Shamolin, M. V. Classes of variable dissipation systems with nonzero mean in the dynamics of a rigid body. (English) Zbl 1140.70456 J. Math. Sci., New York 122, No. 1, 2841-2915 (2004). From the text: The propositions obtained in this paper for variable dissipation systems continue the Poincaré-Bendixson theory for systems on closed two-dimensional manifolds and the topological classification of such systems. The problems considered in the paper foster the development of the qualitative techniques of analysis; therefore, the qualitative theory of variable dissipation systems evolves in a natural way. In chapter 1, we obtain nonlinear dynamical systems describing several versions of the body’s motion. Such systems are variable dissipation systems with zero or nonzero mean. Chapter 2 is devoted to the study of the relative structural stability (relative roughness) of dynamical systems considered not on the whole space of dynamical systems but on certain of its subspace. Moreover, the space of deformations of (dynamical) systems also does not coincide with the whole space of admissible deformations. In particular,we consider systems of differential equations arising in the dynamics of a rigid body interacting with a resisting medium. We show that they are relatively rough; under certain conditions, they can also have nonroughness properties of different degrees. In chapter 3 we have obtained a class of partial solutions for a certain class of systems with nonzero mean and have laid the foundation for carrying out qualitative integration of dynamic equations in the space of quasivelocities. We have also obtained a new two-parametric family of phase portraits on the two-dimensional cylinder. It is shown that this family consists of infinitely many topologically nonequivalent phase portraits with distinct qualitative properties.In chapter 4, we consider in general the possibilities of extending the results of the plane dynamics of a rigid body interacting with the medium to the spatial case. We examine the problem on the spatial motion of a rigid body in a resisting medium. Such a motion is described by variable dissipation dynamical systems with nonzero mean.The fifth chapter (and, in essence, Chapter 3) is devoted to the study of a class of motions of a rigid body, which is interesting from the standpoint of applications. This is a free deceleration in a resisting medium. In fact, this chapter is an introduction to the problem of spatial free deceleration. In this chapter, we obtain some particular solutions of the complete system and prepare the material for the qualitative integration of the dynamical equations in the space of quasi-velocities. The second part of the chapter is devoted to a new two-parametric family of phase portraits in the three-dimensional space. We show that the obtained family consists of an infinite set of topologically nonequivalent phase portraits with different nonlinear qualitative properties.In chapter 6 we consider only the nonlinear model of interaction of a body with a medium in which there is no damping in the system on the part of the medium (in the linear case, \(h = 0\)), which corresponds to the case where the resistance force and its momentum depend only on the angle of attack. In this chapter,we pass to accounting for the additional linear damping acting on the part of the medium. Using nonlinear equations, we study the stability of the rectilinear translation deceleration in the presence of a linear damping momentum. We show that in the framework of the model considered, in principle, there can arise self-oscillations corresponding to limit cycles which arise from a weak focus (the well-known Andronov-Hopf bifurcation). Also, we perform a qualitative analysis of certain nonlinear dynamical systems obtained above but under the condition that there is a linear damping momentum in the system. Depending on the damping coefficient from the medium,w e perform the topological classification of typical topologically nonequivalent phase portraits of the system, which is considered on the phase cylinder of quasi-velocities. We show that under certain conditions, in the system there can arise stable, as well as unstable, self-oscillations. (325 Refs.). Cited in 18 Documents MSC: 70E99 Dynamics of a rigid body and of multibody systems 34C40 Ordinary differential equations and systems on manifolds 37N05 Dynamical systems in classical and celestial mechanics 70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics × Cite Format Result Cite Review PDF Full Text: DOI