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

Abstract

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.

Physically meaningful parameters of viscoelastic models based on the general functional form of the transfer functions

“…Complex materials often exhibit power-law time-dependent relaxation behavior, and their efficient modeling relies on the use of the fractional calculus-based viscoelastic element spring-pot [16]. The thermodynamic compatibility of fractional viscoelastic models has been studied in [38,39], the models' analytical solutions have been provided in [40,41] and their mathematical structures have been discussed from the point of view of differential geometry [42,43]. Nevertheless, the physical interpretation of the constants associated with the spring-pot's constitutive relation remains elusive.…”

Fractional Curvatures of Equiaffine Curves in Three-Dimensional Affine Space

“…These applications offer valuable tools for modeling and analyzing intricate systems. Notably, these applications span across fields such as medicine [11], bioengineering [12], viscoelasticity [13], and dynamical systems [14,15]. In contrast to the straightforward expressions of integer-order derivatives and integrals, various more intricate fractional derivatives and integrals exist.…”

Geometry of Curves with Fractional Derivatives in Lorentz Plane

“…We can also say that a non-local fractional derivative of a function is related to history or a space-range interaction. Furthermore, fractional calculus has many applications to viscoelastic [5][6][7][8][9][10][11], analytical mechanics [12][13][14], and dynamical systems [15][16][17][18][19]. Fractional analysis has also started to be studied from a differential geometry perspective in recent studies.…”