Control Strategy for Generalized Synchrony in Coupled Dynamical Systems

Dynamical systems can be coupled in a manner that is designed to drive the resulting dynamics onto a specified lower dimensional submanifold in the phase space of the combined system. On the submanifold, the variables of the two systems have a well-specified functional relationship. This process can be viewed as a control technique that ensures generalized synchronization. Depending on the nature of the dynamical systems and the specified submanifold, different coupling functions can be derived in order to achieve a desired control objective. We discuss a circuit implementation of this strategy for coupled chaotic Lorenz oscillators, as well as a demonstration of the methodology for designing coordinated motion (swarming) in a set of autonomous drones.

💡 Research Summary

This paper presents a comprehensive framework for achieving generalized synchronization (GS) in coupled dynamical systems, recasting it as a geometric control problem. The core objective is to drive the collective dynamics of two coupled systems onto a pre-specified, lower-dimensional submanifold within the combined phase space. This submanifold is defined by a set of algebraic constraint equations, Φ(x, y) = 0, which represent the desired functional relationship between the variables of the two systems, thereby defining GS.

The proposed methodology is fundamentally geometric. Given the constraint equations, the normal vectors to the target submanifold are computed. The control design then ensures that the flow vector of the coupled system is orthogonal to these normals. This condition leads to a matrix equation for the coupling functions ς(x, y) that need to be added to the intrinsic dynamics of each system. A key strength of this approach is that the governing equation is underdetermined, permitting a large family of possible coupling functions that all satisfy the same geometric constraint. This inherent flexibility allows for the incorporation of additional stabilizing terms (e.g., based on Lyapunov functions) that vanish on the target manifold, enhancing the robustness and adaptability of the control scheme.

The paper elaborates on several concrete scenarios. For a “master-slave” configuration, an explicit coupling function can be derived that forces the slave variables to track a functional transformation of the master’s state, sometimes at the expense of suppressing the slave’s intrinsic dynamics. The framework naturally encapsulates known synchronization types: complete synchronization (Φ: x=y), projective synchronization (Φ: y=A*x), and more complex nonlinear projective synchronization (e.g., Φ: y_3 = x_1^2). The authors provide detailed derivations of coupling functions for these cases applied to coupled chaotic Lorenz oscillators. They demonstrate numerically that different coupling forms (unidirectional master-slave, slave-master, or bidirectional) can confine the dynamics to the same geometric submanifold while resulting in distinct trajectories on that manifold, highlighting the decoupling of the geometric control objective from the specifics of the coupled dynamics.

A significant application is found in the context of swarming or coordinated motion. For the simple case of translational constraints (maintaining a fixed offset, Φ: x_i - y_i = a_i), the control law simplifies dramatically. In a master-slave setup, the slave system essentially replicates the velocity field of the master while adding a corrective term to maintain the offset. This provides a straightforward and elegant control strategy for coordinating multiple autonomous agents, such as drones, to move in formation.

The theoretical framework is supported by two practical implementations. First, an electronic circuit realization of the method is discussed for coupled Lorenz oscillators, validating the approach in an experimental, analog setting. Second, the algorithm is implemented for designing coordinated swarming behavior in a set of autonomous drones, demonstrating its utility in a higher-level, cyber-physical system. The paper concludes by emphasizing the generality of the geometric control viewpoint, its ability to unify various synchronization phenomena under one framework, and its broad applicability across disciplines from nonlinear dynamics and circuit theory to robotics and multi-agent systems.