Abstract
Fractional derivatives and integrals are intrinsically multiscale operators that can act on both space and time dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or variable order (VO) where, in the latter case, the order can also be function of either independent or state variables. When using VO differential governing equations to describe the response of dynamical systems, the order can evolve as a function of the response itself therefore allowing a natural and seamless transition between largely dissimilar dynamics (e.g. linear, nonlinear, and even contact problems). Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. In this study, we present the possible application of VO operators to a class of nonlinear lumped parameter models that has great practical relevance in mechanics and dynamics. Specific examples include hysteresis and contact problems for discrete oscillators. Within this context, we present a methodology to define VO operators capable of capturing such complex physical phenomena. Despite using simplified lumped parameters nonlinear models to present the application of VO operators to mechanics and dynamics, we provide a more qualitative discussion of the possible applications of this mathematical tool in the broader context of continuous multiscale systems.
An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order