{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:49:55Z","timestamp":1772293795987,"version":"3.50.1"},"reference-count":29,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2005,11]]},"abstract":"<jats:p>We prove several results of the orbit behavior of skew product diffeomorphisms generated by quasi-periodic differential systems. The first diffeomorphism is derived from a periodic differential equation on the circle by means of a construction proposed by Z. Opial to get a scalar quasi-periodic equation with all its solutions bounded but without an almost periodic solution. We consider both possible cases for the irrational rotation number, transitive and singular (intransitive). The main result for a transitive case is that the related skew product diffeomorphism has a foliation into invariant curves with pure irrational rotation on each curve (being the same for each curve). For intransitive case, we get invariant sets of two types: a collection of continuous invariant curves and invariant sets being dimensionally inhomogeneous ones.<\/jats:p><jats:p>Section 3 is devoted to perturbations of a skew product diffeomorphism over an irrational rotation being initially foliated into invariant curves. We prove an analog of Poincar\u00e9\u2013Pontryagin theorem which sets conditions when a perturbation of a one-degree-of-freedom Hamiltonian system (given in an annulus and written down in action-angle variables) has limit cycles. Our theorem provides sufficient conditions when a perturbation of a foliated skew product diffeomorphism has isolated invariant curves (asymptotically stable or unstable).<\/jats:p><jats:p>In Sec. 4 we present some results on the geometry of skew product diffeomorphisms derived by a quasi-periodic Riccati equation.<\/jats:p>","DOI":"10.1142\/s0218127405014118","type":"journal-article","created":{"date-parts":[[2006,2,6]],"date-time":"2006-02-06T09:49:18Z","timestamp":1139219358000},"page":"3675-3689","source":"Crossref","is-referenced-by-count":6,"title":["REMARKS ON SKEW PRODUCTS OVER IRRATIONAL ROTATIONS"],"prefix":"10.1142","volume":"15","author":[{"given":"L. M.","family":"LERMAN","sequence":"first","affiliation":[{"name":"Institute for Applied Mathematics &amp; Cybernetics and Department of Differential Equations, University of Nizhny Novgorod, 10 Ul'yanov St., Nizhny Novgorod, 603005, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","first-page":"1259","volume":"37","author":"Anosov D. V.","journal-title":"Izvestiya AN SSSR, Ser. Mat."},{"key":"rf2","volume-title":"Asymptotic Methods in the Theory of Nonlinear Oscillations","author":"Bogolyubov N. N.","year":"1963"},{"key":"rf3","unstructured":"I. U.\u00a0Bronshtein, Extensions of Minimal Groups of Transformations (Shtiintsa P.H., Kishinev, 1975)\u00a0p. 311."},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1112\/S0024609396002160"},{"key":"rf5","volume-title":"Theory of Ordinary Differential Equations","author":"Coddington E.","year":"1955"},{"key":"rf6","first-page":"8","volume":"9","author":"Dinaburg E. I.","journal-title":"Funk. Anal. i Priloz."},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.39.2593"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02097013"},{"key":"rf9","volume-title":"Automorphic Functions","author":"Ford L. R.","year":"1929"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2004.4.457"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100030644"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(92)90107-X"},{"key":"rf13","volume-title":"Intorduction to the Modern Theory of Dynamical Systems","author":"Katok A.","year":"1985"},{"key":"rf14","volume-title":"General Topology","author":"Kelley J. L.","year":"1975"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.2307\/3062098"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1063\/1.165887"},{"key":"rf18","unstructured":"L. M.\u00a0Lerman, Methods of Qualitative Theory of Differential Equations, ed. E. A.\u00a0Leontovich-Andronova (N.I. Lobachevsky Gorky State University, Gorky, 1980)\u00a0pp. 166\u2013185."},{"key":"rf19","volume-title":"Almost Periodic Functions","author":"Levitan B. M.","year":"1953"},{"key":"rf20","doi-asserted-by":"crossref","DOI":"10.1080\/00207179208934253","volume-title":"The General Problem of the Stability of Motion","author":"Lyapunov A. M.","year":"1992"},{"key":"rf21","first-page":"688","volume":"20","author":"Markley N. G.","journal-title":"Proc. London Math. Soc."},{"key":"rf22","doi-asserted-by":"crossref","first-page":"239","DOI":"10.36045\/bbms\/1105540796","volume":"3","author":"Miklaszewski D.","journal-title":"Bull. Belg. Math. Soc. \u2014 Simon Stevin"},{"key":"rf23","first-page":"1979","volume":"5","author":"Millionschikov V. M.","journal-title":"Differ. Uravneniya"},{"key":"rf24","first-page":"1340","volume":"223","author":"Morozov A. D.","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"rf25","volume-title":"Differential Dynamics. An Introduction to the Orbit Structure of Diffeomorphisms","author":"Nitecki Z.","year":"1971"},{"key":"rf26","first-page":"673","volume":"9","author":"Opial Z.","journal-title":"Bull. Acad. Polon. Sci. Ser. Sci.: Math. Astron. Phys."},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127401004029"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1063\/1.166074"},{"key":"rf29","first-page":"883","volume":"4","author":"Pontryagin L. S.","journal-title":"Zhurnal Eksper. Teor. Fiziki"},{"key":"rf31","first-page":"241","volume":"127","author":"Sell G.","journal-title":"Trans. A.M.S."}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127405014118","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,12]],"date-time":"2020-04-12T13:17:03Z","timestamp":1586697423000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127405014118"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,11]]},"references-count":29,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2005,11]]}},"alternative-id":["10.1142\/S0218127405014118"],"URL":"https:\/\/doi.org\/10.1142\/s0218127405014118","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,11]]}}}