{"title":"Parametric Resonance of the Non-Markovian Oscillator","authors":"O. L. Sirotkin, N. A. Koplevatsky","doi":"10.1007/s11141-024-10294-y","DOIUrl":null,"url":null,"abstract":"<p>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.</p>","PeriodicalId":748,"journal":{"name":"Radiophysics and Quantum Electronics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Radiophysics and Quantum Electronics","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s11141-024-10294-y","RegionNum":4,"RegionCategory":"地球科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
引用次数: 0
Abstract
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.
期刊介绍:
Radiophysics and Quantum Electronics contains the most recent and best Russian research on topics such as:
Radio astronomy;
Plasma astrophysics;
Ionospheric, atmospheric and oceanic physics;
Radiowave propagation;
Quantum radiophysics;
Pphysics of oscillations and waves;
Physics of plasmas;
Statistical radiophysics;
Electrodynamics;
Vacuum and plasma electronics;
Acoustics;
Solid-state electronics.
Radiophysics and Quantum Electronics is a translation of the Russian journal Izvestiya VUZ. Radiofizika, published by the Radiophysical Research Institute and N.I. Lobachevsky State University at Nizhnii Novgorod, Russia. The Russian volume-year is published in English beginning in April.
All articles are peer-reviewed.