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Some questions of qualitative theory in dynamics of systems with the variable dissipation

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Abstract

In this work, we consider some problems of the qualitative theory of ordinary differential equations; the study of dissipative systems, as well as variable dissipation system considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and in the oscillation theory, depends on solutions of these problems. We consider such problems as existence and uniqueness problems for trajectories having infinitely remote points as limit sets for systems on the plane, elements of qualitative theory of monotone vector fields, and also existence problems for families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system and the comparison system to the many-dimensional case.

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References

  1. S. A. Agafonov, D. V. Georgievskii, and M. V. Shamolin, “Certain topical problems of geometry and mechanics,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), p. 34.

  2. D. V. Georgievskii and M. V. Shamolin, “On kinematics of a rigid body with fixed point in ℝn,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Fundam. Prikl. Mat. [in Russian], 7, No. 1 (2001), p. 315.

  3. D. V. Georievskii and M. V. Shamolin, “Kinematics and mass geometry of a rigid body with a fixed point in ℝn,” Dokl. Ross. Akad. Nauk, 380, No. 1, 47–50 (2001).

    Google Scholar 

  4. D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with fixed point in ℝn,” Dokl. Ross. Akad. Nauk, 383, No. 5, 635–637 (2002).

    MathSciNet  Google Scholar 

  5. D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion of a generalized gyroscope in ℝn,” Vestn. MGU, Mat., Mekh., No. 5, 37–41 (2003).

  6. D. V. Georgievskii and M. V. Shamolin, “Valerii Vladimirovich Trofimov,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), pp. 5–6.

  7. D. V. Georgievskii and M. V. Shamolin, “On kinematics of a rigid body with fixed point in ℝn,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), pp. 24–25.

  8. D. V. Georgievskii and M. V. Shamolin, “Generalized dynamical Euler equations for a rigid body with fixed point in ℝn,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamantal Directions [in Russian], 23 (2007), p. 30.

  9. D. V. Georgievskii and M. V. Shamolin, “First integrals of equations of motion of a generalized gyroscope in n-dimensional space,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), p. 31.

  10. D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “Geometry and mechanics: Problems, approaches, and methods,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Fund. Prikl. Mat. [in Russian], 7, No. 1 (2001), p. 301–752.

    Google Scholar 

  11. D. V. Georgievskii, V. V. Trofimov, and M. V. Shamolin, “Geometry and mechanics: Problems, approaches, and methods,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), p. 16.

  12. M. V. Shamolin, “The problem of four-dimensional rigid body motion in a resisting medium and one integrability case,” In: Book of Abstracts, Third Int. Conf. “Differential Equations and Applications,” St. Petersburg, Russia, June 1217, 2000 [in Russian]; StPbGTU, St. Petersburg (2000), p. 198.

  13. M. V. Shamolin, “Jacobi integrability of the problem of four-dimensional rigid body motion in a resisting medium,” In: Abstracts of Reports of the International Conference on Differential Equations and Dynamical Systems (Suzdal, 2126.08.2000) [in Russian], Vladimir State University, Vladimir (2000), pp. 196–197.

  14. M. V. Shamolin, “Comparison of certain integrable cases from two-, three-, and four-dimensional dynamics of a rigid body interacting with a medium,” In: Abstracts of Reports of the Vth Crimean International Mathematical School “Lyapunov Function Method and Its Applications” (MFL2000) (Crimea, Alushta, 0513.09.2000) [in Russian], Simferopol (2000), p. 169.

  15. M. V. Shamolin, “On a certain Jacobi integrability case in the dynamics of a four-dimensional rigid body interacting with a medium,” In: Abstracts of Reports of the International Conference on Differential and Integral Equations (Odessa, 1214.09.2000) [in Russian], AstroPrint, Odessa (2000), pp. 294–295.

  16. M. V. Shamolin, “Jacobi integrability in the problem of four-dimensional rigid-body motion in a resisting medium,” Dokl. Ross. Akad. Nauk, 375, No. 3, 343–346 (2000).

    Google Scholar 

  17. M. V. Shamolin, “Integrability of the problem of four-dimensional rigid-body motion in a resisting medium,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Fundam. Prikl. Mat. [in Russian], 7, No. 1 (2001), p. 309.

  18. M. V. Shamolin, “New integrable cases in the dynamics of a four-dimensional rigid body interacting with a medium,” In: Dynamical Systems Modeling and Stability Investigation. Scientific Conference (2225.5.2001) [in Russian], Abstracts of Conference Reports, Kiev (2001), p. 344.

  19. M. V. Shamolin, “New Jacobi integrable cases in the dynamics of a two-, three-, and fourdimensional rigid body interacting with a medium,” In: Abstracts of Reports of the VIIIth All-Russian Session in Theoretical and Applied Mechanics (Perm, 2329.08.2001) [in Russian], Ural Department of the Russian Academy of Sciences, Ekaterinburg (2001), pp. 599–600.

  20. M. V. Shamolin, “New integrable cases in the dynamics of a two-, three-, and four-dimensional rigid body interacting with a medium,” In: Abstracts of Reports of the International Conference on Differential Equations and Dynamical Systems (Suzdal, 0106.07.2002) [in Russian], Vladimir State University, Vladimir (2002), pp. 142–144.

  21. M. V. Shamolin, “On a certain integrable case in dynamics on so(4)×ℝn,” In: Abstracts of Reports of the All-Russian Conference “Differential Equations and Their Applications” (SamDif2005), Samara, June 27 July 2, 2005 [in Russian], Univers-grup, Samara (2005), pp. 97–98.

  22. M. V. Shamolin, “On a certain integrable case of dynamics equations on so(4) × ℝn,” Usp. Mat. Nauk, 60, No. 6, 233234 (2005).

    Google Scholar 

  23. M. V. Shamolin, “On a case of complete integrability in the dynamics of a four-dimensional rigid body,” In: Abstracts of Reports on Differential Equations and Dynamical Systems, Vladimir, 1015.07.2006 [in Russian], Vladimir State University, Vladimir (2006), pp. 226–228.

  24. M. V. Shamolin, “Integrability of the problem of four-dimensional rigid-body motion in a resisting medium,” In: Abstracts of Sessions of the Workshop Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), p. 21.

  25. M. V. Shamolin, “New integrable cases in the dynamics of a four-dimensional rigid body interacting with a medium,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), p. 27.

  26. M. V. Shamolin, “On integrability of motion of a four-dimensional rigid body, a pendulum, being in over-run medium flow,” In: Abstracts of Sessions of the Workshop “Topical Problems of Geometry and Mechanics,” Contemporary Mathematics. Fundamental Directions [in Russian], 23 (2007), p. 37.

  27. M. V. Shamolin, “A case of complete integrability in the dynamics of a four-dimensional rigid body in a nonconservative force field,” In: Abstracts of Reports of the International Congress “Nonlinear Dynamical Analysis-2007,” St. Petersburg, June 48, 2007 [in Russian], St. Petersburg State University, St. Petersburg (2007), p. 178.

  28. M. V. Shamolin, “Case of complete integrability in the dynamics of a four-dimensional rigid body in a nonconsevative force field,” In: Abstracts of Reports of the International Conference “Analysis and Singularities,” Devoted to the 70th Anniversary of V. I. Arnold, Moscow, August 2024, 2007 [in Russian], MIAN, Moscow (2007), pp. 110112.

  29. M. V. Shamolin, “Methods of analysis of dynamics of a 2D- 3D- or 4D-rigid body with a medium,” In: Abstr. Short Commun. Post. Sess. of ICM2002, Beijing, 2002, August 2028, Higher Education Press, Beijing, China (2002), p. 268.

  30. M. V. Shamolin, “4D rigid body and some cases of integrability,” In: Abstracts of ICIAM07, Zurich, Switzerland, June 1620, 2007, ETH Zurich (2007), p. 311.

  31. M. V. Shamolin, “Cases of integrability in 2D-, 3D- and 4D-rigid body,” In: Abstr. of Brief Commun. and Post. of Int. Conf. “Dynamical Methods and Mathematical Modeling,” Valladolid, Spane (Sept. 1822, 2007), ETSII, Valladolid (2007), p. 31.

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  1. M. V. Lomonosov Moscow State University, Moscow, Russia

    M. V. Shamolin

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Correspondence to M. V. Shamolin.

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 78, Partial Differential Equations and Optimal Control, 2012.

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Shamolin, M.V. Some questions of qualitative theory in dynamics of systems with the variable dissipation. J Math Sci 189, 314–323 (2013). https://doi.org/10.1007/s10958-013-1189-9

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