The Kuramoto Model on Oriented and Signed Graphs

Many real-world systems of coupled agents exhibit directed interactions, meaning that the influence of an agent on another is not reciprocal. Furthermore, interactions usually do not have identical amplitude and/or sign. To describe synchronization phenomena in such systems, we use a generalized Kuramoto model with oriented, weighted and signed interactions. Taking a bottom-up approach, we investigate the simplest possible oriented networks, namely acyclic oriented networks and oriented cycles. These two types of networks are fundamental building blocks from which many general oriented networks can be constructed. For acyclic, weighted and signed networks, we are able to completely characterize synchronization properties through necessary and sufficient conditions, which we show are optimal. Additionally, we prove that if it exists, a stable synchronous state is unique. In oriented, weighted and signed cycles with identical natural frequencies, we show that the system globally synchronizes and that the number of stable synchronous states is finite.

[1]  B. Ermentrout,et al.  An adaptive model for synchrony in the firefly Pteroptyx malaccae , 1991 .

[2]  Hyunsuk Hong,et al.  Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. , 2011, Physical review letters.

[3]  Noël Janssens,et al.  Loop flows in a ring AC power system , 2003 .

[4]  Hayato Chiba,et al.  Bifurcations in the Kuramoto model on graphs. , 2018, Chaos.

[5]  Wiesenfeld,et al.  Synchronization transitions in a disordered Josephson series array. , 1996, Physical review letters.

[6]  W. Rudin.  Principles of mathematical analysis , 1964 .

[7]  Tommaso Coletta,et al.  Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks , 2015, 1512.04266.

[8]  Seung-Yeal Ha,et al.  Complete synchronization of Kuramoto oscillators with hierarchical leadership , 2014 .

[9]  Florian Dörfler,et al.  Synchronization in complex networks of phase oscillators: A survey , 2014, Autom..

[10]  Hal L. Smith,et al.  Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions , 1995 .

[11]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[12]  Marc Timme,et al.  Cycle flows and multistability in oscillatory networks. , 2016, Chaos.

[13]  Michael Chertkov,et al.  Synchronization in complex oscillator networks and smart grids , 2012, Proceedings of the National Academy of Sciences.

[14]  Solomon Lefschetz,et al.  Stability by Liapunov's Direct Method With Applications , 1962 .

[15]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion With Limited Communication , 2008, IEEE Transactions on Automatic Control.

[16]  Pablo Monzon,et al.  Local implications of almost global stability , 2009 .

[17]  Yoshiki Kuramoto,et al.  Self-entrainment of a population of coupled non-linear oscillators , 1975 .

[18]  Seung-Yeal Ha,et al.  On the Basin of Attractors for the Unidirectionally Coupled Kuramoto Model in a Ring , 2012, SIAM J. Appl. Math..

[19]  Edward Ott,et al.  Synchronization in large directed networks of coupled phase oscillators. , 2005, Chaos.

[20]  Nancy Kopell,et al.  Synchronization in Networks of Excitatory and Inhibitory Neurons with Sparse, Random Connectivity , 2003, Neural Computation.

[21]  Elena Panteley,et al.  Asymptotic phase synchronization of Kuramoto model with weighted non-symmetric interconnections: A case study , 2013, 52nd IEEE Conference on Decision and Control.

[22]  Michelle Girvan,et al.  Resynchronization of circadian oscillators and the east-west asymmetry of jet-lag. , 2016, Chaos.

[23]  G. Ermentrout.  The behavior of rings of coupled oscillators , 1985, Journal of mathematical biology.

[24]  S. Strogatz,et al.  Frequency locking in Josephson arrays: Connection with the Kuramoto model , 1998 .

[25]  O Mason,et al.  Graph theory and networks in Biology. , 2006, IET systems biology.

[26]  J. Buck.  Synchronous Rhythmic Flashing of Fireflies. II. , 1938, The Quarterly Review of Biology.

[27]  D. Aeyels,et al.  Stability of phase locking in a ring of unidirectionally coupled oscillators , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[28]  L. G. Khazin,et al.  Stability of Critical Equilibrium States , 1991 .

[29]  P. Olver.  Nonlinear Systems , 2013 .

[30]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[31]  Ao Tang,et al.  Synchronization of weakly coupled oscillators: coupling, delay and topology , 2013, 1303.7248.

[32]  R. Spigler,et al.  The Kuramoto model: A simple paradigm for synchronization phenomena , 2005 .

[33]  Oleksandr Burylko.  Competition and Bifurcations in Phase Oscillator Networks with Positive and Negative Couplings , 2012 .

[34]  E. Ott,et al.  Onset of synchronization in large networks of coupled oscillators. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  J. L. Hemmen,et al.  Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators , 1993 .

[36]  Elena Panteley,et al.  Synchronization of phase oscillators with attractive and repulsive interconnections , 2013, 2013 18th International Conference on Methods & Models in Automation & Robotics (MMAR).

[37]  Mauricio Barahona,et al.  Graph partitions and cluster synchronization in networks of oscillators , 2016, Chaos.

[38]  J. Buck.  Synchronous Rhythmic Flashing of Fireflies , 1938, The Quarterly Review of Biology.

[39]  Dane Taylor,et al.  Optimal synchronization of directed complex networks. , 2016, Chaos.

[40]  R. Sepulchre,et al.  Oscillator Models and Collective Motion , 2007, IEEE Control Systems.