Dynamical effects of integrative time-delay coupling.

We study coupled dynamical systems wherein the influence of one system on the other is cumulative: coupling signals are integrated over a time interval τ. A major consequence of integrative coupling is that amplitude death occurs over a wider range and in a single region in parameter space. For coupled limit cycle oscillators (the Landau-Stuart model) we obtain an analytic estimate for the boundary of this region while for coupled chaotic Lorenz oscillators numerical results are presented. For given τ we find that there is a critical coupling strength at which the frequency of oscillations changes discontinuously.

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

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

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

[4]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[5]  Smith,et al.  Phase locking of relativistic magnetrons. , 1989, Physical review letters.

[6]  A. Winfree.  The geometry of biological time , 1991 .

[7]  Luke F. Lester,et al.  Frequency entrainment in optically injected semiconductor lasers , 1997 .

[8]  P. Bak,et al.  Self-organized criticality. , 1988, Physical review. A, General physics.

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

[10]  Kurt Wiesenfeld,et al.  Phase locking of Josephson junction arrays , 1988 .

[11]  Ramakrishna Ramaswamy,et al.  Phase-flip bifurcation induced by time delay. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Hadley,et al.  Phase locking of Josephson-junction series arrays. , 1988, Physical review. B, Condensed matter.

[13]  I. Schreiber,et al.  Strange attractors in coupled reaction-diffusion cells , 1982 .

[14]  Gabriel B. Mindlin,et al.  LOW-FREQUENCY FLUCTUATIONS IN SEMICONDUCTOR LASERS WITH OPTICAL FEEDBACK ARE INDUCED WITH NOISE , 1998 .

[15]  Alex Zettl,et al.  Charge density wave conduction: A novel collective transport phenomenon in solids , 1985 .

[16]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[17]  Michael F. Crowley,et al.  Experimental and theoretical studies of a coupled chemical oscillator: phase death, multistability, and in-phase and out-of-phase entrainment , 1989 .

[18]  Awadhesh Prasad,et al.  Complicated basins in external-cavity semiconductor lasers , 2003 .

[19]  Ramakrishna Ramaswamy,et al.  Universal occurrence of the phase-flip bifurcation in time-delay coupled systems. , 2008, Chaos.

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

[21]  Thomas Erneux,et al.  LOCALIZED SYNCHRONIZATION IN TWO COUPLED NONIDENTICAL SEMICONDUCTOR LASERS , 1997 .

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

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

[24]  R. Macarthur,et al.  Graphical Representation and Stability Conditions of Predator-Prey Interactions , 1963, The American Naturalist.

[25]  Steven H. Strogatz,et al.  Nonlinear dynamics: Death by delay , 1998, Nature.

[26]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

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

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

[29]  Bruce W. Knight,et al.  Dynamics of Encoding in a Population of Neurons , 1972, The Journal of general physiology.

[30]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[31]  Awadhesh Prasad,et al.  Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators , 2005 .

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

[33]  Delisle,et al.  Second-order differential-delay equation to describe a hybrid bistable device. , 1987, Physical review. A, General physics.