Geometric Structures of Fractional Dynamical Systems in Non-Riemannian Space: Applications to Mechanical and Electromechanical Systems

摘要

Based on a non-Riemannian treatment of geometric objects, the geometric structures of fractional-order dynamical systems are investigated. A fractional derivative describes non-local effects across a space or a history encoded in memory features of the system. A system of fractional-order differential equations is formulated in film space that includes fictitious forces. Film space is a geometric space whose coordinates comprise time, and the geometric quantities vary in time. Fractional-order torsion tensors that appear are related to the dissipated energy and the energy conversions between subsystems and power of the system. The geometric treatment is then applied to damped-harmonic and fractional oscillators and the hybrid electromechanical Rikitake system. The damped-harmonic oscillator is characterized by two torsion tensors, whereas the fractional oscillator is characterized by one torsion tensor. Herein, the fractional order of the derivative of the metric tensor is used to characterize the damping of the fractional oscillator. The energy conversions between electromechanical subsystems in the Rikitake system are characterized by the torsion tensor. These results suggest that the non-Riemannian geometric objects can represent the non-local properties of fractional-order dynamical systems.