Open Access

Small C1-smooth perturbations of skew products and the partial integrability property

   | Nov 30, 2020

Cite

V.E. Adler. (2000), Discretizations of the Landau-Lifshits equation, TMPh, 124:1, 897–908.AdlerV.E.2000Discretizations of the Landau-Lifshits equationTMPh124189790810.1007/BF02551066Search in Google Scholar

V.E. Adler, A.I. Bobenko, Yu.B. Suris. (2009), Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases, Funct. Anal. Appl., 43, 3–17.AdlerV.E.BobenkoA.I.SurisYu.B.2009Discrete Nonlinear Hyperbolic Equations. Classification of Integrable CasesFunct. Anal. Appl.4331710.1007/s10688-009-0002-5Search in Google Scholar

V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov. (1982), Attractive nonrough limit sets of Lorenz-attractor type. Trudy Moskovskogo Matematicheskogo Obshchestva 44, 150–212.AfraimovichV.S.BykovV.V.ShilnikovL.P.1982Attractive nonrough limit sets of Lorenz-attractor typeTrudy Moskovskogo Matematicheskogo Obshchestva44150212Search in Google Scholar

S.S. Belmesova, L.S. Efremova. (2015), On the Concept of Integrability for Discrete Dynamical Systems. Investigation of Wandering Points of Some Trace Map, Nonlinear Maps and their Applic. Springer Proc. in Math. and Statist., 112, 127–158.BelmesovaS.S.EfremovaL.S.2015On the Concept of Integrability for Discrete Dynamical Systems. Investigation of Wandering Points of Some Trace MapNonlinear Maps and their Applic. Springer Proc. in Math. and Statist.11212715810.1007/978-3-319-12328-8_7Search in Google Scholar

V.N. Belykh. (1984), On bifurcations of saddle separatrixes of Lorenz system” (Russian), Differential Equations, 20, 1666–1674.BelykhV.N.1984On bifurcations of saddle separatrixes of Lorenz system” (Russian)Differential Equations2016661674Search in Google Scholar

G.D. Birkhoff. (1927), Dynamical systems, Amer. Math. Soc. Colloq. Publ., 9, Amer. Math. Soc., New York.BirkhoffG.D.1927Dynamical systemsAmer. Math. Soc. Colloq. Publ.9Amer. Math. Soc.New York10.1090/coll/009Search in Google Scholar

S.V. Bolotin, V.V. Kozlov. (2017), Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom, Izv. Math., 81, 671–687.BolotinS.V.KozlovV.V.2017Topology, singularities and integrability in Hamiltonian systems with two degrees of freedomIzv. Math.8167168710.1070/IM8600Search in Google Scholar

P. Brandão. (2014), On the structure of Lorenz maps, Preprint, arXiv:1402.2862.BrandãoP.2014On the structure of Lorenz mapsPreprint, arXiv:1402.2862.Search in Google Scholar

L.S. Efremova. (2001), On the Concept of the Ω-Function for the Skew Product of Interval Mappings, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), [English translation], Journ. Math. Sci. (N.-Y.), 105, 1779–1798, original work published 1999.EfremovaL.S.2001On the Concept of the Ω-Function for the Skew Product of Interval Mappings, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), [English translation]Journ. Math. Sci. (N.-Y.)10517791798original work published 199910.1023/A:1011311512743Search in Google Scholar

L.S. Efremova. (2010), Space of C1-smooth skew products of maps of an interval, TMPh, [English translation], Theoretical and Mathematical Physics, 164, 1208–1214.EfremovaL.S.2010Space of C1-smooth skew products of maps of an interval, TMPh, [English translation]Theoretical and Mathematical Physics1641208121410.1007/s11232-010-0102-7Search in Google Scholar

L.S. Efremova. (2013), A decomposition theorem for the space of C1-smooth skew products with complicated dynamics of the quotient map, Mat.Sb., [English translation], Sb. Math., 204, 1598–1623.EfremovaL.S.2013A decomposition theorem for the space of C1-smooth skew products with complicated dynamics of the quotient map, Mat.Sb., [English translation]Sb. Math.2041598162310.1070/SM2013v204n11ABEH004351Search in Google Scholar

L.S. Efremova. (2014), Remarks on the nonwandering set of skew products with a closed set of periodic points of the quotient map, Nonlinear Maps and their Applic. Springer Proc. in Math. and Statist., 57, 39–58.EfremovaL.S.2014Remarks on the nonwandering set of skew products with a closed set of periodic points of the quotient mapNonlinear Maps and their Applic. Springer Proc. in Math. and Statist.57395810.1007/978-1-4614-9161-3_6Search in Google Scholar

L.S. Efremova. (2016), Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, [English translation], Russian Math., 60, 77–81.EfremovaL.S.2016Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, [English translation],Russian Math.60778110.3103/S1066369X16020122Search in Google Scholar

L.S. Efremova. (2016), Nonwandering sets of C1-smooth skew products of interval maps with complicated dynamics of quotient map, Problemy matem. analiza, Novosibirsk, [English translation], Journ. Math. Sci. (New York), 219, 86–98.EfremovaL.S.2016Nonwandering sets of C1-smooth skew products of interval maps with complicated dynamics of quotient map, Problemy matem. analiza, Novosibirsk, [English translation]Journ. Math. Sci. (New York)219869810.1007/s10958-016-3085-6Search in Google Scholar

L.S. Efremova. (2016), Stability as a whole of a family of fibers maps and Ω-stability of C1-smooth skew products of maps of an interval, J. Phys.: Conf. Ser., 692, 012010.EfremovaL.S.2016Stability as a whole of a family of fibers maps and Ω-stability of C1-smooth skew products of maps of an intervalJ. Phys.: Conf. Ser.69201201010.1088/1742-6596/692/1/012010Search in Google Scholar

L.S. Efremova. (2017), Dynamics of skew products of maps of an interval, Uspekhi Mat. Nauk, [English translation], Russian Math. Surveys, 72, 101–178.EfremovaL.S.2017Dynamics of skew products of maps of an interval, Uspekhi Mat. Nauk, [English translation]Russian Math. Surveys7210117810.1070/RM9745Search in Google Scholar

L.S. Efremova. (2018), The Trace Map and Integrability of the Multifunctions, J. Phys.: Conf. Ser., 990, 012003.EfremovaL.S.2018The Trace Map and Integrability of the MultifunctionsJ. Phys.: Conf. Ser.99001200310.1088/1742-6596/990/1/012003Search in Google Scholar

L.S. Efremova. (2020), Small Perturbations of Smooth Skew Products and Sharkovsky's Theorem, JDEA, https://doi.org/10.1080/10236198.2020.1804556EfremovaL.S.2020Small Perturbations of Smooth Skew Products and Sharkovsky's TheoremJDEAhttps://doi.org/10.1080/10236198.2020.180455610.1080/10236198.2020.1804556Search in Google Scholar

L.S. Efremova. (2020), Periodic behavior of maps obtained by small perturbations of smooth skew products, Discontinuity, Nonlinearity, Complexity, 9(4), 519–523.EfremovaL.S.2020Periodic behavior of maps obtained by small perturbations of smooth skew productsDiscontinuity, Nonlinearity, Complexity9451952310.5890/DNC.2020.12.004Search in Google Scholar

R.I. Grigorchuk, A. Žuk. (2001), The Lamplighter group as a group generated by a 2-state automata, and its spectrum, Geometriae Dedicata, 87, 209–244.GrigorchukR.I.ŽukA.2001The Lamplighter group as a group generated by a 2-state automata, and its spectrumGeometriae Dedicata8720924410.1023/A:1012061801279Search in Google Scholar

M.V. Jakobson. (1971), Smooth mappings of the circle into itself, Mat. Sb. (N.S.), [English translation], Mathematics of the USSR – Sbornik, 14, 161–185.JakobsonM.V.1971Smooth mappings of the circle into itself, Mat. Sb. (N.S.), [English translation]Mathematics of the USSR – Sbornik1416118510.1070/SM1971v014n02ABEH002611Search in Google Scholar

A.B. Katok, B. Hasselblatt. (1995), Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge.KatokA.B.HasselblattB.1995Introduction to the modern theory of dynamical systemsEncyclopedia Math. Appl.54Cambridge Univ. PressCambridge10.1017/CBO9780511809187Search in Google Scholar

V.V. Kozlov. (2019), Tensor invariants and integration of differential equations, Russian Math. Surveys, 74, 111–140.KozlovV.V.2019Tensor invariants and integration of differential equationsRussian Math. Surveys7411114010.1070/RM9866Search in Google Scholar

V.V. Kozlov, D.V. Treschev. (2016), Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics, Sb. Math., 207, 1435–1449.KozlovV.V.TreschevD.V.2016Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamicsSb. Math.2071435144910.1070/SM8786Search in Google Scholar

K. Kuratowski. (1966), Topology, 1, Academic Press, New York-London, 1966.KuratowskiK.1966Topology1Academic PressNew York-London196610.1016/B978-0-12-429201-7.50005-3Search in Google Scholar

M.I. Malkin. (1991), Rotation intervals and the dynamics of Lorenz type mappings, Selecta Math. Sovietica, 10, 265–275.MalkinM.I.1991Rotation intervals and the dynamics of Lorenz type mappingsSelecta Math. Sovietica10265275Search in Google Scholar

W. de Melo, S. van Strien. (1993), One-dimensional dynamics, A series of modern surveys in mathematics, Springer-Verlag.de MeloW.van StrienS.1993One-dimensional dynamics, A series of modern surveys in mathematicsSpringer-Verlag10.1007/978-3-642-78043-1Search in Google Scholar

V.V. Nemytskii, V.V. Stepanov. (1960), Qualitative theory of differential equations, Princeton Univ. Press, Princeton, NJ, 1960, original work published 1947.NemytskiiV.V.StepanovV.V.1960Qualitative theory of differential equationsPrinceton Univ. PressPrinceton, NJ1960original work published 1947.Search in Google Scholar

C. Robinson. (1989), Homoclinic bifurcation to a transitive attractor of the Lorenz type, Nonlinearity, 495–518.RobinsonC.1989Homoclinic bifurcation to a transitive attractor of the Lorenz type, Nonlinearity49551810.1088/0951-7715/2/4/001Search in Google Scholar

C. Robinson. (1992), Homoclinic bifurcation to a transitive attractor of the Lorenz type, II, SIAM J. Math. Anal., 23, 1255–1268.RobinsonC.1992Homoclinic bifurcation to a transitive attractor of the Lorenz type, IISIAM J. Math. Anal.231255126810.1137/0523070Search in Google Scholar

M. Rychlik. (1990), Lorenz attractors through Sil’nikov type bifurcations, Part I, Ergodic Theory Dynamical Systems, 10, 793–822.RychlikM.1990Lorenz attractors through Sil’nikov type bifurcations, Part IErgodic Theory Dynamical Systems1079382210.1017/S0143385700005915Search in Google Scholar

A.N. Sharkovskii. (1964), Coexistence of cycles of a continuous map of the line into itself (Russian), Ukrain. Mat. Zh. 16, 61–71, [English translation], (1995) Bifurcation and Chaos, 5, 1263 – 1273.SharkovskiiA.N.1964Coexistence of cycles of a continuous map of the line into itself (Russian)Ukrain. Mat. Zh.166171[English translation], (1995) Bifurcation and Chaos, 5, 1263 – 1273.10.1142/S0218127495000934Search in Google Scholar

A.N. Sharkovsky, Yu.L. Maistrenko, and E.Yu. Romanenko. (1993), Difference equations and their applications, Naukova Dumka, Kyev, [English translation], Math. Appl., 250, Kluwer Acad. Publ., Dordrecht, original work published 1986.SharkovskyA.N.MaistrenkoYu.L.RomanenkoE.Yu.1993Difference equations and their applications, Naukova Dumka, Kyev, [English translation]Math. Appl.250Kluwer Acad. Publ.Dordrechtoriginal work published 1986.Search in Google Scholar

M.V. Shashkov, L.P. Shilnikov. (1994), On the existence of a smooth invariant foliation in Lorenz-type mappings, Differ. Equ., 30, 536–544.ShashkovM.V.ShilnikovL.P.1994On the existence of a smooth invariant foliation in Lorenz-type mappingsDiffer. Equ.30536544Search in Google Scholar

Yu. B. Suris. (1989), Integrable mappings of the standard type, Funct. Anal. Appl., 23, 74–76.SurisYu. B.1989Integrable mappings of the standard typeFunct. Anal. Appl.23747610.1007/BF01078586Search in Google Scholar

A.P. Veselov. (1991), Integrable maps, Russian Math. Survey, 46, 1–51.VeselovA.P.1991Integrable mapsRussian Math. Survey4615110.1070/RM1991v046n05ABEH002856Search in Google Scholar

eISSN:
2444-8656
Language:
English
Publication timeframe:
Volume Open
Journal Subjects:
Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics