A Simple Fractional Model With Complex Dynamics in the Order of the Derivative

Fractional variable order systems with complex dynamics in the order is a little studied topic. In this research, we present three examples of a very simple fractional system with complex dynamics in the order of the derivative. 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 orders of the derivatives. 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 with complex behavior can reveal through the order of their derivatives.