# American Institute of Mathematical Sciences

## 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, doi: 10.3934/dcdsb.2018233
##### References:
 [1] S. Al-Azzawi, L. Jicheng and L. Xianming, Convergence rate of synchronization of systems with additive noise, Discrete & Continuous Dynamical Systems - Series B, 22 (2017), 227-245. doi: 10.3934/dcdsb.2017012. [2] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153. doi: 10.1016/j.physrep.2008.09.002. [3] S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares and C. S. Zhou, The synchronization of chaotic systems, Physics Reports, 366 (2002), 1-101. doi: 10.1016/S0370-1573(02)00137-0. [4] I. Chueshov, P. E. Kloeden and Y. Meihua, Synchronization in couples sine-Gordon wave model, Discrete & Continuous Dynamical Systems - Series B, 21 (2016), 2969-2990. doi: 10.3934/dcdsb.2016082. [5] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Transactions on Circuits and Systems - Part Ⅱ: Analog and Digital Signal Processing, 40 (1993), 626-633. doi: 10.1109/82.246163. [6] F. Dörfler, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010. doi: 10.1073/pnas.1212134110. [7] Z. Cai, M. S. de Queiroz and D. M. Dawson, Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance, IEEE Transactions on Automatic Control, 51 (2006), 524-529. doi: 10.1109/TAC.2005.864204. [8] S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Fundamentals, Journal of Systems Science and Complexity, 29 (2016), 22-40. doi: 10.1007/s11424-015-4063-7. [9] S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207-222. doi: 10.1016/j.cnsns.2016.11.025. [10] J. M. González Miranda, Synchronization and Control of Chaos, Imperial College Press, London, 2004. [11] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume Ⅰ, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [12] J. M. Lee, Riemannian Manifolds, Vol. 176 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [13] T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S. Williams, E. Ott and R. Roy, Complex dynamics and synchronization of delayed-feedback nonlinear oscillators, Philosophical Transactions of the Royal Society A, 368 (2010), 343-366. doi: 10.1098/rsta.2009.0225. [14] J. H. Park, Chaos synchronization of a chaotic system via nonlinear control, Chaos, Solitons and Fractals, 27 (2006), 1369-1375. doi: 10.1016/j.chaos.2005.05.001. [15] L. M. Pecora and T. L. Carroll, Synchronization of chaotic systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, (1996), 142-145. doi: 10.1016/B978-012396840-1/50040-0. [16] X. Pennec, Barycentric subspaces and affine spans in manifolds, Geometric Science of Information, Lecture Notes in Comput. Sci., Springer, Cham, 9389 (2015), 12-21. doi: 10.1007/978-3-319-25040-3_2. [17] A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76. doi: 10.1137/060673400. [18] J.-P. Yeh and K.-L. Wu, A simple method to synchronize chaotic systems and its application to secure communications, Mathematical and Computer Modelling, 47 (2008), 894-902. doi: 10.1016/j.mcm.2007.06.021. [19] C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific Publishing Co. Pte. Ltd, 2007. doi: 10.1142/6570. [20] X. Wu, C. Xu and J. Feng, Complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems and coupling time delays, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1004-1014. doi: 10.1016/j.cnsns.2014.07.003.

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##### References:
 [1] S. Al-Azzawi, L. Jicheng and L. Xianming, Convergence rate of synchronization of systems with additive noise, Discrete & Continuous Dynamical Systems - Series B, 22 (2017), 227-245. doi: 10.3934/dcdsb.2017012. [2] A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno and C. Zhou, Synchronization in complex networks, Physics Reports, 469 (2008), 93-153. doi: 10.1016/j.physrep.2008.09.002. [3] S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares and C. S. Zhou, The synchronization of chaotic systems, Physics Reports, 366 (2002), 1-101. doi: 10.1016/S0370-1573(02)00137-0. [4] I. Chueshov, P. E. Kloeden and Y. Meihua, Synchronization in couples sine-Gordon wave model, Discrete & Continuous Dynamical Systems - Series B, 21 (2016), 2969-2990. doi: 10.3934/dcdsb.2016082. [5] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz, Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE Transactions on Circuits and Systems - Part Ⅱ: Analog and Digital Signal Processing, 40 (1993), 626-633. doi: 10.1109/82.246163. [6] F. Dörfler, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010. doi: 10.1073/pnas.1212134110. [7] Z. Cai, M. S. de Queiroz and D. M. Dawson, Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance, IEEE Transactions on Automatic Control, 51 (2006), 524-529. doi: 10.1109/TAC.2005.864204. [8] S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Fundamentals, Journal of Systems Science and Complexity, 29 (2016), 22-40. doi: 10.1007/s11424-015-4063-7. [9] S. Fiori, Nonlinear damped oscillators on Riemannian manifolds: Numerical simulation, Communications in Nonlinear Science and Numerical Simulation, 47 (2017), 207-222. doi: 10.1016/j.cnsns.2016.11.025. [10] J. M. González Miranda, Synchronization and Control of Chaos, Imperial College Press, London, 2004. [11] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume Ⅰ, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. [12] J. M. Lee, Riemannian Manifolds, Vol. 176 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1997. doi: 10.1007/b98852. [13] T. E. Murphy, A. B. Cohen, B. Ravoori, K. R. B. Schmitt, A. V. Setty, F. Sorrentino, C. R. S. Williams, E. Ott and R. Roy, Complex dynamics and synchronization of delayed-feedback nonlinear oscillators, Philosophical Transactions of the Royal Society A, 368 (2010), 343-366. doi: 10.1098/rsta.2009.0225. [14] J. H. Park, Chaos synchronization of a chaotic system via nonlinear control, Chaos, Solitons and Fractals, 27 (2006), 1369-1375. doi: 10.1016/j.chaos.2005.05.001. [15] L. M. Pecora and T. L. Carroll, Synchronization of chaotic systems, Chaos: An Interdisciplinary Journal of Nonlinear Science, (1996), 142-145. doi: 10.1016/B978-012396840-1/50040-0. [16] X. Pennec, Barycentric subspaces and affine spans in manifolds, Geometric Science of Information, Lecture Notes in Comput. Sci., Springer, Cham, 9389 (2015), 12-21. doi: 10.1007/978-3-319-25040-3_2. [17] A. Sarlette and R. Sepulchre, Consensus optimization on manifolds, SIAM Journal on Control and Optimization, 48 (2009), 56-76. doi: 10.1137/060673400. [18] J.-P. Yeh and K.-L. Wu, A simple method to synchronize chaotic systems and its application to secure communications, Mathematical and Computer Modelling, 47 (2008), 894-902. doi: 10.1016/j.mcm.2007.06.021. [19] C. W. Wu, Synchronization in Complex Networks of Nonlinear Dynamical Systems, World Scientific Publishing Co. Pte. Ltd, 2007. doi: 10.1142/6570. [20] X. Wu, C. Xu and J. Feng, Complex projective synchronization in drive-response stochastic coupled networks with complex-variable systems and coupling time delays, Communications in Nonlinear Science and Numerical Simulation, 20 (2015), 1004-1014. doi: 10.1016/j.cnsns.2014.07.003.
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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