Amplitude death in networks of delay-coupled delay oscillators

Amplitude death is a dynamical phenomenon in which a network of oscillators settles to a stable state as a result of coupling. Here, we study amplitude death in a generalized model of delay-coupled delay oscillators. We derive analytical results for degree homogeneous networks which show that amplitude death is governed by certain eigenvalues of the network's adjacency matrix. In particular, these results demonstrate that in delay-coupled delay oscillators amplitude death can occur for arbitrarily large coupling strength k. In this limit, we find a region of amplitude death which already occurs at small coupling delays that scale with 1/k. We show numerically that these results remain valid in random networks with heterogeneous degree distribution.

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

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

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

[4]  S. Strogatz,et al.  Amplitude death in an array of limit-cycle oscillators , 1990 .

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

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

[7]  C. Masoller,et al.  Synchronizability of chaotic logistic maps in delayed complex networks , 2008, 0805.2420.

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

[9]  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.

[10]  Fatihcan M. Atay.  Complex Time-Delay Systems: Theory and Applications , 2010 .

[11]  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.

[12]  K. Bar-Eli,et al.  On the stability of coupled chemical oscillators , 1985 .

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

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

[15]  Fatihcan M. Atay,et al.  Complex Time-Delay Systems , 2010 .

[16]  D. V. Reddy,et al.  Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks , 1999, chao-dyn/9908005.

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

[18]  Thilo Gross,et al.  Generalized Models Reveal Stabilizing Factors in Food Webs , 2009, Science.

[19]  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.

[20]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

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

[22]  Awadhesh Prasad,et al.  Amplitude death in coupled chaotic oscillators. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Shuguang Guan,et al.  Stability of the steady state of delay-coupled chaotic maps on complex networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  P. Hövel,et al.  Control of unstable steady states in neutral time-delayed systems , 2008, 1201.5964.

[25]  J. Kurths,et al.  Synchronization of time-delayed systems. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[27]  Chuandong Li,et al.  Chaotic lag synchronization of coupled time-delayed systems and its applications in secure communication , 2004 .

[28]  G. Ermentrout.  Oscillator death in populations of “all to all” coupled nonlinear oscillators , 1990 .

[29]  Dirk Roose,et al.  Numerical stability analysis and computation of Hopf bifurcation points for delay differential equations , 1996 .

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

[31]  J. D. Farmer,et al.  Chaotic attractors of an infinite-dimensional dynamical system , 1982 .

[32]  K. Ikeda,et al.  High-dimensional chaotic behavior in systems with time-delayed feedback , 1987 .

[33]  M. Shiino,et al.  Synchronization of infinitely many coupled limit-cycle type oscillators , 1989 .

[34]  Y. Lai,et al.  Amplitude modulation in a pair of time-delay coupled external-cavity semiconductor lasers , 2003 .

[35]  Voss,et al.  Anticipating chaotic synchronization , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[36]  E. M. Shahverdiev,et al.  Generalized synchronization in time-delayed systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  I Fischer,et al.  Amplitude and phase effects on the synchronization of delay-coupled oscillators. , 2010, Chaos.

[38]  Thilo Gross,et al.  Engineering mesoscale structures with distinct dynamical implications , 2012, New Journal of Physics.

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

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

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

[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]  Kestutis Pyragas.  SYNCHRONIZATION OF COUPLED TIME-DELAY SYSTEMS : ANALYTICAL ESTIMATIONS , 1998 .

[44]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[45]  Thilo Gross,et al.  Generalized models as a universal approach to the analysis of nonlinear dynamical systems. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[47]  Meng Zhan,et al.  Complete synchronization and generalized synchronization of one-way coupled time-delay systems. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.