Nonlinear Second-Order Dynamical Systems on Riemannian Manifolds

Linear as well as non-linear mathematical systems that exhibit an oscillatory behavior are ubiquitous in sciences and engineering. Such mathematical systems have been used to model the behavior of biological structures, such as the pulsating contraction of cardiac cells, as well as the behavior of electrical and mechanical components. Chaotic oscillators are currently being used in the secure transmission of information. The state of such classical dynamical systems evolve in the Euclidean space R^n (typically, n=1,2,3). The current paper aims at proposing a principled mathematical technique to design second-order nonlinear dynamical systems over curved Riemannian manifolds and to discuss a numerical simulation framework that is compatible with the structure of such spaces.