The present research outlines a non-linear control theory, based on the PID regulation scheme, to synchronize two second-order dynamical systems insisting on a Riemannian manifold. Also, this document presents some numerical recipes to implement the systems, as well as the control scheme, on a computing platform and a large number of numerical simulation results focused on the synchronization of Duffing-like non-linear oscillators on the unit sphere.
Nonlinear dynamical systems on manifolds constitute a challenging and exciting research topic at the cross border of applied mathematics and theoretical sciences. Indeed, these dynamical systems have appeared in a number of subjects in pure mathematics as well as applied sciences. Nonlinear dynamical systems on manifolds have been studied in the scientific literature because they arise naturally from the modeling of complex physical structures and because such dynamical systems constitute the basis for several modern applications. Such kind of dynamical systems are currently being investigated upon in at least two different contexts, according to their realm of application:
- Synthetic (or algorithmic) systems: These systems arise as extensions to manifold of classical nonlinear oscillators, such as the Duffing and the van Der Pol systems, or are designed specifically to generate complicated trajectories. These nonlinear systems are used to test non-linear signal processing algorithm and nonlinear control strategies, such as in the case of secure transmission of information by signal masking.
- Models of physical phenomena: These systems arise upon a modeling complicated physical phenomena, such as the rotational motion of a satellite or a drone and in general relativity.