{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,7]],"date-time":"2026-01-07T23:07:49Z","timestamp":1767827269298,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T00:00:00Z","timestamp":1497484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A semi-analytical method is proposed to calculate stochastic quasi-periodic responses of limit cycles in non-equilibrium dynamical systems excited by periodic forces and weak random fluctuations, approximately. First, a kind of 1\/N-stroboscopic map is introduced to discretize the quasi-periodic torus into closed curves, which are then approximated by periodic points. Using a stochastic sensitivity function of discrete time systems, the transverse dispersion of these circles can be quantified. Furthermore, combined with the longitudinal distribution of the circles, the probability density function of these closed curves in stroboscopic sections can be determined. The validity of this approach is shown through a van der Pol oscillator and Brusselator.<\/jats:p>","DOI":"10.3390\/e19060280","type":"journal-article","created":{"date-parts":[[2017,6,15]],"date-time":"2017-06-15T10:07:33Z","timestamp":1497521253000},"page":"280","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Approximation of Stochastic Quasi-Periodic Responses of Limit Cycles in Non-Equilibrium Systems under Periodic Excitations and Weak Fluctuations"],"prefix":"10.3390","volume":"19","author":[{"given":"Kongming","family":"Guo","sequence":"first","affiliation":[{"name":"Research Center of Applied Mechanics, School of Mechano-Electronic Engineering, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Jiang","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Strength and Vibration, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yalan","family":"Xu","sequence":"additional","affiliation":[{"name":"Research Center of Applied Mechanics, School of Mechano-Electronic Engineering, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,15]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2199","DOI":"10.3390\/e12102199","article-title":"Thermodynamics and fluctuations far from equilibrium","volume":"12","author":"Ross","year":"2010","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"011133","DOI":"10.1103\/PhysRevE.86.011133","article-title":"Stochastic resonance in multistable systems: The role of dimensionality","volume":"86","author":"Nicolis","year":"2012","journal-title":"Phys. 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