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Trichotomous noise induced resonance behavior for a fractional oscillator with random mass. (English) Zbl 1396.70026
Summary: We investigate the stochastic resonance (SR) phenomenon in a fractional oscillator with random mass under the external periodic force. The fluctuations of the mass are modeled as a trichotomous noise. Applying the Shapiro-Loginov formula and the Laplace transform technique, we obtain the exact expression of the first moment of the system. The non-monotonic behaviors of the spectral amplification (SPA) versus the driving frequency indicate that the bona fide SR appears. The necessary and sufficient conditions for the emergence of the generalized stochastic resonance phenomena on the noise flatness and on the noise intensity in the particular case of \(\Omega =\omega_0\), \(v\to 0\) are established. Particularly, the hypersensitive response of the SPA to the noise intensity is found, which is previously reported and believed to be absent in the case of dichotomous noise.

70L05 Random vibrations in mechanics of particles and systems
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Gitterman, M; Shapiro, I, Stochastic resonance in a harmonic oscillator with random mass subject to asymmetric dichotomous noise, J. Stat. Phys., 144, 139-149, (2011) · Zbl 1225.82047
[2] Klafter, J; Sokolov, IM, Anomalous diffusion spreads its wings, Phys. World, 18, 29-32, (2005)
[3] Goychuk, I, Subdiffusive Brownian ratchets rocked by a periodic force, Chem. Phys., 375, 450-457, (2010)
[4] Metzler, R; Barkai, E; Klafter, J, Deriving fractional Fokker-Planck equations from a generalized master equation, Europhys. Lett., 46, 431-436, (1999)
[5] Metzler, R; Klafter, J, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77, (2000) · Zbl 0984.82032
[6] Metzler, R; Klafter, J, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37, r161-r208, (2004) · Zbl 1075.82018
[7] Lutz, E, Fractional Langevin equation, Phys. Rev. E, 64, 051106, (2001)
[8] Burov, S; Barkai, E, Critical exponent of the fractional Langevin equation, Phys. Rev. Lett., 100, 070601, (2008)
[9] Deng, WH; Barkai, E, Ergodic properties of fractional Brownian-Langevin motion, Phys. Rev. E, 79, 011112, (2009)
[10] Jeon, JH; Metzler, R, Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement, Phys. Rev. E, 85, 021147, (2012)
[11] Kneller, GR; Baczynski, K; Pasenkiewicz-Gierula, M, Communication: consistent picture of lateral subdiffusion in lipid bilayers: molecular dynamics simulation and exact results, J. Chem. Phys., 135, 141105, (2011)
[12] Jeon, JH; Leijnse, N; Oddershede, LB; Metzler, R, Anomalous diffusion and power-law relaxation of the time averaged Mean squared displacement in worm-like micellar solutions, New J. Phys., 15, 045011, (2013)
[13] Zhong, SC; Wei, K; Gao, SL; Ma, H, Stochastic resonance in a linear fractional Langevin equation, J. Stat. Phys., 150, 867-880, (2013) · Zbl 1266.82046
[14] Lizana, L; Ambjörnsson, T; Taloni, A; Barkai, E; Lomholt, MA, Foundation of fractional Langevin equation: harmonization of a many-body problem, Phys. Rev. E, 81, 051118, (2010)
[15] Kou, SC; Xie, XS, Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. Rev. Lett., 93, 180603, (2004)
[16] Mandelbrot, BB; Ness, JW, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437, (1968) · Zbl 0179.47801
[17] Jeon, JH; Metzler, R, Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries, Phys. Rev. E, 81, 021103, (2010)
[18] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[19] Kubo, R, The fluctuation-dissipation theorem, Rep. Prog. Phys., 29, 255-284, (1966) · Zbl 0163.23102
[20] Rekker, A; Mankin, R, Energetic instability of a fractional oscillator, WSEAS Trans. Syst., 9, 203-212, (2010)
[21] Mason, TG; Weitz, DA, Optical measurements of the linear viscoelastic moduli of complex fluids, Phys. Rev. Lett., 74, 1250-1253, (1995)
[22] Guigas, G; Kalla, C; Weiss, M, Probing the nanoscale viscoelasticity of intracellular fluids in living cells, Biophys. J., 93, 316-323, (2007)
[23] Gotze, W; Sjogren, L, Relaxation processes in supercooled liquids, Rep. Prog. Phys., 55, 241, (1992)
[24] Gu, Q; Schiff, EA, Non-Gaussian transport measurements and the Einstein relation in amorphous silicon, Phys. Rev. Lett., 76, 3196-3199, (1996)
[25] Soika, E; Mankin, R; Lumi, N, Parametric resonance of a harmonic oscillator with fluctuating mass, 233, (2012), New York
[26] Gitterman, M, New type of Brownian motion, J. Stat. Phys., 146, 239-243, (2012) · Zbl 1235.82053
[27] Benzi, R; Sutera, A; Vulpiani, A, The mechanism of stochastic resonance, J. Phys. A, 14, l453-457, (1981)
[28] Benzi, R; Parisi, G; Sutera, A; Vulpiani, A, Stochastic resonance in climatic change, Tellus, 34, 10-16, (1982)
[29] Nicolis, C, Stochastic aspects of climatic transitions: response to a periodic forcing, Tellus, 34, 1, (1982)
[30] Hänggi, P; Inchiosa, ME; Fogliatti, D; Bulsara, AR, Nonlinear stochastic resonance: the saga of anomalous output-input gain, Phys. Rev. E, 62, 6155-6163, (2000)
[31] Mankin, R; Ainsaar, A; Reiter, E, Trichotomous noise-induced transitions, Phys. Rev. E., 60, 1374, (1999)
[32] Lang, RL; Yang, L; Qin, HL; Di, GH, Trichotomous noise induced stochastic resonance in a linear system, Nonlinear Dyn., 69, 1423-1427, (2012)
[33] Sauga, A., Mankin, R., Ainsaar, A.: AIP Conference Proceedings. Resonant behavior of a fractional oscillator with fluctuating mass, pp. 224-232. AIP, New York (2012)
[34] Soika, E; Mankin, R; Priimets, J, Generalized Langevin equation with multiplicative trichotomous noise, Proc. Estonian Acad. Sci., 61, 113-127, (2012) · Zbl 1256.34046
[35] Shapiro, VE; Loginov, VM, “formulae of differentiation” and their use for solving stochastic equations, Physica A, 91, 563-574, (1978)
[36] Laas, K., Mankin, R., Reiter, E.: Stochastic resonance in the case of a generalized Langevin equation with a Mittag-Leffler friction kernel. In: Proceeding MACMESE 10 Proceedings of the 12th WSEAS International Conference on Mathematical and Computational Methods in Science and Engineering, pp. 313-318 (2010). ISBN: 978-960-474-243-1
[37] Soika, E; Mankin, R; Priimets, J, Response of a generalized Langevin system to a multiplicative trichotomous noise, 87-93, (2011), Singapore
[38] Oppenheim, A.V., Willsky, A.S., Nawab, S.H.: Signals and Systems. Prentice Hall, Shangai (2005)
[39] Soika, E; Mankin, R; Ainsaar, A, Resonant behavior of a fractional oscillator with fluctuating frequency, Phys. Rev. E, 81, 011141, (2010)
[40] Gammaitoni, L; Marchesoni, F; Santucci, S, Stochastic resonance as a Bona fide resonance, Phys. Rev. Lett., 74, 1052-1055, (1995)
[41] Mankin, R; Lass, K; Laas, T; Reiter, E, Stochastic multiresonance and correlation-time-controlled stability for a harmonic oscillator with fluctuating frequency, Phys. Rev. E., 78, 031120, (2008)
[42] Gitterman, M, Classical harmonic oscillator with multiplicative noise, Physica A, 352, 309-334, (2005)
[43] Laas, K., Mankin, R., Rekker, A.: Hypersensitive response of a harmonic oscillator with fluctuating frequency to noise amplitude. In: Proceedings of the 5th WSEAS International Conference on Mathematical Biology and Ecology, pp. 15-20 (2009). ISBN: 978-960-474-038-3 · Zbl 1225.82047
[44] Laas, K; Mankin, R; Rekker, A, Constructive influence of noise flatness and friction on the resonant behavior of a harmonic oscillator with fluctuating frequency, Phys. Rev. E., 79, 051128, (2009)
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