Investigation into the Local and Global Bifurcations of the Whirling Planar Pendulum
Hyde, Griffin Nicholas
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This thesis details the investigation into the Whirling Planar Pendulum system. The WPP is a pendulum that is spun around a vertical spin axis at a controllable horizontal offset. This dynamical system exhibits both local and global bifurcations. The local pitchfork bifurcation leads to the splitting of a single stable equilibrium point into three (two stable and one unstable), as the spin rate is increased. The global bifurcations lead to two independent types of chaotic oscillations which are induced by sinusoidal excitations. The types of chaos are each associated with one of two homoclinic orbits in the system's phase portraits. The onset of each type of chaos is investigated through Melnikov's Method applied to the system's Hamiltonian, to find parameters at which the stable and unstable manifolds intersect transversely, indicating the onset of chaotic motion. These results are compared to simulation results, which suggest chaotic motion through the appearance of strange attractors in the Poincaré maps. Additionally, evidence of the WPP system experiencing both types of chaos simultaneously was found, resulting in a merger of two distinct types of strange attractor.
General Audience Abstract
This report details the investigation into the Whirling Planar Pendulum system. The WPP is a pendulum that is spun around a vertical spin axis at a controllable horizontal offset. This system can be used to investigate what are known as local and global bifurcations. A local bifurcation occurs when the single equilibrium state (corresponding to the pendulum hanging straight down) when spun at low speeds, bifurcates into three equilibria when the spin rate is increased beyond a certain value. The global bifurcations occur when the system experiences sinusoidal forcing near certain equilibrium conditions. The resulting chaotic oscillations are investigated using Melnikov’s method, which determines when the sinusoidal forcing results in chaotic motion. This chaotic motion comes in two types, which cause the system to behave in different ways. Melnikov’s method, and results from a simulation were used to determine the parameter values in which the pendulum experiences each type of chaos. It was seen that at certain parameter values, the WPP experiences both types of chaos, supporting the observation that these types of chaos are not necessarily independent of each other, but can merge and interact.
- Masters Theses