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Dynamical systems with variable dissipation: approaches, methods, and applications. (English. Russian original) Zbl 1189.37022
J. Math. Sci., New York 162, No. 6, 741-908 (2009); translation from Fundam. Prikl. Mat. 14, No. 3, 3-237 (2008).
The main results of this works are as follows:
1. To present a relatively simple methodology for finding dimensionless parameters of the medium action on a rigid body under the quasi-stationarity conditions, which is successfully applied for studying the motion of bodies having a simple form, the circular cylinders entering the water.
2. To elaborate new qualitative methods for studying variable dissipation systems. Obtained the conditions for the existence of the bifurcation of stable and unstable auto-oscillation birth. Presented the conditions for the absence of such trajectories. The theory of Poincare plane topographical systems and comparison systems is extended to the spatial case. A sufficiently simple methodology for proving the Poisson stability of unclosed trajectories of dynamical systems is presented. The definitions of relatively structural stability and relative structural instability of various degrees are introduced.
3. New integrable cases and families of phase portraits in the plane rigid body dynamics are discovered. Some model variants of the body motion in a resisting medium are qualitatively studied and have been integrated. It is shat the first integrals of the corresponding systems are transcendental functions and that are expressed through elementary functions. New two-parameter families of topologically nonequivalent phase portraits are constructed.
4. New integrable cases and families of many-dimensional phase portraits in the spatial rigid body dynamics are discovered. A new two-parameter family of phase portraits in the problem of the spatial free drag of a body is obtained.

##### MSC:
 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
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##### References:
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Shamolin, ”A new two-parameter family of phase portraits with limit cycles in rigid body dynamics interacting with a medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 15–19, 1995. Abstracts of Reports (Study of Systems) [in Russian], Kiev (1995), p. 125. [271] M. V. Shamolin, ”New two-parameter families of the phase patterns on the problem of a body motion in a resisting medium,” in: ICIAM’95, Book of Abstracts. Hamburg, 3–7 July, 1995, Hamburg, Germany (1995), p. 436. [272] M. V. Shamolin, ”On relative structural stability of dynamical systems in problem of body motion in a resisting medium,” the abstract of a talk at the Chebyshev Readings, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 6, 17 (1995). [273] M. V. Shamolin, ”Poisson-stable and dense orbits in rigid body dynamics,” in: 3rd Experimental Chaos Conf., Advance Program. Edinburgh, Scotland, August 21–23, 1995, Edinburgh, Scotland (1995), p. 114. [274] M. V. Shamolin, ”Qualitative methods to the dynamic model of interaction of a rigid body with a resisting medium and new two-parametric families of the phase portraits,” in: DynDays’95 (Sixteenth Annual Informal Workshop), Program and Abstracts. Lyon, June 28–July 1, 1995, Lyon, France (1995), p. 185. [275] M. V. Shamolin, ”Relative structural stability of dynamical systems for problem of body motion in a medium,” in: Analytical, Numerical, and Experimental Methods in Mechanics. A Collection of Scientific Works [in Russian], Izd. Mosk. Univ., Moscow (1995), pp. 14–19. [276] M. V. Shamolin, ”Structural optimization of the controlled rigid motion in a resisting medium,” in: WCSMO-1, Extended Abstracts. Posters. Goslar, May 28–June 2, 1995, Goslar, Germany (1995), p. 18–19. [277] M. V. Shamolin, ”A list of integrals of dynamical equations in spatial problem of body motion in a resisting medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 20–24, 1996. Abstracts of Reports (Study of Systems) [in Russian], Kiev (1996), p. 142. [278] M. V. Shamolin, ”Definition of relative roughness and two-parameter family of phase portraits in rigid body dynamics,” Usp. Mat. Nauk, 51, No. 1, 175–176 (1996). · Zbl 0874.70006 [279] M. V. Shamolin, ”Introduction to problem of body drag in a resisting medium and a new two-parameter family of phase portraits,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 4, 57–69 (1996). · Zbl 0923.70008 [280] M. V. Shamolin, ”Introduction to spatial dynamics of rigid body motion in resisting medium,” in: Materials of Int. Conf. and Chebyshev Readings Devoted to the 175th Anniversary of P. L. Chebyshev, Moscow, May 14–19, 1996, Vol. 2 [in Russian], Izd. Mosk. Univ., Moscow (1996), pp. 371–373. [281] M. V. Shamolin, ”On a certain integrable case in dynamics of spatial body motion in a resisting medium,” in: II Symposium in Classical and Celestial Mechanics. Abstracts of Reports. Velikie Luki, August 23–28, 1996 [in Russian], Moscow–Velikie Luki (1996), pp. 91–92. [282] M. V. Shamolin, ”Periodic and Poisson stable trajectories in problem of body motion in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 55–63 (1996). [283] M. V. Shamolin, ”Qualitative methods in dynamics of a rigid body interacting with a medium,” in: II Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 25–30, 1996. Abstracts of Reports, Pt. III [in Russian], Novosibirsk (1996), p. 267. · Zbl 0900.70152 [284] M. V. Shamolin, ”Qualitative methods in interacting with the medium rigid body dynamics,” in: Abstracts of GAMM Wissenschaftliche Jahrestagung’96, 27.–31. May, 1996, Czech Rep., Karls-Universität Prag., Prague (1996), pp. 129–130. [285] M. V. Shamolin, ”Qualitative methods in interacting with the medium rigid body dynamics,” in: Abstracts of XIXth ICTAM, Kyoto, Japan, August 25–31, 1996, Kyoto, Japan (1996), p. 285. · Zbl 0900.70152 [286] M. V. Shamolin, ”Relative structural stability and relative structural instability of different degrees in topological dynamics,” in: Abstracts of Int. Topological Conf. Dedicated to P. S. Alexandroff’s 100th Birthday ”Topology and Applications,” Moscow, May 27–31, 1996, Fazis, Moscow (1996), pp. 207–208. [287] M. V. Shamolin, ”Spatial Poincar’e topographical systems and comparison systems,” in: Abstract of Reports of Math. Conf. ”Erugin Readings,” Brest, May 14–16, 1996 [in Russian], Brest (1996), p. 107. [288] M. V. Shamolin, ”Topographical Poincar’e systems in many-dimensional spaces,” in: Fifth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, July 29–August 2, 1996, Szeged, Hungary (1996), p. 45. [289] M. V. Shamolin, ”Variety of types of phase portraits in dynamics of a rigid body interacting with a resisting medium,” Dokl. Ross. Akad. Nauk, 349, No. 2, 193–197 (1996). · Zbl 0900.70152 [290] M. V. Shamolin, ”Classical problem of a three-dimensional motion of a pendulum in a jet flow,” in: 3rd EUROMECH Solid Mechanics Conf., Book of Abstracts, Stockholm, Sweden, August 18–22, 1997, Royal Inst. of Technology, Stockholm, Sweden (1997), p. 204. [291] M. V. Shamolin, ”Families of three-dimensional phase portraits in dynamics of a rigid body,” in: EQUADIFF 9, Abstracts, Enlarged Abstracts, Brno, Czech Rep., August 25–29, 1997, Masaryk Univ., Brno, Czech Rep. (1997), p. 76. [292] M. V. Shamolin, ”Jacobi integrability of problem of a spatial pendulum placed in over-running medium flow,” in: Modelling and Investigation of System Stability. Sci. Conf., May 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 143. [293] M. V. Shamolin, ”Mathematical modelling of dynamics of a spatial pendulum flowing around by a medium,” in Proc. of VII Int. Symposium ”Methods of Discrete Singularities in Problems of Mathematical Physics,” Feodociya, June 26–29, 1997 [in Russian], Kherson State Technical Univ., Kherson (1997), pp. 153–154. [294] M. V. Shamolin, ”On an integrable case in spatial dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 65–68 (1997). [295] M. V. Shamolin, ”Partial stabilization of body rotational motions in a medium under a free drag,” Abstracts of Reports of All-Russian Conf. with Int. Participation ”Problems of Celestial Mechanics,” St. Petersburg, June 3–6, 1997 [in Russian], Institute of Theoretical Astronomy, Russian Academy of Sciences, St. Petersburg (1997), pp. 183–184. [296] M. V. Shamolin, ”Qualitative methods in dynamics of a rigid body interacting with a medium,” in: YSTM’96: ”Young People, the Third Millenium,” Proc. of Int. Congress (Ser. Professional) [in Russian], Vol. 2, NTA ”APFN,” Moscow (1997), pp. I-4. [297] M. V. Shamolin, ”Spatial dynamics of a rigid body interacting with a medium,” Workshop in Mechanics of Systems and Problems of Motion Control and Navigation, Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 4, 174 (1997). [298] M. V. Shamolin, ”Spatial Poincar’e topographical systems and comparison systems,” Usp. Mat. Nauk, 52, No. 3, 177–178 (1997). · Zbl 0915.58062 [299] M. V. Shamolin, ”Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” in: Proc. of WCSMO-2, Zakopane, Poland, May 26–30, 1997, Zakopane, Poland (1997), pp. 387–392. [300] M. V. Shamolin, ”Three-dimensional structural optimization of controlled rigid motion in a resisting medium,” in: WCSMO-2, Extended Abstracts, Zakopane, Poland, May 26–30, 1997, Zakopane, Poland (1997), pp. 276–277. [301] M. V. Shamolin, ”Absolute and relative structural stability in spatial dynamics of a rigid body interacting with a medium,” in: Proc. of Int. Conf. ”Mathematics in Industry,” ICIM-98, Taganrog, June 29–July 3, 1998 [in Russian], Taganrog State Pedagogical Inst., Taganrog (1998), pp. 332–333. [302] M. V. Shamolin, ”Families of portraits with limit cycles in plane dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 6, 29–37 (1998). [303] M. V. Shamolin, ”Family of three-dimensional phase portraits in spatial dynamics of a rigid body interacting with a medium,” in: III Int. Symposium in Classical and Celestial Mechanics, August 23–27, 1998, Velikie Luki. Abstracts of Reports [in Russian], Computational Center of Russian Academy of Sciences, Moscow–Velikie Luki (1998), pp. 165–167. [304] M. V. Shamolin, ”Lyapunov functions method and many-dimensional topographical systems of Poincaré in rigid body dynamics,” in: Abstract of Reports of IV Crimenian Int. Math. School ”Lyapunov Function Method and Its Applications,” Crimea, Alushta, September 5–12, Simpheropol’ State Univ., Simpheropol’ (1998), p. 80. [305] M. V. Shamolin, ”Many-dimensional topographical Poincaré systems in rigid body dynamics,” in: Abstracts of GAMM Wissenschaftliche Jahrestagung’98, 6.–9. April, 1998, Universität Bremen, Bremen, Germany (1998), p. 128. [306] M. V. Shamolin, ”Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of Int. Congress ”Nonlinear Analysis and Its Applications,” Moscow, September 1–5, 1998 [in Russian], Moscow (1998), p. 131. [307] M. V. Shamolin, ”Methods of nonlinear analysis in dynamics of a rigid body interacting with a medium,” in: CD-Proc. of the Congress ”Nonlinear Analysis and Its Applications,” Moscow, Russia, September 1–5, 1998, Moscow (1999), pp. 497–508. [308] M. V. Shamolin, ”New two-parametric families of the phase portraits in three-dimensional rigid body dynamics,” in: Int. Conf. Devoted to the 90th Anniversary of L. S. Pontryagin, Moscow, August 31–September 6, 1998, Abstract of Reports, Differntial Equations, Izd. Mosk. Univ., Moscow (1998), pp. 97–99. [309] M. V. Shamolin, ”On integrability in transcendental functions,” Usp. Mat. Nauk, 53, No. 3, 209–210 (1998). · Zbl 0925.34003 [310] M. V. Shamolin, ”Qualitative and numerical methods in some problems of spatial dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of 5th Int. Conf.-Workshop ”Engineering-Physical Problems of New Tehnics,” Moscow, May 19–22, 1998 [in Russian], Moscow State Technical Univ., Moscow (1998), pp. 154–155. [311] M. V. Shamolin, ”Some classical problems in three-dimensional dynamics of a rigid body interacting with a medium,” in: Proc. of ICTACEM’98, Kharagpur, India, December 1–5, 1998, Aerospace Engineering Dep., Indian Inst. of Technology, Kharagpur, India (1998), p. 11. [312] M. V. Shamolin, ”Some problems of spatial dynamics of a rigid body interactng with a medium under quasi-stationarity conditions,” in: Abstracts of Reports of All-Russian Sci.-Tech. Conf. of Young Scientists ”Modern Problems of Aero-Cosmos Science,” Zhukovskii, May 27–29, 1998 [in Russian], Central Aero-Hydrodynamical Inst., Moscow (1998), pp. 89–90. [313] M. V. Shamolin, ”Certain classes of partial solutions in dynamics of a rigid body interacting with a medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 178–189 (1999). [314] M. V. Shamolin, ”Families of long-period trajectories in spatial dynamics of a rigid body,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 25–29 1999. Abstracts of Reports [in Russian], Kiev (1999), p. 60. [315] M. V. Shamolin, ”Integrability in terms of transcendental functions in rigid body dynamics,” in: Book of Abstracts of GAMM Annual Meeting, April 12–16, 1999, Metz, France, Université de Metz, Metz, France (1999), p. 144. [316] M. V. Shamolin, ”Long-periodic trajectories in rigid body dynamics,” in: Sixth Colloquium on the Qualitative Theory of Differential Equations, Bolyai Institute, Regional Committee of the Hungarian Academy of Sciences, August 10–14, 1999, Szeged, Hungary (1999), p. 47. [317] M. V. Shamolin, ”Mathematical modelling in 3D dynamics of a rigid interacting with a medium,” in: Book of Abstracts of the Second Int. Conf. ”Tools for Mathematical Modelling,” Saint-Petersburg, Russia, 14–19 June, 1999, Saint-Petersburg State Tech. Univ., Saint-Petersburg (1999), pp. 122–123. [318] M. V. Shamolin, ”Methods of analysis of a deceleration of a rigid in 3D medium,” in: Contributed Abstracts of 3rd ENOC, Copenghagen (Lyngby), Denmark, August 8–12, 1999, Tech. Univ. of Denmark, Copenghagen (1999). [319] M. V. Shamolin, ”New families of the nonequivalent phase portraits in 3D rigid body dynamics,” in: Abstracts of Second Congress ISAAC 1999, Fukuoka, Japan, August 16–21, 1999, Fukuoka Ins. of Tech., Fukuoka (1999), pp. 205–206. [320] M. V. Shamolin, ”New Jacobi integrable cases in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 364, No. 5, 627–629 (1999). · Zbl 1065.70500 [321] M. V. Shamolin, ”Nonlinear dynamical effects in spatial body drag in a resisting medium,” in: Abstracts of Reports of III Int. Conf. ”Chkalov Readings, Engineering-Physical Problems of Aviation and Cosmos Technics” (June 1–4, 1999) [in Russian], EATK GA, Egor’evsk (1999), pp. 257–258. [322] M. V. Shamolin, ”On roughness of dissipative systems and relative roughness and nonroughness of variable dissipation systems,” Usp. Mat. Nauk, 54, No. 5, 181–182 (1999). · Zbl 0968.34039 [323] M. V. Shamolin, ”Properties of integrability of systems in terms of transcendental functions,” in: Final Progr. and Abstracts of Fifth SIAM Conf. on Appl. of Dynamic. Syst., May 23–27, 1999, Snowbird, Utah, USA, SIAM (1999), p. 60. [324] M. V. Shamolin, ”Some properties of transcendental integrable dynamical systems,” in: Book of Abstracts of EQUADIFF 10, Berlin, August 1–7, 1999, Free Univ. of Berlin, Berlin (1999), pp. 286–287. [325] M. V. Shamolin, ”Structural stability in 3D dynamics of a rigid body,” in: WCSMO-3, Short Paper Proc., Buffalo, NY, May 17–21, 1999, Vol. 2, Buffalo (1999), pp. 475–477. [326] M. V. Shamolin, ”Structural stability in 3D dynamics of a rigid body,” in: CD-Proc. of WCSMO-3, Buffalo, NY, May 17–21, 1999, Buffalo (1999), p. 6. · Zbl 1065.70500 [327] M. V. Shamolin, ”A new family of phase portraits in spatial dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 371, No. 4, 480–483 (2000). [328] M. V. Shamolin, ”About interaction of a rigid body with a resisting medium under an assumption of a jet flow,” in: Book of Abstracts II (General sessions) of 4th EUROMECH Solid Mech. Conf., Metz, France (June 26–30, 2000), Univ. of Metz (2000), p. 703. [329] M. V. Shamolin, ”Comparison of certain integrability cases from two-, three-, and four-dimensional dynamics of a rigid body interacting with a medium,” in: Abstracts of Reports of V Crimeanian Int. Math. School ”Lyapunov Function Method and Its Application,” (MLF-2000), Crimea, Alushta, September 5–13, 2000 [in Russian], Simpheropol’ (2000), p. 169. [330] M. V. Shamolin, ”Integrability and nonintegrability in terms of transcendental functions,” in: CD-Abstracts of 3rd ECM (Poster sessions), Barcelona, Spain, June 10–14, 2000, Poster No. 36. [331] M. V. Shamolin, ”Jacobi integrability of problem of four-dimensional body motion in a resisting medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, August 21–26, 2000 [in Russian], Vladimir State Univ., Vladimir (2000), pp. 196–197. [332] M. V. Shamolin, ”Jacobi integrability in problem of four-dimensional rigid body motion in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000). [333] M. V. Shamolin, ”Many-dimensional Poincaré systems and transcendental integrability,” in: IV Siberian Congress in Applied and Industrial Mathematics, Novosibirsk, June 26–July 1, 2000. Abstracts of Reports, Pt. I [in Russian], Novosibirsk, Institute of Mathematics (2000), pp. 25–26. [334] M. V. Shamolin, ”Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” In: Book of Abstracts of ECCOMAS 2000, Barcelona, Spain, 11–14 September, Barcelona (2000), p. 495. [335] M. V. Shamolin, ”Mathematical modelling of interaction of a rigid body with a medium and new cases of integrability,” in: CD-Proc. of ECCOMAS 2000, Barcelona, Spain, 11–14 September, Barcelona (2000), p. 11. [336] M. V. Shamolin, ”Methods of analysis of dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2000 at the Univ. of Göttingen, 2–7 April, 2000, Univ. of Göttingen (2000), p. 144. [337] M. V. Shamolin, ”New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of 16th IMACS World Congress 2000, Lausanne, Switzerland, August 21–25, 2000, EPFL (2000), p. 283. [338] M. V. Shamolin, ”New families of many-dimensional phase portraits in dynamics of a rigid body interacting with a medium,” in: CD-Proc. of 16th IMACS World Congress 2000, Lausanne, Switzerland, August 21–25, 2000, EPFL (2000). [339] M. V. Shamolin, ”On a certain case of Jacobi integrability in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. in Differential and Integral Equations, Odessa, September 12–14, 2000 [in Russian], AstroPrint, Odessa (2000), pp. 294–295. [340] M. V. Shamolin, ”On limit sets of differential equations near singular points,” Usp. Mat. Nauk, 55, No. 3, 187–188 (2000). · Zbl 0968.34021 [341] M. V. Shamolin, ”On roughness of disspative systems and relative roughness of variable dissipation systems,” the abstract of a talk at the Workshop in Vector and Tensor Analysis Named after P. K. Rashevskii, Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 63 (2000). [342] M. V. Shamolin, ”Problem of four-dimensional body motion in a resisting medium and one case of integrability,” in: Book of Abstracts of the Third Int. Conf. ”Differential Equations and Applications,” St. Petersburg, Russia, June 12–17, 2000 [in Russian], St. Petersburg State Univ., St. Petersburg (2000), p. 198. [343] M. V. Shamolin, ”Comparison of some cases of integrability in dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2001, ETH Zurich, 12–15 February, 2001, ETH, Zurich (2001), p. 132. [344] M. V. Shamolin, ”Complete integrability of equations for motion of a spatial pendulum in over-running medium flow,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 5, 22–28 (2001). [345] M. V. Shamolin, ”Diagnosis problem as the main problem of general differential diagnosis problem,” in: Book of Abstracts of the Third Int. Conf. ”Tools for Mathematical Modelling,” St. Petersburg, Russia, June 18–23, 2001 [in Russian], St. Petersburg State Technical Univ., St. Petersburg (2001), p. 121. [346] M. V. Shamolin, ”Integrability cases of equations for spatial dynamics of a rigid body,” Prikl. Mekh., 37, No. 6, 74–82 (2001). · Zbl 1010.70520 [347] M. V. Shamolin, ”Integrability of a problem of four-dimensional rigid body in a resisting medium,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” Fund. Prikl. Mat., 7, No. 1, 309 (2001). [348] M. V. Shamolin, ”New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Sci. Conf., May 22–25, 2001 [in Russian], Kiev (2001), p. 344. [349] M. V. Shamolin, ”New Jacobi integrable cases in dynamics of two-, three-, and four-dimensional rigid body interacting with a meduium,” in: Abstracts of Reports of VIII All-Russian Congress in Theoretical and Applied Mechanics, Perm’, August 23–29, 2001 [in Russian], Ural Department of the Russian Academy of Sciences, Ekaterinburg (2001), pp. 599–600. [350] M. V. Shamolin, ”On stability of motion of a body twisted around its longitudinal axis in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 189–193 (2001). [351] M. V. Shamolin, ”Pattern recognition in the model of the interaction of a rigid body with a resisting medium,” in: Col. of Abstracts of First SIAM–EMS Conf. ”Applied Mathematics in Our Changing World,” Berlin, Germany, September 2–6, 2001, Springer, Birkh”auser (2001), p. 66. [352] M. V. Shamolin, ”Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” Fund. Prikl. Mat., 7, No. 1, 302–303 (2001). [353] M. V. Shamolin, ”Dynamical systems with variable dissipation in 3D dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of 4th ENOC, Moscow, Russia, August 19–23, 2002, Inst. Probl. Mech. Russ. Acad. Sci., Moscow (2002), p. 109. [354] M. V. Shamolin, ”Foundations in differential and topological diagnostics,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2002, Univ. of Augsburg, March 25–28, 2002, Univ. of Augsburg (2002), p. 154. [355] M. V. Shamolin, ”Methods of analysis of dynamics of a 2D- 3D-, or 4D-rigid body with a medium,” in: Abstracts, Short Communications, Poster Sessions of ICM-2002, Beijing, August 20–28, 2002, Higher Education Press, Beijing, China (2002), p. 268. · Zbl 1006.34035 [356] M. V. Shamolin, ”New integrable cases in dynamics of a two-, three-, and four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, July 1–6, 2002 [in Russian], Vladimir State Univ., Vladimir (2002), pp. 142–144. [357] M. V. Shamolin, ”On integrability of certain classes of nonconservative systems,” Usp. Mat. Nauk, 57, No. 1, 169–170 (2002). [358] M. V. Shamolin, ”Some questions of the qualitative theory of ordinary differential equations and dynamics of a rigid body interacting with a medium,” J. Math. Sci., 110, No. 2, 2526–2555 (2002). · Zbl 1006.34035 [359] M. V. Shamolin, ”Foundations of differential and topological diagnostics,” J. Math. Sci., 114, No. 1, 976–1024 (2003). · Zbl 1067.93020 [360] M. V. Shamolin, ”Global structural stability in dynamics of a rigid body interacting with a medium,” in: 5th ICIAM, Sydney, Australia, 7–11 July, 2003, Univ. of Technology, Sydney (2003), p. 306. [361] M. V. Shamolin, ”Integrability and nonintegrability in terms of transcendental functions,” in: Book of Abstracts of Annual Sci. Conf. GAMM 2003, Abano Terme–Padua, Italy, 24–28 March, 2003, Univ. of Padua (2003), p. 77. [362] M. V. Shamolin, ”Integrability in transcendental functions in rigid body dynamics,” in: Abstracts of Reports of Sci. Conf. ”Lomonosov Readings,” Sec. Mechanics, April 17–27, 2003, Moscow, M. V. Lomonosov Moscow State Univ. [in Russian], MGU, Noscow (2003), p. 130. [363] M. V. Shamolin, ”New integrable cases and families of portraits in the plane and spatial dynamics of a rigid body interacting with a medium,” J. Math. Sci., 114, No. 1, 919–975 (2003). · Zbl 1067.70006 [364] M. V. Shamolin, ”On a certain spatial problem of rigid body motion in a resisting medium,” in: Abstracts of Reports of Int. Sci. Conf. ”Third Polyakhov Readings,” St. Petersburg, February 4–6, 2003 [in Russian], NIIKh St. Petersburg Univ., St. Petersburg (2003), pp. 170–171. [365] M. V. Shamolin, ”On integrability of nonconservative dynamical systems in transcendental functions,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 27–30, 2003, Abstracts of Reports [in Russian], Kiev (2003), p. 277. [366] M. V. Shamolin, ”Some questions of differential and topological diagnostics,” in: Book of Abstracts of 5th European Solid Mech. Conf. (ESMC-5), Thessaloniki, Greece, August 17–22, 2003, Aristotle Univ. Thes. (AUT), European Mech. Sc. (EUROMECH) (2003), p. 301. · Zbl 1067.93020 [367] M. V. Shamolin, ”Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body,” J. Math. Sci., 122, No. 1, 2841–2915 (2004). · Zbl 1140.70456 [368] M. V. Shamolin, ”Geometric representation of motion in a certain problem of body interaction with a medium,” Prikl. Mekh., 40, No. 4, 137–144 (2004). · Zbl 1116.74378 [369] M. V. Shamolin, ”Integrability of nonconservative systems in elementary functions,” in: X Math. Int. Conf. Named after Academician M. Kravchuk, May 13–15, 2004, Kiev [in Russian], Kiev (2004), p. 279. · Zbl 1140.70456 [370] M. V. Shamolin, Methods for Analysis of Classes of Nonconservative Systems in Dynamics of a Rigid Body Interacting with a Medium [in Russian], Doctoral Dissertation, MGU, Moscow (2004). · Zbl 1140.70456 [371] M. V. Shamolin, Methods for Analysis of Classes of Nonconservative Systems in Dynamics of a Rigid Body Interacting with a Medium [in Russian], Theses of Doctoral Dissertation, MGU, Moscow (2004). · Zbl 1140.70456 [372] M. V. Shamolin, ”Some cases of integrability in dynamics of a rigid body interacting with a resisting medium,” in: Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Suzdal’, July 5–10, 2004, Vladimir State Univ., Vladimir (2004), pp. 296–298. [373] M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2004). · Zbl 1140.70456 [374] M. V. Shamolin, ”A case of complete integrability in spatial dynamics of a rigid body interacting with a medium taking account of rotational derivatives of force moment in angular velocity,” Dokl. Ross. Akad. Nauk, 403, No. 4, 482–485 (2005). [375] M. V. Shamolin, ”Cases of complete integrability in dynamics of a four-dimensional rigid body interacting with a medium,” in: Abstracts of Reports of Int. Conf. ”Functional Spaces, Approximation Theory, and Nonlinear Analysis” Devoted to the 100th Anniversary of A. M. Nikol’skii, Moscow, May 23–29, 2005 [in Russian], V. A. Steklov Math. Inst. of the Russian Academy of Sciences, Moscow (2005), p. 244. [376] M. V. Shamolin, ”Comparison of Jacobi integrable cases of plane and spatial body motions in a medium under streamline flow-around,” Prikl. Mat. Mekh., 69, No. 6, 1003–1010 (2005). · Zbl 1100.74546 [377] M. V. Shamolin, ”Integrability in transcendental functions in rigid body dynamics,” in: Math. Conf. ”Modern Problems of Applied Mathematics and Mathematical Modelling,” Voronezh, December 12–17, 2005 [in Russian], Voronezh State Acad., Voronezh (2005), p. 240. [378] M. V. Shamolin, ”Mathematical model of interaction of a rigid body with a resisting medium in a jet flow,” in: Abstracts. Pt. 1. 76 Annual Sci. Conf. (GAMM), Luxembourg, March 28–April 1, 2005, Univ. du Luxembourg (2005), pp. 94–95. [379] M. V. Shamolin, ”On a certain integrable case in dynamics on so(4){$$\times$$}$$\mathbb{R}$$4,” in: Abstracts of Reports of All-Russian Conf. ”Differential Equations and Their Applications,” (SamDif-2005), Samara, June 27–July 2, 2005 [in Russian], Univers-Grupp, Samara (2005), pp. 97–98. [380] M. V. Shamolin, ”On a certain integrable case of equations of dynamics in so(4) {$$\times$$} $$\mathbb{R}$$4,” Usp. Mat. Nauk, 60, No. 6, 233–234 (2005). [381] M. V. Shamolin, ”On body motion in a resisting medium taking account of rotational derivatives of aerodynamic force moment in angular velocity,” in: Abstracts of Reports of Sci. Conf. ”Lomonosov Readings-2005,” Sec. Mechanics, April, 2005, Moscow, M. V. Lomonosov Moscow State Univ. [in Russian], MGU, Moscow (2005), p. 182. [382] M. V. Shamolin, ”On rigid body motion in a resisting medium taking account of rotational derivatives of areodynamical force moment in angular velocity,” in: Modelling and Studying of Systems, Sci. Conf., May 23–25, 2005. Abstracts of Reports [in Russian], Kiev (2005), p. 351. [383] M. V. Shamolin, ”Some cases of integrability in 3D dynamics of a rigid body interacting with a medium,” in: Book of Abstracts. IMA Int. Conf. ”Recent Advances in Nonlinear Mechanics,” Aberdeen, Scotland, August 30–September 1, 2005, Aberdeen (2005), p. 112. [384] M. V. Shamolin, ”Structural stable vector fields in rigid body dynamics,” in: Abstracts of 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005), Lodz, Poland, December 12–15, 2005, Tech. Univ. Lodz (2005), p. 78. · Zbl 1100.74546 [385] M. V. Shamolin, ”Structural stable vector fields in rigid body dynamics,” in: Proc. of 8th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2005), Lodz, Poland, December 12–15, 2005, Vol. 1, Tech. Univ. Lodz (2005), pp. 429–436. [386] M. V. Shamolin, ”Variable dissipation dynamical systems in dynamics of a rigid body interacting with a medium,” in: Differential Equations and Computer Algebra Tools, Materials of Int. Conf., Brest, October 5–8, 2005, Pt. 1 [in Russian], BGPU, Minsk (2005), pp. 231–233. [387] M. V. Shamolin, ”Almost conservative systems in dynamics of a rigid body,” in: Book of Abstracts, 77th Annual Meeting of GAMM, March 27–31, 2006, Technische Univ. Berlin, Technische Univ. Berlin (2006), p. 74. [388] M. V. Shamolin, ”Model problem of body motion in a resisting medium taking account of dependence of resistance force on angular velocity,” in: Scientifuc Report of Institute of Mechanics, Moscow State Univ. [in Russian], No. 4818, Institute of Mechanics, Moscow State Univ., Moscow (2006), p. 44. [389] M. V. Shamolin, ”On a case of complete integrability in four-dimensional rigid body dynamics,” Abstracts of Reports of Int. Conf. in Differential Equations and Dynamical Systems, Vladimir, July 10–15, 2006 [in Russian], Vladimir State Univ., Vladimir (2006), pp. 226–228. [390] M. V. Shamolin, ”On trajectories of characteristic points of a rigid body moving in a medium,” in: Int. Conf. ”Fifth Okunev Readings,” St. Petersburg, June 26–30, 2006. Abstracts of Reports [in Russian], Balt. State Tech. Univ., St. Petersburg (2006), p. 34. [391] M. V. Shamolin, ”Spatial problem on rigid body motion in a resistingmedium,” in: VIII Crimeanian Int. Math. School ”Lyapunov Function Method and Its Applications,” Abstracts of Reports, Alushta, September 10–17, 2006, Tavriya National Univ. [in Russian], DiAiPi, Simpheropol’ (2006), p. 184. [392] M. V. Shamolin, ”To problem on rigid body spatial drag in a resisting medium,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 3, 45–57 (2006). [393] M. V. Shamolin, ”To spatial problem of rigid body interaction with a resisting medium,” in: Abstracts of Reports of IX All-Russian Congress in Theoretical and Applied Mechanics, Nizhnii Novgorod, August 22–28, 2006, Vol. I [in Russian], N. I. Lobachevskii Nizhegodskii State Univ., Niznii Novgorod (2006), p. 120. [394] M. V. Shamolin, ”Variable dissipation systems in dynamics of a rigid body interacting with a medium,” Fourth Polyakhov Readings, Abstracts of Reports of Int. Sci. Conf. in Mechanics, St. Petersburg, February 7–10, 2006 [in Russian], VVM, St. Petersburg (2006), p. 86. [395] M. V. Shamolin, ”4D rigid body and some cases of integrability,” in: Abstracts of ICIAM07, Zurich, Switzerland, June 16–20, 2007, ETH, Zurich (2007), p. 311. [396] M. V. Shamolin, ”A case of complete integrability in dynamics on a tangent bundle of two-dimensional sphere,” Usp. Mat. Nauk, 62, No. 5, 169–170 (2007). · Zbl 1137.37325 [397] M. V. Shamolin, ”Case of complete integrability in dynamics of a four-dimensional rigid body in nonconcervative force field,” in: ”Nonlinear Dynamical Analysis-2007,” Abstracts of Reports of Int. Congress, St. Petersburg, June 4–8, 2007 [in Russian], St. Petersburg State Univ., St. Petersburg (2007), p. 178. [398] M. V. Shamolin, ”Cases of complete integrability in dynamics of a rigid body interacting with a medium,” Abstracts of Reports of All-Russian Conf. ”Modern Problems of Continuous Medium Mechanics” Devoted to Memory of L. I. Sedov in Connection with His 100th Anniversary, Moscow, November 12–14, 2007 [in Russian], MIAN, Moscow (2007), pp. 166–167. [399] M. V. Shamolin, ”Cases of complete integrability in dynamics of a four-dimensional rigid body in a nonconservative force field,” in: Abstract of Reports of Int. Conf. ”Analysis and Singularities,” Devoted to 70th Anniversary of V. I. Arnol’d, August 20–24, 2007, Moscow [in Russian], MIAN, Moscow (2007), pp. 110–112. [400] M. V. Shamolin, ”Cases of complete integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Abstracts of Reports of Int. Conf. ”Classical Problems of Rigid Body Dynamics,” June 9–13, 2007 [in Russian], Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk (2007), pp. 81–82. [401] M. V. Shamolin, ”Complete integrability of equations of motion for a spatial pendulum in medium flow taking account of rotational derivatives of moments of its action force,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, 3, 187–192 (2007). [402] M. V. Shamolin, ”Integrability in elementary functions of variable dissipation systems,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 38. [403] M. V. Shamolin, ”Integrability of problem of four-dimensional rigid body motion in a resisting medium,” the abstract of a talk at theWorkshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 21. [404] M. V. Shamolin, ”Integrability of strongly nonconservative systems in transcendental elementary functions,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 40. [405] M. V. Shamolin, Methods of Analysis of Variable Dissipation Dynamical Systems in Rigid Body Dynamics [in Russian], Ekzamen, Moscow (2007). · Zbl 1334.70001 [406] M. V. Shamolin, ”New integrable cases in dynamics of a four-dimensional rigid body interacting with a medium,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 27. [407] M. V. Shamolin, ”On account of rotational derivatives of a force moment of action of the medium in angular velocity of the rigid body on body motion,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 44. [408] M. V. Shamolin, ”On account of rotational derivatives of aerodynamical force moment on body motion in a resisting medium,” the abstract of a talk at theWorkshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 26. [409] M. V. Shamolin, ”On integrability in elementary functions of certain classes of nonconservative dynamical systems,” in: Modelling and Study of Systems, Sci. Conf., May 22–25, 2007. Abstracts of Reports [in Russian], Kiev (2007), p. 249. [410] M. V. Shamolin, ”On integrability in transcendental functions,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 34. [411] M. V. Shamolin, ”On integrability of motion of four-dimensional body-pendulum situated in over-running medium flow,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 37. [412] M. V. Shamolin, ”On rigid body motion in a resisting medium taking account of rotational derivatives of aerodynamic force moment in angular velocity,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 44. [413] M. V. Shamolin, ”On stability of a certain regime of rigid body motion in a resisting medium,” in: Abstracts of Reports of Sci. Conf. ”Lomonosov Readings-2007,” Sec. Mechanics, Moscow, Moscow State Univ., April, 2007 [in Russian], MGU, Moscow (2007), p. 153. [414] M. V. Shamolin, ”On the problem of a symmetric body motion in a resisting medium,” in: Book of Abstracts of EMAC-2007 (1–4 July, 2007, Hobart, Australia), Univ. Tasmania, Hobart, Australia (2007), p. 25. [415] M. V. Shamolin, ”On work of All-Russian Conference ’Differential Equations and Their Applications,’ Samara, June 27–July 2, 2005,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 45. [416] M. V. Shamolin, Some Problems of Differential and Topological Diagnosis [in Russian], Ekzamen, Moscow (2007). · Zbl 1164.74395 [417] M. V. Shamolin, ”The cases of complete integrability in dynamics of a rigid body interacting with a medium,” in: Book of Abstracts of Int. Conf. on the Occasion of the 150th Birthday of A. M. Lyapunov (June 24–30, 2007, Kharkiv, Ukraine), Verkin Inst. Low Temper. Physics Engineer. NASU, Kharkiv (2007), pp. 147–148. [418] M. V. Shamolin, ”The cases of integrability in 2D-, 3D-, and 4D-rigid body,” in: Abstracts of Short Communications and Posters of Int. Conf. ”Dynamical Methods and Mathematical Modelling,” Valladolid, Spain (September 18–22, 2007), ETSII, Valladolid (2007), p. 31. [419] M. V. Shamolin, ”The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Abstracts of 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007), Lodz, Poland, December 17–20, 2007, Tech. Univ. Lodz (2007), p. 115. [420] M. V. Shamolin, ”The cases of integrability in terms of transcendental functions in dynamics of a rigid body interacting with a medium,” in: Proc. of 9th Conf. on Dynamical Systems (Theory and Applications) (DSTA 2007), Lodz, Poland, December 17–20, 2007, Vol. 1, Tech. Univ. Lodz (2007), pp. 415–422. [421] M. V. Shamolin, ”Variety of types of phase portraits in dynamics of a rigid body interacting with a medium,” the abstract of a talk at the Workshop ”Actual Problems of Geometry and Mechanics,” in: Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences, Moscow (2007), p. 17. [422] M. V. Shamolin, ”Three-parameter family of phase portraits in dynamics of a rigid body interacting with a medium,” Dokl. Ross. Akad. Nauk, 418, No. 1, 46–51 (2008). · Zbl 1257.70015 [423] M. V. Shamolin and D. V. Shebarshov, ”Projections of Lagrangian tori of a biharmonic oscillator on position state and dynamics of a rigid body interacting with a medium,” in: Modelling and Study of Stability of Systems, Sci. Conf., May 19–23, 1997. Abstracts of Reports [in Russian], Kiev (1997), p. 142. [424] M. V. Shamolin and D. V. Shebarshov, ”Lagrange tori and the Hamilton–Jacobi equation,” in: Book of Abstracts of Conf. ”Partial Differential Equations: Theory and Numerical Solutions” (Praha, August 10–16, 1998), Charles Univ., Praha (1998), p. 88. [425] M. V. Shamolin and D. V. Shebarshov, ”Certain problems of differential diagnosis,” in: Dynamical Systems Modelling and Stability Investigation. Sci. Conf., May 25–29, 1999. Abstracts of Reports (System Modelling) [in Russian], Kiev (1999), p. 61. [426] M. V. Shamolin and D. V. Shebarshov, ”Methods for solving main problem of differential diagnosis,” Deposit. at VINITI, No. 1500-V99 (1999). [427] M. V. Shamolin and D. V. 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