Control Strategy for Generalized Synchrony in Coupled Dynamical Systems

TL;DR

This paper presents a geometric control framework to enforce generalized synchronization by constraining the joint dynamics of coupled nonlinear systems to a specified submanifold defined by $\Phi(\mathbf{X})=0$. By making the flow orthogonal to the manifold normals, the authors derive coupling terms that satisfy $\epsilon\mathfrak{N}\bm{\varsigma}=-\mathfrak{N}\mathbf{F}$, enabling flexible master-slave and translational constraint implementations. They validate the approach with practical circuit implementations of Lorenz oscillators exhibiting projective synchrony and nonlinear scaling, and extend the methodology to a swarm algorithm for autonomous drones that maintains prescribed separations. Overall, the work provides a hardware-friendly, versatile method for engineering generalized synchronization across electronic and robotic platforms, with implications for secure communication, coordinated motion, and multi-agent control.

Abstract

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.

Figures (14)

• Figure 1: (Colour online) Transition to complete synchronization as a function of coupling strength $\epsilon$ in the coupled Lorenz system; see Eq. (\ref{['pecora_carroll']}). (a) The two largest transverse Lyapunov exponents, and (b) the order parameter $\Delta$ that measures deviations from the synchronization submanifold. The case discussed in pc1990 was for $\epsilon$=1.
• Figure 2: (Colour online) Projective synchronisation with $\alpha_1 = 1$, $\alpha_2 = 2$ and $\alpha_3 = 3$. The blue dots are for unidirectional (master-slave) coupling while red dots show the dynamics with bi-directional coupling. While the dynamics in either case is confined to the same plane, the trajectories occupy different parts of the specified submanifold. Here $\epsilon = 1$, but the dynamics reaches the submanifold for smaller $\epsilon$ in both coupling cases.
• Figure 3: (Colour online) Projection of the dynamics in the coupled system, now confined to the subspace defined by $x_1=y_1, x_2=y_2, x_3=y_3^2$. The different coupling schemes bring the dynamics to different regions within this submanifold while retaining the characteristics of the two oscillators, namely their chaotic nature. The value of $\epsilon$ is 1. See text for details.
• Figure 4: (Colour online) The transition to nonlinear projective synchrony using the three different coupling forms, as a function of the strength $\epsilon$, as seen in terms of the order parameter $\Delta$ defined in Eq. (\ref{['op']}). For the (i) Master-Slave coupling Eq. (\ref{['nonlinear_ms']}) (red dashed line), (ii) Slave-Master coupling Eq. (\ref{['nonlinear_sm']}) (black dotted line), and (iii) bidirectional coupling, Eq. (\ref{['nonlinear_bi']}) (solid blue line). Note the logarithmic scale on the ordinate. In all three cases the systems show GS only for $\epsilon = 1$.
• Figure 5: Circuit diagram for the projective synchronization ($x_i-\alpha y_i$ =0), where $\alpha=2.0$. Values of the resistors and capacitors are given in the text, and connections between the two oscillators are shown by the nodes ($N_i$) for simplicity. The respective paired nodes (say N$_1$-N$_1$) are connected during the real-time hardware experiment.