Dynamical systems with variable dissipation: approaches, methods, and applications
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.
scientific article; zbMATH DE number 5660185
scientific article