Fractional Dynamics: A Comprehensive Exploration of Non-integer Order Systems

This article delves into the applications of fractional calculus, an extension of classical calculus that introduces non-integer derivative orders. The primary focus of this research is to present a methodology for simulating fractional differential equations and to explore the effects of fractional order in two well-known dynamic systems: the Van der Pol and Duffing systems. These systems are known for their nonlinear characteristics and, in certain cases, exhibit complex and rich dynamic behaviors. Initially, the Grunwald-Letnikov definition of fractional derivatives is introduced, followed by the general numerical solution for a fractional differential equation. The Van der Pol system is modified by the inclusion of a fractional time derivative of order q, reducing the integer order of the system to 1 + q, while the Duffing system is modified in terms of viscous damping, by the add of a fractional damping, which is now related to the fractional variation in displacement. The dynamics of the systems are characterized using classical methods of nonlinear dynamics, such as time-response, Poincaré sections, bifurcation diagrams and fast Fourier transform, as well as more advanced approaches, such as the continuous wavelet transform (CWT) and the Hilbert-Huang transform (HHT).