×

zbMATH — the first resource for mathematics

Generalized stochastic resonance for a fractional noisy oscillator with random mass and random damping. (English) Zbl 1436.34061
Summary: In this paper, we consider the random dichotomous fluctuations on both mass and damping in a fractional oscillator, which is subject to an additive fractional Gaussian noise and driven by a periodic force. In order to investigate the generalized stochastic resonance (GSR) phenomena, we acquire the exact expression of the first-order moment of system’s steady response by applying the generalized fractional Shapiro-Loginov formula and Laplace transform. Additionally, we discuss the evolutions of the output amplitude amplification (OAA) with driving frequency, noise parameters, fractional order, and damping strength. It is observed that the non-monotonic resonance behaviors of one-peak GSR, double-peak GSR and triple-peak GSR exist in this fractional system. Moreover, the interplay of mass fluctuation, damping fluctuation, and memory effect can generate a rich variety of non-equilibrium cooperation phenomena, especially the stochastic multi-resonance (SMR) behaviors. It is worth emphasizing that the triple-peak GSR was not observed in previously proposed fractional oscillator subjected to dichotomous noise. Finally, the numerical simulations are also carried out based on predictor-corrector approach to verify the effectiveness of analytic result.
MSC:
34F15 Resonance phenomena for ordinary differential equations involving randomness
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
34F05 Ordinary differential equations and systems with randomness
37C60 Nonautonomous smooth dynamical systems
34C60 Qualitative investigation and simulation of ordinary differential equation models
Software:
FODE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benzi, R.; Sutera, A.; Vulpliani, A., The mechanism of stochastic resonance, J. Phys. A: Math. Gen., 14, L453-457 (1981)
[2] Gammaitoni, L.; Hänggi, P.; Jung, P.; Marchesoni, F., Stochastic resonance, Rev. Mod. Phys., 70, 223-287 (1998)
[3] Wellens, T.; Shatokhin, V.; Buchleitners, A., Stochastic resonance, Rep. Prog. Phys., 67, 45-105 (2004)
[4] Chapeau-Blondeau, F.; Rousseau, D., Noise improvements in stochastic resonance: from signal amplification to optimal detection, Fluct. Noise Lett., 2, L221-233 (2002)
[5] Ai, B.; Liu, L., Stochastic resonance in a stochastic bistable system, J. Stat. Mech., 2007, P02019 (2007) · Zbl 07120390
[6] Gitterman, M., Classical harmonic oscillator with multiplicative noise, Physica A, 352, 309-334 (2005)
[7] McNamara, B.; Wiesenfeld, K.; Roy, R., Observation of stochastic resonance in a ring laser, Phys. Rev. Lett., 60, 2626-2629 (1988)
[8] Metzler, R.; Klafter, J., The restaurant at the end of the randomwalk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A: Math. Gen., 37, R161C208 (2004)
[9] Kou, SC, Stochastic modeling in nanoscale biophysics: subdiffusion within proteins, Ann. Appl. Stat., 2, 501C535 (2008)
[10] Deng, W.; Barkai, E., Ergodic properties of fractional Brownian Langevinmotion, Phys. Rev. E, 79, 011112 (2009)
[11] Bao, J.; Zhuo, Y., Investigation on anomalous diffusion for nuclear fusion reactions, Phys. Rev. C, 67, 233-234 (2003)
[12] Goychuk, I., Anomalous relaxation and dielectric response, Phys. Rev. E, 76, 040102 (2007)
[13] Lin, L.; Zhou, X.; Ma, H., Subdiffusive transport of fractional two-headed molecular motor, Acta Phys. Sin., 62, 240501 (2013) · Zbl 1289.37017
[14] Achar, BN Narahari; Hanneken, JW; Enck, T.; Clarke, T., Dynamics of the fractional oscillator, Physica A, 297, 361-367 (2001) · Zbl 0969.70511
[15] Ryabov, YE; Puzenko, A., Damped oscillations in view of the fractional oscillator equation, Phys. Rev. B, 66, 553-562 (2002)
[16] Sauga, A.; Mankin, R.; Ainsaar, A., Resonant behavior of a fractional oscillator with fluctuating mass, AIP Conf. Proc., 1487, 224 (2012)
[17] Zhong, S.; Wei, K.; Gao, S.; Ma, H., Trichotomous noise induced resonance behavior for a fractional oscillator with random mass, J. Stat. Phys., 159, 195-209 (2015) · Zbl 1396.70026
[18] Zhong, S.; Yang, J.; Zhang, L.; Ma, H.; Luo, M., Resonant behavior of a harmonic oscillator with fluctuating mass driven by a Mittag-Leffler noise, J. Stat. Mech., 2017, 023211 (2017)
[19] Huang, Z.; Guo, F., Stochastic resonance in a fractional linear oscillator subject to random viscous damping and signal-modulated noise, Chin. J. Phys., 54, 69-76 (2016)
[20] Ren, R.; Luo, M.; Deng, K., Stochastic resonance in a fractional oscillator subjected to multiplicative trichotomous noise, Nonlinear Dyn., 90, 379-390 (2017)
[21] Mankin, R.; Kekker, A., Memory-enhanced energetic stability for a fractional oscillator with fluctuating frequency, Phys. Rev. E, 81, 041122 (2010)
[22] Zhong, S.; Zhang, L.; Wang, H.; Ma, H.; Luo, M., Nonlinear effect of time delay on the generalized stochastic resonance in a fractional oscillator with multiplicative polynomial noise, Nonlinear Dyn., 89, 1327-1340 (2017)
[23] Tian, Y.; Zhong, L.; He, G.; Yu, T.; Luo, M.; Stanley, HE, The resonant behavior in the oscillator with double fractional-order damping under the action of nonlinear multiplicative noise, Physica A, 490, 845-856 (2018)
[24] He, G.; Tian, Y.; Wang, Y., Stochastic resonance in a fractional oscillator with random damping strength and random spring stiffness, J. Stat. Mech., 2013, P09026 (2013)
[25] Lin, L.; Chen, C.; Wang, H., Trichotomous noise induced stochastic resonant in a fractional oscillator with random damping and random frequency, J. Stat. Mech., 2016, 023201 (2016)
[26] Burov, S.; Gitterman, M., Noisy oscillator: random mass and random damping, Phys. Rev. E, 94, 052144 (2016)
[27] Landau, LD; Lifshitz, EM, Statistical Physics (1958), London: Pergamon, London
[28] Kubo, R., The fluctuation-dissipation theorem, Rep. Prog. Phys., 29, 255-284 (1966) · Zbl 0163.23102
[29] Gitterman, M., New type of Brownian motion, J. Stat. Phys., 146, 239-243 (2012) · Zbl 1235.82053
[30] Gitterman, M., Stochastic oscillator with random mass: new type of Brownian motion, Physica A, 395, 11-21 (2014) · Zbl 1395.82181
[31] Wang, H.; Ni, F.; Lin, L.; Lv, W.; Zhu, H., Transport behaviors of locally fractional coupled Brownian motors with fluctuating interactions, Physica A, 505, 124-135 (2018)
[32] Abdalla, MS, Time-dependent harmonic oscillator with variable mass under the action of a driving force, Phys. Rev. A, 34, 4598-4605 (1986)
[33] Ausloos, M.; lambiotte, R., Brownian particle having a fluctuating mass, Phys. Rev. E, E73, 011105 (2006)
[34] Dykman, MI; Khasin, M.; Portman, J.; Shaw, SW, Spectrum of an oscillator with jumping frequency and the interference of partial susceptibilities, Phys. Rev. Lett., 105, 230601 (2010)
[35] Gadomski, A.; Siódmiak, J.; Santamarìa-Holek, I.; Rubì, JM; Ausloos, M., Kinetics of growth process controlled by mass-convective fluctuations and finite-size curvature effects, Acta Phys. Polon. B, 36, 1537-1559 (2005)
[36] West, BJ; Seshadri, V., Model of gravity wave growth due to fluctuations in the air-sea coupling parameter, J. Geophys. Res., 86, 4293-4298 (1981)
[37] Chomaz, JM; Couarion, A., Against the wind, Phys. Fluids, 11, 2977-2983 (1999) · Zbl 1149.76343
[38] Helot, F.; Libchaber, A., Unidirectional crystal growth and crystal anisotropy, Phys. Scr, T9, 126-129 (1985)
[39] Saul, A.; Showalter, K., Oscillations and travel waves in chemical systems (1985), New York: Wiley, New York
[40] Lutz, E., Fractional Langevin equation, Phys. Rev. E, 64, 051106 (2001)
[41] Burov, S.; Barkai, E., Fractional Langevin equation: overdamped, underdamped and critical behaviors, Phys. Rev. E, 78, 031112 (2008)
[42] Ghosh, SK; Cherstvy, AG; Metzler, R., Non-universal tracer difussion in crowded media of non-inert obstacles, Phys. Chem. Chem. Phys., 17, 1847-1858 (2015)
[43] Liu, L.; Cherstvy, AG; Metzler, R., Facilitated diffusion of transcription factor proteins with anomalous bulk diffusion, J. Phys. Chem. B., 121, 1284-1289 (2017)
[44] Yu, T.; Luo, M.; Hua, Y., The resonant behavior of fractional harmonic oscillator with fluctuating mass, Acta Phys. Sin., 62, 210503 (2013)
[45] Laas, K.; Mankin, R., Resonant behavior of a fractional oscillator with random damping, AIP Conf. Proc., 1404, 131-138 (2011)
[46] Kou, SC; Xie, X., Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. Rev. Lett., 93, 180603 (2004)
[47] Caputo, M., Linear models of dissipation whose Q is almost frequency independent, Geophys. J. R. Astron. Soc., 13, 529-539 (1967)
[48] Podlubny, I., Fractional Differential Equations (1999), San Diego: Academic Press, San Diego · Zbl 0918.34010
[49] Shapiro, VE; Loginov, VM, Formulae of differentiation and their use for solving stochastic equations, Physica A, 91, 563-574 (1978)
[50] Oppenheim, AV; Willsky, AS; Nawab, SH, Signals and Systems (2012), Xi’an: Prentice Hall, Xi’an
[51] Burada, PS; Schmid, G.; Reguera, D.; Rubi, JM; Hänggi, P., Double entropic stochastic resonance, Europhys. Lett., 87, 50003 (2009)
[52] Deng, WH, Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227, 1510-1522 (2007) · Zbl 1388.35095
[53] Deng, WH; Barkai, E., Ergodic properties of fractional Brownian-Langevin motion, Phys. Rev. E, 79, 011112 (2009)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.