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Prokhorov M. D., Ponomarenko V. I. Reconstruction of ensembles of coupled time-delay systems from time series. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 5, pp. 3-16. DOI:


Reconstruction of ensembles of coupled time-delay systems from time series

Prokhorov Mihail Dmitrievich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

The methods for the reconstruction of model delay-differential equations for ensembles of coupled time-delay systems from their time series are proposed. The methods efficiency is illustrated using chaotic and periodic time series from chains of diffusively coupled model and experimental time-delay systems for the cases of unidirectional andmutual coupling.


1. Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston: Academic Press, 1993. 2. Glass L., Mackey M.C. From Clocks to Chaos: The Rhythms of Life. Princeton: Princeton University Press, 1988. 3. Mokhov I.I., Smirnov D.A. El Nino Southern Oscillation drives North Atlantic Oscillation as revealed with nonlinear techniques from climatic indices // Geophys. Res. Lett. 2006. Vol. 33. L03708. 4. Ikeda K. Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system // Opt. Commun. 1979. Vol. 30. P. 257. 5. Lang R., Kobayashi K. External optical feedback effects on semiconductor injection lasers // IEEE J. Quantum Electron. 1980. Vol. 16. P. 347. 6. Peil M., Jacquot M., Chembo Y.K., Larger L., Erneux T. Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic oscillators // Phys. Rev. E. 2009. Vol. 79. 026208. 7. Hale J.K., Lunel S.M.V. Introduction to Functional Differential Equations. New York: Springer, 1993. 8. Рубаник В.П. Колебания квазилинейных систем с запаздыванием. М.: Наука, 1969. 9. Fowler A.C., Kember G. Delay recognition in chaotic time series // Phys. Lett. A. 1993. Vol. 175. P. 402. 10. Hegger R., Bunner M.J., Kantz H., Giaquinta A.  ? Identifying and modeling delay feedback systems // Phys. Rev. Lett. 1998. Vol. 81. P. 558. 11. Udaltsov V.S., Goedgebuer J.-P., Larger L., Cuenot J.-B., Levy P., Rhodes W.T. Cracking chaos-based encryption systems ruled by nonlinear time delay differential equations // Phys. Lett. A. 2003. Vol. 308. P. 54. 12. Tian Y.-C., Gao F. Extraction of delay information from chaotic time series based on information entropy // Physica D. 1997. Vol. 108. P. 113. 13. Kaplan D.T., Glass L. Coarse-grained embeddings of time series: Random walks, gaussian random process, and deterministic chaos // Physica D. 1993. Vol. 64. P. 431. 14. Bunner M.J., Popp M., Meyer Th., Kittel A., Rau U., Parisi J.  ? Recovery of scalar time-delay systems from time series // Phys. Lett. A. 1996. Vol. 211. P. 345. 15. Bunner M.J., Ciofini M., Giaquinta A., Hegger R., Kantz H., Meucci R., Politi A.  ? Reconstruction of systems with delayed feedback: (I) Theory // Eur. Phys. J. D. 2000. Vol. 10. P. 165. 16. Voss H., Kurths J. Reconstruction of non-linear time delay models from data by the use of optimal transformations // Phys. Lett. A. 1997. Vol. 234. P. 336. 17. Ellner S.P., Kendall B.E., Wood S.N., McCauley E., Briggs C.J. Inferring mechanism from time-series data: Delay differential equations // Physica D. 1997. Vol. 110. P. 182. 18. Voss H.U., Schwache A., Kurths J., Mitschke F. Equations of motion from chaotic data: A driven optical fiber ring resonator // Phys. Lett. A. 1999. Vol. 256. P. 47. 19. Horbelt W., Timmer J., Voss H.U. Parameter estimation in nonlinear delayed feedback systems from noisy data // Phys. Lett. A. 2002. Vol. 299. P. 513. 20. Ort ?in S., Gutierrez J.M., Pesquera L., Vasquez H.  ? Nonlinear dynamics extraction for time-delay systems using modular neural networks synchronization and prediction // Physica A. 2005. Vol. 351. P. 133. 21. Bezruchko B.P., Karavaev A.S., Ponomarenko V.I., Prokhorov M.D. Reconstruction of time-delay systems from chaotic time series // Phys. Rev. E. 2001. Vol. 64. 056216. 22. Prokhorov M.D., Ponomarenko V.I., Karavaev A.S., Bezruchko B.P. Reconstruction of time-delayed feedback systems from time series // Physica D. 2005. Vol. 203. P. 209. 23. Mensour B., Longtin A. Synchronization of delay-differential equations with application to private communication // Phys. Lett. A. 1998. Vol. 244. P. 59. 24. Shahverdiev E.M., Sivaprakasam S., Shore K.A. Parameter mismatches and perfect anticipating synchronization in bidirectionally coupled external cavity laser diodes // Phys. Rev. E. 2002. Vol. 66. 017206. 25. Bocharov G.A., Rihan F.A. Numerical modelling in biosciences using delay differential equations // J. Comp. Appl. Math. 2000. Vol. 125. P. 183. 26. Kotani K., Takamasu K., Ashkenazy Y., Stanley H.E., Yamamoto Y. Model for cardio-respiratory synchronization in humans // Phys. Rev. E. 2002. Vol. 65. 051923. 27. Yanchuk S., Perlikowski P. Delay and periodicity // Phys. Rev. E. 2009. Vol. 79. 046221. 28. Prokhorov M.D., Ponomarenko V.I. Estimation of coupling between time-delay systems from time series // Phys. Rev. E. 2005. Vol. 72. 016210. 29. Prokhorov M.D., Ponomarenko V.I. Reconstruction of time-delay systems using small impulsive disturbances // Phys. Rev. E. 2009. Vol. 80. 066206. 30. Pyragas K. Synchronization of coupled time-delay systems: Analytical estimations // Phys. Rev. E. 1998. Vol. 58. P. 3067. 31. Buric N., Vasovi  ? c N.  ? Global stability of synchronization between delay-differential systems with generalized diffusive coupling // Chaos, Solitons and Fractals. 2007. Vol. 31. P. 336. 32. Ikeda K., Matsumoto K. High-dimensional chaotic behavior in systems with time-delayed feedback // Physica D. 1987. Vol. 29. P. 223.

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