BHATTACHARYA, T. A. P. O. B. R. A. T. A. (2015). Chaos in Non-Euclidean Geometries Arising in Mechanics. Unc Charlotte Electronic Theses And Dissertations.
Chaotic systems are a class of nonlinear dynamical systems that have captured the attention of mathematicians and engineers for long. Even though chaotic systems are deterministic in nature, long term prediction of behavior of such systems is difficult. Most of the studies on the evolution of such systems, have been restricted to the Euclidean manifold. The proposed dissertation will- examine the dependence of elementary concepts from dynamical systems theory, particularly those that pertain to stability and measures ofchaos, on the metric structure assigned to the space or manifold in which a dynamical system evolves.-examine the way in which the input-output relationships defined by mechanical control systems with nonlinear dynamics, particularly systemsrepresenting problems in robotic locomotion, viewed as transformations between differentiable manifolds, affect measures of chaos.The present research summarizes background material and preliminary results and explores chaos in the context of non-Euclidean manifolds. It also focuses on mechanical systems that have the ability to locomote by internal shape change. Such systems can be modeled by principal connections using geometric methods. The study involves investigating the flow of the differential equations that govern the motion of the body under chaotic actuation of the shape parameters.
