Stabilization of a steady state in oscillators coupled by a digital delayed connection

The present paper deals with stabilization of a steady state in a networked oscillator system that consists of oscillators coupled by a digital delayed connection. This connection is realized by a first-in, first-out queue. The semi-discretization technique allows us to obtain a simple characteristic equation for steady-state stability. This equation can be expressed by real polynomials whose coefficients depend on the network topology. The stability analysis based on the characteristic polynomials reveals that the digital delayed connection better facilitates the stabilization of the steady state compared to the well-known continuous-time delayed connection.

[1]  Akio Ushida,et al.  Spatio-temporal chaos in simple coupled chaotic circuits , 1995 .

[2]  Thilo Gross,et al.  Stability of networks of delay-coupled delay oscillators , 2011 .

[3]  Junzhong Yang,et al.  Transitions to amplitude death in a regular array of nonlinear oscillators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Keiji Konishi,et al.  Limitation of time-delay induced amplitude death , 2003 .

[5]  Wei Zhang,et al.  Stability of networked control systems , 2001 .

[6]  W. Zou,et al.  Eliminating delay-induced oscillation death by gradient coupling. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  D. V. Reddy,et al.  Time delay effects on coupled limit cycle oscillators at Hopf bifurcation , 1998, chao-dyn/9810023.

[8]  Fatihcan M. Atay,et al.  Stability of Coupled Map Networks with Delays , 2006, SIAM J. Appl. Dyn. Syst..

[9]  Fatihcan M. Atay,et al.  Total and partial amplitude death in networks of diffusively coupled oscillators , 2003 .

[10]  G. Stépán,et al.  Subcritical Hopf bifurcations in a car-following model with reaction-time delay , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  Pi,et al.  Experimental observation of the amplitude death effect in two coupled nonlinear oscillators , 2000, Physical review letters.

[12]  G. Ermentrout,et al.  Amplitude response of coupled oscillators , 1990 .

[13]  Y. Yamaguchi,et al.  Theory of self-synchronization in the presence of native frequency distribution and external noises , 1984 .

[14]  Keiji Konishi,et al.  Stability analysis and design of amplitude death induced by a time-varying delay connection , 2010 .

[15]  Gábor Stépán,et al.  Semi‐discretization method for delayed systems , 2002 .

[16]  Tanu Singla,et al.  Exploring the dynamics of conjugate coupled Chua circuits: simulations and experiments. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Awadhesh Prasad,et al.  Amplitude death in the absence of time delays in identical coupled oscillators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Patrick Celka,et al.  Experimental verification of pyragas's chaos control method applied to chua's circuit , 1994 .

[19]  Ramana Dodla,et al.  Phase-locked patterns and amplitude death in a ring of delay-coupled limit cycle oscillators. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Henning U. Voss,et al.  Real-Time Anticipation of Chaotic States of an Electronic Circuit , 2002, Int. J. Bifurc. Chaos.

[21]  Chang-Yuan Cheng.  Induction of Hopf bifurcation and oscillation death by delays in coupled networks , 2009 .

[22]  John Guckenheimer,et al.  The Dynamics of Legged Locomotion: Models, Analyses, and Challenges , 2006, SIAM Rev..

[23]  Kentaro Hirata,et al.  An Experimental Suppression of Spatial Instability in One-Way Open Coupled Chua's Circuits , 2002, Int. J. Bifurc. Chaos.

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

[25]  Masahiro Shimizu,et al.  A modular robot that exhibits amoebic locomotion , 2006, Robotics Auton. Syst..

[26]  Keiji Konishi,et al.  Topology-free stability of a steady state in network systems with dynamic connections. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  E Schöll,et al.  Control of unstable steady states by time-delayed feedback methods. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Keiji Konishi,et al.  Time-delay-induced stabilization of coupled discrete-time systems. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Sen,et al.  Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators , 1998, Physical review letters.

[30]  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.

[31]  Keiji Konishi,et al.  Amplitude death in time-delay nonlinear oscillators coupled by diffusive connections. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  Gábor Stépán,et al.  On the higher-order semi-discretizations for periodic delayed systems , 2008 .

[33]  Janos Turi,et al.  Delayed feedback of sampled higher derivatives , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[35]  Tetsuro Endo,et al.  Mode analysis of a multimode ladder oscillator , 1976 .

[36]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[37]  Henk Nijmeijer,et al.  Synchronization of delay-coupled nonlinear oscillators: an approach based on the stability analysis of synchronized equilibria. , 2009, Chaos.

[38]  Awadhesh Prasad,et al.  Nature of the phase-flip transition in the synchronized approach to amplitude death. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Yuan Yuan,et al.  Stability Switches and Hopf Bifurcations in a Pair of Delay-Coupled Oscillators , 2007, J. Nonlinear Sci..

[40]  Fatihcan M. Atay,et al.  Oscillator death in coupled functional differential equations near Hopf bifurcation , 2006 .

[41]  Gábor Stépán,et al.  Approximate stability charts for milling processes using semi-discretization , 2006, Appl. Math. Comput..

[42]  K. Konishi.  Amplitude death in oscillators coupled by a one-way ring time-delay connection. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Meng Zhan,et al.  Partial time-delay coupling enlarges death island of coupled oscillators. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  Abhijit Sen,et al.  Death island boundaries for delay-coupled oscillator chains , 2006 .

[45]  Anna Scaglione,et al.  A scalable synchronization protocol for large scale sensor networks and its applications , 2005, IEEE Journal on Selected Areas in Communications.

[46]  Keiji Konishi,et al.  Amplitude Death Induced by a Global Dynamic Coupling , 2007, Int. J. Bifurc. Chaos.

[47]  Keiji Konishi,et al.  Amplitude death induced by dynamic coupling. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.