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Fundam. Prikl. Mat., 2008, Volume 14, Issue 3, Pages 3–237 (Mi fpm1121)  

This article is cited in 26 scientific papers (total in 26 papers)

Dynamical systems with variable dissipation: Approaches, methods, and applications

M. V. Shamolin

M. V. Lomonosov Moscow State University

Abstract: This work is devoted to the development of qualitative methods in the theory of nonconservative systems that arise, e.g., in such fields of science as the dynamics of a rigid body interacting with a resisting medium, oscillation theory, etc. This material can call the interest of specialists in the qualitative theory of ordinary differential equations, in rigid body dynamics, as well as in fluid and gas dynamics since the work uses the properties of motion of a rigid body in a medium under the streamline flow around conditions.
The author obtains a full spectrum of complete integrability cases for nonconservative dynamical systems having nontrivial symmetries. Moreover, in almost all cases of integrability, each of the first integrals is expressed through a finite combination of elementary functions and is a transcendental function of its variables, simultaneously. In this case, the transcendence is meant in the complex analysis sense, i.e., after the continuation of the functions considered to the complex domain, they have essentially singular points. The latter fact is stipulated by the existence of attracting and repelling limit sets in the system considered (for example, attracting and repelling foci).
The author obtains new families of phase portraits of systems with variable dissipation on lower- and higher-dimensional manifolds. He discusses the problems of their absolute or relative roughness. He discovers new integrable cases of the rigid body motion, including those in the classical problem of motion of a spherical pendulum placed in the over-running medium flow.

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English version:
Journal of Mathematical Sciences (New York), 2009, 162:6, 741–908

Bibliographic databases:

UDC: 517.925+531.01+531.552

Citation: M. V. Shamolin, “Dynamical systems with variable dissipation: Approaches, methods, and applications”, Fundam. Prikl. Mat., 14:3 (2008), 3–237; J. Math. Sci., 162:6 (2009), 741–908

Citation in format AMSBIB
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\by M.~V.~Shamolin
\paper Dynamical systems with variable dissipation: Approaches, methods, and applications
\jour Fundam. Prikl. Mat.
\yr 2008
\vol 14
\issue 3
\pages 3--237
\mathnet{http://mi.mathnet.ru/fpm1121}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2482029}
\zmath{https://zbmath.org/?q=an:1189.37022}
\elib{http://elibrary.ru/item.asp?id=12174967}
\transl
\jour J. Math. Sci.
\yr 2009
\vol 162
\issue 6
\pages 741--908
\crossref{https://doi.org/10.1007/s10958-009-9657-y}
\elib{http://elibrary.ru/item.asp?id=15307901}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70350662331}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Shamolin M.V., “New cases of integrability in the spatial dynamics of a rigid body”, Dokl. Phys., 55:3 (2010), 155–159  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. Shamolin M.V., “Complete list of first integrals in the problem on the motion of a 4D solid in a resisting medium under assumption of linear damping”, Dokl. Phys., 56:9 (2011), 498–501  crossref  mathscinet  mathscinet  adsnasa  isi  elib  elib
    3. Shamolin M.V., “A new case of integrability in spatial dynamics of a rigid solid interacting with a medium under assumption of linear damping”, Dokl. Phys., 57:2 (2012), 78–80  crossref  mathscinet  mathscinet  adsnasa  isi  elib  elib
    4. M. V. Shamolin, “Zadacha o dvizhenii tela v soprotivlyayuscheisya srede s uchetom zavisimosti momenta sily soprotivleniya ot uglovoi skorosti”, Matem. modelirovanie, 24:10 (2012), 109–132  mathnet  mathscinet
    5. M. V. Shamolin, “Complete list of first integrals for dynamic equations of motion of a solid body in a resisting medium with consideration of linear damping”, Moscow University Mechanics Bulletin, 67:4 (2012), 92–95  mathnet  crossref
    6. Chistyakov V.V., “Opredelenie traektorii svobodnogo dvizheniya girostabilizirovannogo tela cherez proektivno-dvoistvennye peremennye”, Vestnik rossiiskogo universiteta druzhby narodov. seriya: matematika, informatika, fizika, 2013, no. 1, 212–223  elib
    7. V. V. Chistyakov, “Ob odnom sposobe chislennogo integrirovaniya uravnenii svobodnogo ploskoparallelnogo dvizheniya operennogo snaryada v soprotivlyayuscheisya srede”, Izv. IMI UdGU, 2013, no. 1(41), 96–108  mathnet
    8. Bryukhov D., “On Modified Quaternionic Analysis, Irrotational Velocity Fields and New Gradient Dynamical Systems in R-3”, 11th International Conference of Numerical Analysis and Applied Mathematics 2013, Pts 1 and 2 (Icnaam 2013), AIP Conference Proceedings, 1558, eds. Simos T., Psihoyios G., Tsitouras C., Amer Inst Physics, 2013, 485–488  crossref  mathscinet  isi
    9. M. V. Shamolin, “Integrable cases in the dynamics of a multi-dimensional rigid body in a nonconservative field in the presence of a tracking force”, J. Math. Sci., 214:6 (2016), 865–891  mathnet  crossref  mathscinet
    10. A. V. Andreev, M. V. Shamolin, “Matematicheskoe modelirovanie vozdeistviya sredy na tverdoe telo i novoe dvukhparametricheskoe semeistvo fazovykh portretov”, Vestn. SamGU. Estestvennonauchn. ser., 2014, no. 10(121), 109–115  mathnet
    11. M. V. Shamolin, “Sluchai integriruemosti, sootvetstvuyuschie dvizheniyu mayatnika na ploskosti”, Vestn. SamGU. Estestvennonauchn. ser., 2015, no. 10(132), 91–113  mathnet  elib
    12. Shamolin M.V., “Complete List of the First Integrals of Dynamic Equations of a Multidimensional Solid in a Nonconservative Field Under the Assumption of Linear Damping”, Dokl. Phys., 60:10 (2015), 471–475  crossref  isi  elib
    13. M. V. Shamolin, “Integrable variable dissipation systems on the tangent bundle of a multi-dimensional sphere and some applications”, J. Math. Sci., 230:2 (2018), 185–353  mathnet  crossref  elib
    14. M. V. Shamolin, “On the problem of free deceleration of a rigid body with the cone front part in a resisting medium”, Math. Models Comput. Simul., 9:2 (2017), 232–247  mathnet  crossref  zmath  elib
    15. M. V. Shamolin, “Sluchai integriruemosti, sootvetstvuyuschie dvizheniyu mayatnika v trekhmernom prostranstve”, Vestn. SamU. Estestvennonauchn. ser., 2016, no. 3-4, 75–97  mathnet  elib
    16. M. V. Shamolin, “Integrable systems on the tangent bundle of a multi-dimensional sphere”, J. Math. Sci. (N. Y.), 234:4 (2018), 548–590  mathnet  crossref
    17. M. V. Shamolin, “Integrable systems in dynamics on a tangent foliation to a sphere”, Moscow University Mechanics Bulletin, 71:2 (2016), 27–32  mathnet  crossref  isi  elib
    18. M. V. Shamolin, “Sluchai integriruemosti, sootvetstvuyuschie dvizheniyu mayatnika v chetyrekhmernom prostranstve”, Vestn. SamU. Estestvennonauchn. ser., 2017, no. 1, 41–58  mathnet  elib
    19. M. V. Shamolin, “Auto-oscillations under the braking of a rigid body in a resisting medium”, J. Appl. Industr. Math., 11:4 (2017), 572–583  mathnet  crossref  crossref  elib
    20. M. V. Shamolin, “O dvizhenii mayatnika v mnogomernom prostranstve. Chast 1. Dinamicheskie sistemy”, Vestn. SamU. Estestvennonauchn. ser., 2017, no. 3, 41–64  mathnet  crossref  elib
    21. Luque A., Barbancho J., Fernandez Canete J., Cordoba A., “Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systemsphase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems”, Int. J. Bifurcation Chaos, 27:14 (2017), 1730051  crossref  mathscinet  zmath  isi  scopus
    22. Shamolin M.V., “New Cases of Integrable Systems With Dissipation on the Tangent Bundle of a Three-Dimensional Manifold”, Dokl. Phys., 62:11 (2017), 517–521  crossref  mathscinet  isi  scopus
    23. Shamolin M.V., “New Cases of Integrable Systems With Dissipation on a Tangent Bundle of a Two-Dimensional Manifold”, Dokl. Phys., 62:8 (2017), 392–396  crossref  mathscinet  isi  scopus
    24. Shamolin M.V., “New Cases of Integrable Systems With Dissipation on a Tangent Bundle of a Multidimensional Sphere”, Dokl. Phys., 62:5 (2017), 262–265  crossref  mathscinet  isi  scopus
    25. M. V. Shamolin, “Simulation of the spatial action of a medium on a body of conical form”, J. Appl. Industr. Math., 12:2 (2018), 347–354  mathnet  crossref  crossref  elib  elib
    26. M. V. Shamolin, “A new case of an integrable system with dissipation on the tangent bundle of a multidimensional sphere”, Moscow University Mechanics Bulletin, 73:3 (2018), 51–59  mathnet  crossref  isi
  • Фундаментальная и прикладная математика
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