Reconstruction of time-delayed feedback systems from time series

For various classes of time-delay systems we propose the methods of their model delay-differential equation reconstruction from time series. The methods are based on the characteristic location of extrema in the time series of time-delay systems and the projection of infinite-dimensional phase space of these systems to suitably chosen low-dimensional subspaces. We verify our methods by using them for the recovery of time-delay differential equations from their chaotic solutions and for modelling experimental systems with delay-induced dynamics from their chaotic time series.

[1]  Leon Glass,et al.  Coarse-grained embeddings of time series: random walks, Gaussian random processes, and deterministic chaos , 1993 .

[2]  Uwe Rau,et al.  Recovery of scaler time-delay systems from time series , 1996 .

[3]  Holger Kantz,et al.  Identifying and Modeling Delay Feedback Systems. , 1998, chao-dyn/9907019.

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

[5]  Laurent Larger,et al.  Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations , 2003 .

[6]  C Zhou,et al.  Extracting messages masked by chaotic signals of time-delay systems. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[7]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[8]  H. Voss,et al.  Reconstruction of non-linear time delay models from data by the use of optimal transformations , 1997 .

[9]  Fathalla A. Rihan,et al.  Numerical modelling in biosciences using delay differential equations , 2000 .

[10]  Stephen P. Ellner,et al.  Inferring mechanism from time-series data: delay-differential equations , 1997 .

[11]  K. Ikeda.  Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system , 1979 .

[12]  André Longtin,et al.  Synchronization of delay-differential equations with application to private communication , 1998 .

[13]  V I Ponomarenko,et al.  Extracting information masked by the chaotic signal of a time-delay system. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[16]  B. Silverman,et al.  Nonparametric regression and generalized linear models , 1994 .

[17]  H. Kantz,et al.  Reconstruction of systems with delayed feedback: I. Theory , 2000 .

[18]  Furong Gao,et al.  Extraction of delay information from chaotic time series based on information entropy , 1997 .

[19]  M D Prokhorov,et al.  Reconstruction of time-delay systems from chaotic time series. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Parisi,et al.  Tool to recover scalar time-delay systems from experimental time series. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  A. Fowler,et al.  Delay recognition in chaotic time series , 1993 .

[22]  Th. Meyer,et al.  Recovery of the time-evolution equation of time-delay systems from time series , 1997, chao-dyn/9907009.

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