Generalization of KCC-theory to fractional dynamical systems and application to viscoelastic oscillations

Abstract

As a geometrical method for analyzing the stability of dynamical systems, Kosambi–Cartan–Chern (KCC)-theory determines the Jacobi stability of a perturbation given to a trajectory based on the deviation curvature tensor. In this study, KCC-theory is generalized to dynamical systems with fractional derivatives, and the deviation curvature tensor is obtained from the fractional differential equation under the condition that the order of the fractional derivatives is close to an integer. Fractional KCC-theory is then applied to a one-dimensional spring–mass–damper model in which the damping force is expressed as a fractional derivative. In this case, three solution types are obtained depending on the damping coefficient and the fractional derivative order. One is damped oscillation with overshoot, which also appears in models with integer order derivatives, and another is damped oscillation without overshoot. The remaining one is overdamping, which neither oscillates nor overshoots. The deviation curvature tensor obtained from the fractional oscillatory model shows that both the damped oscillation without overshoot and the overdamping are Jacobi stable. When a discontinuous Jacobi instability appears on the solution curve, the solution type is classified as damped oscillation with overshoot. This result indicates that damped oscillation with overshoot and other solutions can be classified according to Jacobi stability.