRhaverelated to the case where the external noise and the weakperiodic force are introduced additively However Dykmanet al [58] studied the case where the signal is multiplied tonoise namely the noise is modulated by a signal They foundthatwhen an asymmetric bistable system is driven by a signal-modulated noise stochastic resonance appeared in contrastto the additive noise new characteristics emerge and theirresults were in agreement with experiments FurthermoreCao and Wu [59] studied the SR characteristics of a linearsystem driven by a signal-modulated noise and an additivenoise It seems that a periodically modulated noise is notuncommon and arises for example at the output of anyamplifier (optics or radio astronomy) whose amplificationfactor varies periodically with time

Due to the synergy of generalized friction kernel of a GLEand the periodically modulated noise the stochastic resonantbehaviors of a GLE can be influenced In contrast to thecase that has been investigated before new dynamic char-acteristics emerge Motivated by the above discussions wewould like to explore the stochastic resonance phenomenonin a generalized harmonic oscillator with multiplicative andperiodically modulated noises Moreover we consider theGLE is driven by a fractional Gaussian noise and a Mittag-Leffler noise respectively in this paper We focus on thevarious nonmonotonic behaviors of the output amplitude 119860with the systemparameters and the parameters of the internaldriven noise

The physical motivations of this paper are as follows (1)in view of the importance of stochastic generalized harmonicoscillator (linear oscillator) with memory in physics chem-istry and biology and due to the periodicallymodulated noisearising at the output of the amplifier of the optics deviceand radio astronomy device to establish a physical model inwhich the SR can contain the effects of the two factors thelinearity of the system and the periodical modulation of thenoise (2) The second one is to give a theoretical foundationfor the study of SR characteristic features of a generalizedharmonic oscillator subject to multiplicative periodicallymodulated noises and external periodic force Our studyshows that such a model leads to stochastic resonancephenomenon Meanwhile an exact analytic expression of theoutput amplitude is obtained Based on it some characteristicfeatures of SR are revealed

The paper is organized as follows Section 2 gives theintroductions of the generalized Langevin equation the frac-tional Gaussian noise and the Mittag-Leffler noise Section 3gives analytical expression of the output amplitude of the



2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2 System Model


(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + 119889119880 (119909)119889119909 = 120578 (119905) (1)


⟨120578 (119905) 120578 (119904)⟩ = 119862 (|119905 minus 119904|) = 119896119861119879 sdot 119870 (|119905 minus 119904|) (2)




(1) 119861119867 (0) = 0(2) ⟨119861119867 (119905)⟩ = 0 119905 gt 0(3) ⟨119861119867 (119905) 119861119867 (119904)⟩ = 12 (|119905|2119867 + |119904|2119867 minus |119905 minus 119904|2119867) 119905 119904 gt 0

(3)


120576119867 (119905) = radic2119896119861119879119889119861119867 (119905)119889119905 119905 gt 0 (4)


119862 (119905) = ⟨120576119867 (0) 120576119867 (119905)⟩ = 2119896119861119879sdot lim119904rarr0+


119904rarr0+




(5)



119862 (119905) = 119896119861119879 1120591120572119864120572 [minus(|119905|120591 )120572] (6)


119864120572 (119910) = infinsum119895=0

119910119895Γ (120572119895 + 1) (7)


119862 (119905) = 1198621120591 exp(minus|119905|120591 ) (8)




(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905) + 120574int1199050119870 (119905 minus 119906) (119906) 119889119906 + [12059620 + 1205851 (119905)] 119909

= 1198601 sin (Ω119905) + 1198602 sin (Ω119905) 1205852 (119905) + 120578 (119905) (9)


⟨1205851 (119905) 1205851 (119904)⟩ = 1198862 exp (minusV |119905 minus 119904|) (10)




3 First Moment


⟨1205851 119889119899119909119889119905119899 ⟩ = ( 119889119889119905 + V)119899 ⟨1205851119909⟩ (12)


0119870 (119905 minus 119906) 119889 ⟨119909 (119906)⟩119889119906 119889119906 + 12059620 ⟨119909⟩ + ⟨1205851119909⟩

= 1198601 sin (Ω119905) (13)



⟨1205851 (119905) 11988921199091198891199052 ⟩ + 120574int1199050119870 (119905 minus 119906)⟨1205851 (119905) 119889119909119889119906⟩119889119906

+ 12059620 ⟨1205851119909⟩ + 1198862 ⟨119909⟩ = 1198602119863 sin (Ω119905) (14)


[1198892 ⟨1205851119909⟩1198891199052 + 2V119889 ⟨1205851119909⟩119889119905 + V2 ⟨1205851119909⟩] + 12059620 ⟨1205851119909⟩+ 1198862 ⟨119909⟩ + 120574119890minusV119905 int119905

0119870 (119905 minus 119906)

sdot [119889 ⟨1205851 (119906) 119909 (119906)⟩119889119906 + V ⟨1205851 (119906) 119909 (119906)⟩] 119890V119906119889119906= 1198602119863 sin (Ω119905)

(15)




[1199042 + 120574 (119904) 119904 + 12059620]1198831 (119904) + 1198832 (119904)= LT [1198601 sin (Ω119905)]

11988621198831 (119904) + [(119904 + V)2 + 12059620 + 120574 (119904 + V) (119904 + V)]1198832 (119904)= LT [1198602119863 sin (Ω119905)]

(16)




(17)



11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











11 (119904) = 1 (119904) [12059620 + (119904 + V)2 + 120574 (119904 + V) (119904 + V)] 21 (119904) = minus1 (119904)

(18)





(19)






Ω119905


(20)

A1 sin (Ωt)

A2Dsin (Ωt)

H11(t)

H21(t)

A11 sin (Ωt + 12059311)

A21 sin (Ωt + 12059321)





(21)


(22)


11986011 = 1198601 1003816100381610038161003816100381611986711 (119890119895Ω)10038161003816100381610038161003816 = 1198601radic11986021 + 1198612111986221 + 11986321 11986022 = 1198602119863 1003816100381610038161003816100381611986721 (119890119895Ω)10038161003816100381610038161003816 = 1198602119863radic11986022 + 1198612211986222 + 11986322

(23)


119860 ≜ radic119860211 + 119860221 + 21198601111986021 cos (1205931112059321) (24)


0 lt 1198862 le 1198862cr = 12059620 [12059620 + V2 + 120574V sdot (V)] (25)




119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870 (119905) = 119862 (119905)119896119861119879 = 2119867 (2119867 minus 1) 1199052119867minus2 (26)




(28)

12059321 = arctan(minus11989141198913) (29)

where




119887 = radicV2 + Ω2120579 = arctan(Ω

V)


(30)


119870 (119905) = 119862 (119905)119896119861119879 = 1120591120572119864120572 [minus(|119905|120591 )120572] (31)




(33)

where



+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+11987221198724 cos(2120579 + 1205872 120572)+119872211987241198725 cos [(2 + 120572) 120579 + 1205872 120572]






119887 = radicV2 + Ω2120579 = arctan(Ω

V)

1198720 = 12059620 minus Ω21198721 = 120574Ω1205721198722 = 1198872

1198723 = 1205741198871205721198724 = (120591Ω)120572 1198725 = (120591119887)120572 1198726 = 11987201198724 +11987211198727 = 119872512059620 +11987231198728 = 120596201198720 minus 11988621198729 = 11987201198727 minus1198725119886211987210 = 11987241198728 +119872112059620 11987211 = 1198722119872611987212 = 1198725 (11987241198728 + 120596201198721) +1198723119872611987213 = 119872211987251198726

(34)









X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X = 09Y = 02Level = 1

X = 13Y = 06Level = 2

X = 03Y = 095Level = 0

(i)

(ii)

(0)

02 04 06 08 1 12 14 16 18 20D

0

01

02

03

04Ω

05

06

07

08

09

1


05

1

15

2

25

3

35

4

45

5

A

05 10a2

(a)

1

2

3

4

5

6

7

8

9

10

A

05 10a2

(b)

3

35

4

45

5

55

6

65

A

05 10a2

(c)







X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X = 04Y = 07Level = 2

X = 11Y = 025Level = 1

X = 01Y = 005Level = 0

(ii)

(i)

(0)

0

01

02

03

04Ω

05

06

07

08

09

1

02 04 06 08 1 12 14 16 18 20D





0 lt 1198862 le 1198862crMLn = 12059620 [12059620 + V2 + 120574V1205721 + (120591V)120572 ] (36)






5 Conclusions



0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

A

05 10a2

(a)

2

3

4

5

6

7

8

9

A

05 10a2

(b)

15

2

25

3

35

4

45

A

05 10a2

(c)




Competing Interests


Acknowledgments


References
























































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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Research Article The Stochastic Resonance Behaviors of a ...downloads.hindawi.com/journals/amp/2016/5164575.pdf · Harmonic Oscillator Subject to Multiplicative and Periodically Modulated