About asymptotic behavior of the mathematical expectation of the total energy of the harmonic oscillator with random perturbation
DOI:
https://doi.org/10.17721/1812-5409.2018/3.6Abstract
In this article, the asymptotic behavior of the mathematical expectation of the total energy of a harmonic oscillator without friction under the influence of an energy pump with a controlling element of the form of a stochastic harmonic oscillator without friction with a white noise perturbation in resonant and non-resonant cases is found. During the analytical solving the problem of finding the mathematical expectation of the total energy of a harmonic oscillator with random perturbation, the properties of the Wiener process, the stochastic Ito integral, and the mathematical expectation of the product of stochastic integrals are used.
Key words: harmonic oscillator, total energy, white noise.
Pages of the article in the issue: 47 - 52
Language of the article: Ukrainian
References
BANDERSKYI, M. and PASTUR, L. (1975) Ob asymptotike uravnenyi 2-go poryadka so sluchaynyim koeffitsientom Teoriya funktsyi, funktsion. analiz i ikh pril., 22, p.3-14.
GIKHMAN, I. and SKOROKHOD, A. Vvedenie v teoriyu sluchaynyikh protsessov. Moskva: Nauka.
DIVNICH, M., KIROVSKYI, D. and YERSHOV, A. (2005) Asymptotichnyi analiz matematichnogo spodivannya povnoyi energiyi vipadkovogo garmonichnogo oscillyatora Visn. Kyiv. Un-ty. Mathematika, mekhanika., 3, p.104-112.
KULINICH, G. and BERNATSKAYA, YU. (1998) O phazovom “portrete” garmonicheskogo oscillatora s treniem, vozmushennogo sluchaynyim protsessom tipa “belogo shuma” Math. Zap., 68 (4), p. 862-869.
KULINICH, G. (1997) Yakisnyi analiz vplivu na garmonichnyi oscillator z tertyam vipadkovikh zburen’ tipu “bilogo shumu” vzdovzh vektora phazovoyi shvidkosti Ukr. Math. Journ., 49 (1), p.36-47.
KULINICH, G. (1998) Yakisnyi analiz povedinki garmonichnogo oscillyatora pid vplivom vipadkovikh zburen’ parametriv protsessami tipu “bilogo I drobovogo shumiv” Teoriya imovirnostey i math. statistika, 58, p. 81-91.
KULINICH, G., KUROVSKYI, D. and PETRUSENKO, D. (2006) Pro asymptotichnu povedinku garmonichnogo ostsillatora mathematichnogo spodivannya povnoyi energiyi stokhastichnogo garmonichnogo ostsillatora Math. XI Mizhnar. Nauk. Conf. Im. Akad. M. Kravtchuka. Kyiv, p.719.
KULINICH, G. (1985) Pro granichnu povedinku vipadkovogo oscillatora Visn. Kyiv. Un-ty. Mathematika, mekhanika., 25, p.108-113.
LEVAKOV, A. (2010) Vvedenie v teoriyu integralov Lebega I Ito. Posobie dlya studentov prikladnoy mathematiki i informatiki [Online] p.18-22. – Avialable from: http://elib.bsu.by/bitstream/123456789/1631/1/metmain.pdf.
EL’TSGOL’TS, L. (1965) Differentsial’nie uravneniya i variatsionnoe ischisleniye. Moskva: Nauka.
KULINICH, G. (1995) On the limiting behaviors of a harmonic oscillator with random external disturbance Y.A.M.S.A., 8, p.265-274.
KULINICH, G. (1995) Qualitative analisys of the influence of random perturbations of the phase velocity of the garmonic oscillator Random Oper. And Stochast. Equat., 3 (2), p.141-152.
PAPANIROLAN, G. (1973) Stohastic equations and their applications Amer. Math. Mon., 5, p.526-545.
Downloads
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
