Synchronization in networks with heterogeneous coupling delays.

Synchronization in networks of identical oscillators with heterogeneous coupling delays is studied. A decomposition of the network dynamics is obtained by block diagonalizing a newly introduced adjacency lag operator which contains the topology of the network as well as the corresponding coupling delays. This generalizes the master stability function approach, which was developed for homogenous delays. As a result the network dynamics can be analyzed by delay differential equations with distributed delay, where different delay distributions emerge for different network modes. Frequency domain methods are used for the stability analysis of synchronized equilibria and synchronized periodic orbits. As an example, the synchronization behavior in a system of delay-coupled Hodgkin-Huxley neurons is investigated. It is shown that the parameter regions where synchronized periodic spiking is unstable expand when increasing the delay heterogeneity.

[1]  G. Stépán.  Retarded dynamical systems : stability and characteristic functions , 1989 .

[2]  L. Pecora.  Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems , 1998 .

[3]  C. Mirasso,et al.  Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers. , 2001, Physical review letters.

[4]  R. E. Amritkar,et al.  General mechanism for amplitude death in coupled systems. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Eckehard Schöll,et al.  Some basic remarks on eigenmode expansions of time-delay dynamics , 2007 .

[6]  Gábor Stépán,et al.  Improved prediction of stability lobes with extended multi frequency solution , 2013 .

[7]  Philipp Hövel,et al.  Time-delayed feedback in neurosystems , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  I Kanter,et al.  Synchronization of chaotic networks with time-delayed couplings: an analytic study. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Gábor Stépán,et al.  Stability chart for the delayed Mathieu equation , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  Wolfgang Kinzel,et al.  Synchronisation and scaling properties of chaotic networks with multiple delays , 2013 .

[11]  Valentin Flunkert,et al.  Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases , 2012 .

[12]  R. Bellman,et al.  Differential-Difference Equations , 1967 .

[13]  W. Marsden.  I and J , 2012 .

[14]  L. Glass.  Synchronization and rhythmic processes in physiology , 2001, Nature.

[15]  D. V. Reddy,et al.  Experimental Evidence of Time Delay Induced Death in Coupled Limit Cycle Oscillators , 2000 .

[16]  A. Pluchino,et al.  CHANGING OPINIONS IN A CHANGING WORLD: A NEW PERSPECTIVE IN SOCIOPHYSICS , 2004 .

[17]  G. M. Schoen,et al.  Stability and stabilization of time-delay systems , 1995 .

[18]  D. Bachrathy,et al.  Bisection method in higher dimensions and the efficiency number , 2012 .

[19]  G. Hill.  On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon , 1886 .

[20]  A. Gjurchinovski,et al.  Amplitude death in oscillator networks with variable-delay coupling. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Andrew G. Glen,et al.  APPL , 2001 .

[22]  Guanrong Chen,et al.  Complexity and synchronization of the World trade Web , 2003 .

[23]  A. Pikovsky,et al.  Synchronization: Theory and Application , 2003 .

[24]  Maurizio Porfiri,et al.  Analysis of parameter mismatches in the master stability function for network synchronization , 2011, 1102.1709.

[25]  L. Shampine.  Solving ODEs and DDEs with residual control , 2005 .

[26]  C Masoller,et al.  Random delays and the synchronization of chaotic maps. , 2005, Physical review letters.

[27]  Andreas Otto.  Frequency domain methods for the analysis of time delay systems , 2016 .

[28]  Jürgen Jost,et al.  Delays, connection topology, and synchronization of coupled chaotic maps. , 2004, Physical review letters.

[29]  Keiji Konishi,et al.  Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Philipp Hövel,et al.  Synchronization of Coupled Neural oscillators with Heterogeneous delays , 2012, Int. J. Bifurc. Chaos.

[31]  Eckehard Schöll,et al.  Cluster and group synchronization in delay-coupled networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Jianhong Wu,et al.  Introduction to Functional Differential Equations , 2013 .

[33]  Francesco Sorrentino,et al.  Cluster synchronization and isolated desynchronization in complex networks with symmetries , 2013, Nature Communications.

[34]  I Kanter,et al.  Zero lag synchronization of chaotic systems with time delayed couplings. , 2010, Physical review letters.

[35]  E. Schöll,et al.  Heterogeneous delays in neural networks , 2013, 1311.1919.

[36]  Gábor Orosz,et al.  Decomposing the dynamics of heterogeneous delayed networks with applications to connected vehicle systems. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Steffen Ihlenfeldt,et al.  Stability of milling with non-uniform pitch and variable helix Tools , 2017 .

[38]  S. Gurevich,et al.  Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Zoltan Dombovari,et al.  The effect of serration on mechanics and stability of milling cutters , 2010 .

[40]  S V Gurevich,et al.  Instabilities of localized structures in dissipative systems with delayed feedback. , 2013, Physical review letters.

[41]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[42]  Chyi Hwang,et al.  A note on the use of the Lambert W function in the stability analysis of time-delay systems , 2005, Autom..

[43]  Wolfgang Kinzel,et al.  Zero-lag synchronization and multiple time delays in two coupled chaotic systems. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Joachim Peinke,et al.  Self-organized synchronization and voltage stability in networks of synchronous machines , 2013, ArXiv.

[45]  Sue Omel,et al.  PERIOD , 1996, SIGGRAPH Visual Proceedings.

[46]  Gábor Orosz,et al.  Decomposition of Nonlinear Delayed Networks around Cluster States with Applications to Neurodynamics , 2014, SIAM J. Appl. Dyn. Syst..

[47]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[48]  Tobias Damm,et al.  The Lambert W function and the spectrum of some multidimensional time-delay systems , 2007, Autom..

[49]  Zoltan Dombovari,et al.  The Effect of Helix Angle Variation on Milling Stability , 2012 .

[50]  T. Carroll,et al.  Master Stability Functions for Synchronized Coupled Systems , 1998 .

[51]  Fatihcan M Atay,et al.  The consensus problem in networks with transmission delays , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[52]  S. Strogatz,et al.  Phase diagram for the collective behavior of limit-cycle oscillators. , 1990, Physical review letters.

[53]  Zoltan Dombovari,et al.  Chatter stability of milling in frequency and discrete time domain , 2008 .

[54]  S Yanchuk,et al.  Synchronizing distant nodes: a universal classification of networks. , 2010, Physical review letters.

[55]  F. Atay.  Distributed delays facilitate amplitude death of coupled oscillators. , 2003, Physical review letters.

[56]  Erik M. Bollt,et al.  Master stability functions for coupled nearly identical dynamical systems , 2008, 0811.0649.

[57]  Jianfeng Feng,et al.  Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  Marc Timme,et al.  Self-organized synchronization in decentralized power grids. , 2012, Physical review letters.