On synchronization of coupled Hopf-Kuramoto oscillators with phase delays

This paper presents new methods and results on synchronization of coupled Hopf nonlinear oscillators, which are commonly used as the dynamic model of engineered central pattern generators (CPGs). On balanced graphs, any positive coupling gain is proven to induce almost global asymptotic synchronization, and a threshold value for truly global exponential synchronization is also computed. Furthermore, a hierarchical connection between coupled Hopf oscillators and Kuramoto oscillators is identified. Finally, a new result on the synchronization of Kuramoto oscillators with arbitrary time-varying heterogeneous frequencies and delays is derived.

[1]  Soon-Jo Chung,et al.  CPG-based control of a turtle-like underwater vehicle , 2008, Auton. Robots.

[2]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion: All-to-All Communication , 2007, IEEE Transactions on Automatic Control.

[3]  Winfried Stefan Lohmiller,et al.  Contraction analysis of nonlinear systems , 1999 .

[4]  Florian Dörfler,et al.  Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators , 2009, Proceedings of the 2010 American Control Conference.

[5]  Tetsuya Iwasaki,et al.  Circulant Synthesis of Central Pattern Generators With Application to Control of Rectifier Systems , 2008, IEEE Transactions on Automatic Control.

[6]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

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

[8]  Nicolas Tabareau,et al.  A Contraction Theory Approach to Stochastic Incremental Stability , 2007, IEEE Transactions on Automatic Control.

[9]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[10]  Jean-Jacques E. Slotine,et al.  Stable concurrent synchronization in dynamic system networks , 2005, Neural Networks.

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

[12]  Mark W. Spong,et al.  On Exponential Synchronization of Kuramoto Oscillators , 2009, IEEE Transactions on Automatic Control.

[13]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[14]  Soon-Jo Chung,et al.  Neurobiologically Inspired Control of Engineered Flapping Flight , 2009, 0905.2789.

[15]  Auke Jan Ijspeert,et al.  Simulation and Robotics Studies of Salamander Locomotion Applying Neurobiological Principles to the Control of Locomotion in Robots , 2005 .

[16]  A. Jadbabaie,et al.  On the stability of the Kuramoto model of coupled nonlinear oscillators , 2005, Proceedings of the 2004 American Control Conference.

[17]  A. Papachristodoulou,et al.  Synchonization in Oscillator Networks with Heterogeneous Delays, Switching Topologies and Nonlinear Dynamics , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.