# American Institute of Mathematical Sciences

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April  2019, 24(4): 1725-1741. doi: 10.3934/dcdsb.2018233

## Synchronization of first-order autonomous oscillators on Riemannian manifolds

 Dipartimento di Ingegneria dell'Informazione, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona (Italy)

Received  September 2017 Revised  March 2018 Published  August 2018

The present research work recalls a control-theoretic approach to the synchronization of a first-order master/slave oscillators pair on $\mathbb{R}^3$ and extends such technique to the case of curved Riemannian manifolds. As theoretical results, this paper proves the asymptotic convergence of the feedback controller and studies the entity of the 'control effort'. As a case study, the complete equations for the controller of a slave oscillator on the unit hypersphere $\mathbb{S}^{n-1}$ are laid out and are illustrated by numerical examples for $n = 3$ and $n = 10$, even in the hypothesis of noisy master-system state measurement.

Citation: Simone Fiori. Synchronization of first-order autonomous oscillators on Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1725-1741. doi: 10.3934/dcdsb.2018233
##### References:

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##### References:
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 $\mathbb{M} = \mathbb{R}^n$
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
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
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
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
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
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 $\alpha = 0.1$
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