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

Phase Shadows: An Enhanced Representation of Nonlinear Dynamic Systems

“…But the phase shadow plot has advantages not only when the system's behavior has a limit cycle, but also to display equilibrium points [Shamolin, 2009], bifurcations [Lima & Arun, 2006;Luo & Guo, 2016] and chaotic behavior [Han et al, 2013]. This statement will be justified in the following subsections.…”

“…Duffing oscillator [Rajamani & Rajasekar, 2017;Luo & Huang, 2012;Luo & Min, 2011], energy harvesting from sea waves [De Paula et al, 2012;Lenci, 2014] or rigid bodies interacting with a resisting medium [Shamolin, 2009] show a very similar behavior to the pendulum. Also, in other physical realms, it appears that different oscillators can be easily modeled as pendulum-like systems [Blackburn et al, 1987] or show dynamic behavior resembling pendulums [Ruelas & Rand, 2015].…”

“…One of the most used tools in the description of these nonlinear systems is the phase portrait [Shamolin, 2009], where typical nonlinear behavior can be easily identified, such as multiple equilibrium points, limit cycles, bifurcations and chaos.…”

Spatial motion of a pendulum in a jet flow: qualitative aspects and integrability

“…Consider a moving homogeneous body of mass m. A portion of its surface is a flat disk. A jet flow is past the body [1,2]. The other portion of the body's surface is inside the volume bounded by the jet stalling at the disk edge and is not affected by the medium.…”

“…In this case, the force field is constructed from the reasons for the medium action on the body under the streamline (or separation) flow around and quasi-stationarity conditions. It turns out that studying the motion of such a class of bodies reduces to systems with energy scattering (dissipative systems) or with its pumping (the so-called anti-dissipative systems) [1,2]. Note that similar problems were already appeared in applied aerodynamics in the studies of Central Aero-Hydrodynamic Institute named after professor N. E. Zhukovskii.…”