We develop differential equations for the probability density of the phase coordinates of the dynamic systems with parametric fluctuations in the form of the non-Markovian dichotomic noise, which has arbitrary lifetime distribution functions in the states ±1. As an example, we calculate the first moment of the phase coordinate of the linear oscillator, whose perturbed motion is described by the stochastic analogue of the Mathieu—Hill equation. These calculations aim at demonstrating the fact that in the case of linear dynamic systems, the non-Markovian parametric fluctuations having hidden periodicity are capable of inducing the states absent in the deterministic regime without periodic coefficients. The problem is solved by the method of supplementary variables, which transforms the non-Markovian dichotomic noise to the Markovian noise. It is shown that the amplitude oscillations, which are typical of the parametric resonance are present, when the structure of the dichotomic noise is described by the lifetime distribution function in the states ±1 in the form of the sum of weighted Erlang distribution exponents of the various order and a constant value of +1. The delta-correlated and Gaussian properties of the studied processes are not used. The calculations are performed within the framework of simple differential equations without involving integral operators and the Novikov—Furutsu—Donsker theorem.
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A. A.Kochanov, T.E.Vadivasova, and V. S.Anishchenko, Izvestiya Vyssh. Uchebn. Zaved. Applied Nonlinear Dynamics, 19, 2, 43–55 (2011). https://doi.org/10.18500/0869-6632-2011-19-2-43-55
P. S. Landa and A.A. Zaikin, J. Exp. Theor. Phys., 84, No. 1, 197–208 (1997). https://doi.org/10.1134/1.558137
W. Horsthemke and R. Lefever, Noise-Induced Transitions, Springer (2010). https://doi.org/10.1007/3-540-36852-3
V. V. Bolotin and V. G. Moskvin, Izv. AN SSSR. Mekh. Tverd. Tela, No. 4, 88–94 (1972).
J. McKenna and J.A.Morrison, J. Math. Phys., 11, No. 8, 2348–2360 (1970). https://doi.org/10.1063/1.1665402
O. L. Sirotkin, Izv. Saratov Univ. Physics, 21, No. 4, 343–354 (2021). https://doi.org/10.18500/1817-3020-2021-21-4-343-354
N. G.Van Kampen, Stochastic Processes in Physics and Chemistry, Elsevier, North Holland (2007). https://doi.org/10.1016/B978-0-444-52965-7.X5000-4
D.R. Cox, Proc. Camb. Philos. Soc., 51, No. 3, 433–441 (1955). https://doi.org/10.1017/S0305004100030437
F. Lange, Correlation Electronics [in Russian], Sudpromgiz, Leningrad (1963).
V. I. Tikhonov and M.A.Mironov, Markovian Processes [in Russian], Sov. Radio, Moscow (1977).
V. T.Altarev, G. I. Shakun, and P. I.Trofimov, Failure and Recovery Processes in Data Transmission Systems [in Russian], Svyaz’, Moscow (1977).
I. Goychuk, Phys. Rev. E, 70, No. 1, 016109 (2004). https://doi.org/10.1103/PhysRevE.70.016109
A. N. Morozov and A. V. Skripkin, Non-Markovian Physical Processes [in Russian], Fizmatlit, Moscow (2018).
A. N. Morozov and A. V. Skripkin, Herald of the Bauman Moscow State Techn. Univ. Ser. Natural Sciences, No. 1, 68–77 (2011).
A. N. Morozov, Herald of the Bauman Moscow State Techn. Univ. Ser. Natural Sciences, No. 5, 57–66 (2017). https://doi.org/10.18698/1812-3368-2017-5-57-66
A. S. Kharitonov and L.A. Shelepin, Kratk. Soobsch. Fiz. Lebedev Phys. Inst., No.1, 17–22 (1998).
Yu.A.Kulagin, R. I. Serikov, I.V. Siminovsky, and L.A. Shelepin, Kratk. Soobsch. Fiz. Lebedev Phys. Inst., No. 7, 17–24 (1999).
É.A.Azroyants and L.A. Shelepin, Non-Markovian Processes and Their Applications [in Russian], Preprint No. 58, Lebedev Phys. Inst. Russ. Acad. Sci. (1998).
R. Bobryk and A.Chrzeszczyk, Physica A. Stat. Mech. Appl., 316, 225–232 (2002).https://doi.org/10.1016/S0378-4371(02)01312-2
F.Poulin and G. Flierl, Proc. R. Soc. A, 464, 1885–1904 (2008). https://doi.org/10.1098/rspa.2008.0007
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Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 66, No. 4, pp. 306–316, April 2023. Russian DOI: https://doi.org/10.52452/00213462_2023_66_04_306
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Sirotkin, O.L., Koplevatsky, N.A. Parametric Resonance of the Non-Markovian Oscillator. Radiophys Quantum El 66, 276–285 (2023). https://doi.org/10.1007/s11141-024-10294-y
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DOI: https://doi.org/10.1007/s11141-024-10294-y