A Simple Fractional Model with Unusual Dynamics in the Derivative Order

Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: (1) an integer-order differential equation that includes the state variable, (2) a differential equation that incorporates the state variable and features both integer- and fractional-order derivatives, and (3) fractional variable-order differential equations nested in the derivative orders. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems can reveal through the order of their derivatives.