Complex dynamics in delay-differential equations with large delay

Abstract. We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback.

[1]  S Yanchuk,et al.  Delay and periodicity. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Daniel J. Gauthier,et al.  Stabilizing unstable steady states using extended time-delay autosynchronization. , 1998, Chaos.

[3]  Gavrielides,et al.  Lang and Kobayashi phase equation. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[4]  Philipp Hövel,et al.  Stabilizing continuous-wave output in semiconductor lasers by time-delayed feedback. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Ulrich Parlitz,et al.  Stabilizing unstable steady states using multiple delay feedback control. , 2004, Physical review letters.

[6]  Hartmut Haug,et al.  Theory of laser diodes with weak optical feedback. II. Limit-cycle behavior, quasi-periodicity, frequency locking, and route to chaos , 1993 .

[7]  Giacomelli,et al.  Relationship between delayed and spatially extended dynamical systems. , 1996, Physical review letters.

[8]  J. Kurths,et al.  Analysis and control of complex nonlinear processes in physics, chemistry and biology , 2007 .

[9]  A. Gjurchinovski,et al.  Stabilization of unstable steady states by variable-delay feedback control , 2008, 0805.4064.

[10]  J. L. Hudson,et al.  Adaptive control of unknown unstable steady states of dynamical systems. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Michel Nizette.  Stability of square oscillations in a delayed-feedback system. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[14]  A. Gjurchinovski,et al.  Variable-delay feedback control of unstable steady states in retarded time-delayed systems. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Ulrich Parlitz,et al.  Controlling dynamical systems using multiple delay feedback control. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Ingo Fischer,et al.  Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario. , 2002, Physical review letters.

[17]  Philipp Hövel,et al.  Control of Complex Nonlinear Systems with Delay , 2010 .

[18]  R. Lang,et al.  External optical feedback effects on semiconductor injection laser properties , 1980 .

[19]  M. Rosenblum,et al.  Controlling synchronization in an ensemble of globally coupled oscillators. , 2004, Physical review letters.

[20]  Jesper Mørk,et al.  Chaos in semiconductor lasers with optical feedback: theory and experiment , 1992 .

[21]  T. Erneux.  Applied Delay Differential Equations , 2009 .

[22]  Hartmut Haug,et al.  Theory of laser diodes with weak optical feedback. I. Small-signal analysis and side-mode spectra , 1993 .

[23]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[24]  Matthias Wolfrum,et al.  Instabilities of lasers with moderately delayed optical feedback , 2002 .

[25]  S. A. Kashchenko.  Bifurcation singularities of a singularly perturbed equation with delay , 1999 .

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

[27]  C R Mirasso,et al.  Numerical statistics of power dropouts based on the Lang-Kobayashi model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[28]  Gauthier,et al.  Stabilizing unstable periodic orbits in fast dynamical systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  E Schöll,et al.  All-optical noninvasive control of unstable steady states in a semiconductor laser. , 2006, Physical review letters.

[30]  M Radziunas,et al.  Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Matthias Wolfrum,et al.  Eckhaus instability in systems with large delay. , 2006, Physical review letters.

[32]  Matthias Wolfrum,et al.  A Multiple Time Scale Approach to the Stability of External Cavity Modes in the Lang-Kobayashi System Using the Limit of Large Delay , 2010, SIAM J. Appl. Dyn. Syst..

[33]  H. Schuster,et al.  Nonlinear dynamics of nanosystems , 2010 .

[34]  Philipp Hövel,et al.  Control of unstable steady states by long delay feedback. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Glorieux,et al.  Stabilization and characterization of unstable steady states in a laser. , 1993, Physical review. A, Atomic, molecular, and optical physics.

[36]  Levine,et al.  Diode lasers with optical feedback: Stability of the maximum gain mode. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[37]  J. L. Hudson,et al.  Stabilizing and tracking unknown steady States of dynamical systems. , 2002, Physical review letters.

[38]  K. Blyuss,et al.  Asymptotic properties of the spectrum of neutral delay differential equations , 2009, 1201.5957.

[39]  Philipp Hövel,et al.  Control of unstable steady states by extended time-delayed feedback. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  L. Tuckerman,et al.  Bifurcation analysis of the Eckhaus instability , 1990 .