Synchronization of first-order autonomous oscillators on Riemannian manifolds

Figure 1.  Master/slave/controller configuration: The feedback-type control chain is designed to make the slave oscillator sync asymptotically with the master oscillator. The symbol $\mathbb{M}$ denotes the state space of the master/slave pair which, in the classical setting, is $\mathbb{M} = \mathbb{R}^n$

Figure 2.  Example of dynamics of the system (26) over the sphere $\mathbb{S}^2$. The left-hand panel illustrates the trajectory of the system (the open circle denotes the initial point). The right-hand panels illustrate the kinetic energy $K(t)$ of the system (top panel) and the temporal evolution of the components of the state vector $z$ (bottom panel: $z_1$ in green color, $z_2$ in blue color, $z_3$ in red color). In this simulation, the initial state-vector $z_0$ and the matrix $A$ were chosen randomly

Figure 3.  Example of synchronization of oscillators over the sphere $\mathbb{S}^2$. The top panel on the left-hand side illustrates the components of the vector $z^\textrm{m}(t)$, while the bottom panel on the left-hand side illustrates the components of the state-vector $z^\textrm{s}(t)$. The top panel on the right-hand side illustrates the kinetic energies $K_\textrm{s}(t)$ and $K_\textrm{m}(t)$ of the oscillators. The bottom panel on the right-hand side illustrates the theoretical Lyapunov function versus the actual one

Figure 4.  Example of synchronization of oscillators over the sphere $\mathbb{S}^2$. Synchronizing trajectories over the state-space. Black solid line: Master oscillator. Red solid line: Slave oscillator

Figure 5.  Example of synchronization of oscillators over the sphere $\mathbb{S}^{9}$ subjected to temporary connection loss at $t = 25$. The top panel on the left-hand side illustrates the components of the vector $z^\textrm{m}(t)$, while the bottom panel on the left-hand side illustrates the components of the state-vector $z^\textrm{s}(t)$. The top panel on the right-hand side illustrates the kinetic energies $K_\textrm{s}(t)$ and $K_\textrm{m}(t)$ of the oscillators, from which it is quite apparent the behavior of the control chain during the simulated connection loss. The bottom panel on the right-hand side illustrates the theoretical Lyapunov function versus the actual one

Figure 6.  Example of synchronization of oscillators over the sphere $\mathbb{S}^{2}$ in the presence of an exponentiated-additive disturbance. The top panel on the left-hand side illustrates the components of the observable state-vector $\zeta^\textrm{m}(t)$, while the bottom panel on the left-hand side illustrates the components of the state-vector $z^\textrm{s}(t)$. The top panel on the right-hand side illustrates the kinetic energies $K_\textrm{s}(t)$ and $K_\textrm{m}(t)$ of the oscillators. The bottom panel on the right-hand side illustrates the theoretical Lyapunov function versus the actual one

Figure 7.  Example of synchronization of oscillators over the sphere $\mathbb{S}^2$ in the presence of an exponentiated-additive disturbance. Synchronizing trajectories (noisy-observable master state and slave state) over the state-space obtained when the master oscillator state vector is observed under an exponentiated-additive random disturbance of standard deviation $\alpha = 0.1$