Synchronization of fractional-order chaotic networks in Presnov form via homogeneous controllers.

The Presnov decomposition for continuously differential vector fields consists of a gradient of a scalar field (dissipative term) and a vector field orthogonal to a radial vector of the origin (circulative or non-dissipative term). This paper uses the Presnov decomposition associated with a fractional-order Lorenz family to design algorithms that solve the multi-synchronization problem. Assuming that the whole state is known, two active control solutions are proposed to achieve the desired synchronization. Both control algorithms take advantage of homogeneous functions to have a powerful and simple Lyapunov analysis to show Mittag-Leffler stabilization in the first algorithm. In contrast, the second one achieves synchronization before a predefined time defined by the user. The combination of homogeneous functions and Riemann–Liouville integral in the controllers improve the behavior of the dynamic systems due to the impact on the robustness properties in the control scheme. Numerical simulations are presented to show the reliability of the proposed methods.