ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Sysoev I. V., Ponomarenko V. I., Prokhorov M. D. Reconstruction of model equations of networks of oscillators with delay in node dynamics and couplings between them: Review. Izvestiya VUZ. Applied Nonlinear Dynamics, 2019, vol. 27, iss. 4, pp. 13-51. DOI: 10.18500/0869-6632-2019-27-4-13-51

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Language: 
Russian
Heading: 
Review of Actual Problems of Nonlinear Dynamics
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-2019-27-4-13-51

Reconstruction of model equations of networks of oscillators with delay in node dynamics and couplings between them: Review

Autors: 
Sysoev Ilya Vyacheslavovich, Saratov State University
Ponomarenko Vladimir Ivanovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Prokhorov Mihail Dmitrievich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Abstract: 

The aim of this review is to show the modern level of research in the area of reconstruction of network models from measured time series, for which individual nodes are described by time-delayed equations or there is a delay in coupling. Methods are described for reconstruction of coupling coefficients and functions, nonlinear functions of individual nodes and for detection of superfluous couplings. The techniques for delay time detection are considered separately due to their choice is crucial for success of entire reconstruction procedure. There presented the results of reconstruction from times series of model oscillators with different nonlinear functions, coupling fucntions, with number of nodes in a netwoks ranging widely (from 3 to tens of nodes). In addition, the results of reconstruction of models from different radiophysical experiments are presented. The advantages and shortcomings of proposed approaches are discussed in comparison with other known from literature methods of coupling estimation. The effects of time series length, the amount of a priori information, measurement noise, calculation errors on method efficiency are considered. 

Key words: 
time-delayed equations
networks of coupled oscillators
reconstruction from time series
collective dynamics
Reference: 
  1. Heiligenthal S., J¨ungling T., D’Huys O., Arroyo-Almanza D.A., Soriano M.C., Fischer I., Kanter I., Kinzel W. Strong and weak chaos in networks of semiconductor lasers with time-delayed couplings // Phys. Rev. E. 2013. Vol. 88, no. 1. P. 012902. 
  2. Buri´c N., Vasovi´c N. Global stability of synchronization between delay-differential systems with generalized diffusive coupling // Chaos, Solitons & Fractals. 2007. Vol. 31, no. 2. P. 336–342.
  3. Bindu M. Krishna and Manu P. John and Nandakumaran V.M. Multi-user bidirectional communication using isochronal synchronisation of array of chaotic directly modulated semiconductor lasers // Physics Letters A. 2010. Vol. 374, no. 17. P. 1835–1842. 
  4. Mincheva M., Roussel M.R. Graph-theoretic methods for the analysis of chemical and biochemical networks. II. Oscillations in networks with delays // Journal of Mathematical Biology. 2007. Vol. 55, no. 1. P. 87–104. 
  5. Kuang Y. Delay Differential Equations with Applications in Population Dynamics. Boston: Academic Press, 1993. 398 p. 
  6. Bocharov G.A., Fathalla A.R. Numerical modelling in biosciences using delay differential equations // Journal of Computational and Applied Mathematics. 2000. Vol. 125, no. 1. P. 183–199. 
  7. Orosz G., Moehlis J., Murray R.M. Controlling biological networks by time-delayed signals // Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 2010. Vol. 368, no. 1911. P. 439–454.
  8. Glyzin S.D., Marushkina E.A. Complicated Dynamic Regimes in a Neural Network of Three Oscillators with a Delayed Broadcast Connection // Automatic Control and Computer Sciences. 2018. Vol. 52, no. 7. P. 885-893. 
  9. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh. Self-Sustained Relaxation Oscillations in Time-Delay Neural Systems // Journal of Physics: Conference Series. 2016. Vol. 727, no. 1. P. 012004. 
  10. Glyzin S., Goryunov V., Kolesov A. Spatially inhomogeneous modes of logistic differential equation with delay and small diffusion in a flat area // Lobachevskii Journal of Mathematics. 2017. Vol. 38, no. 5. P. 898–905. 
  11. Packard N., Crutchfield J., Farmer J., Shaw R. Geometry from a Time Series // Phys. Rev. Lett. 1980. Vol. 45. P. 712–716. 
  12. Fowler A.C., Kember G. Delay recognition in chaotic time series // Physics Letters A. 1993. Vol. 175, no. 6. P. 402–408. 
  13. Hegger R., B¨unner M.J., Kantz H., Giaquinta A. Identifying and Modeling Delay Feedback Systems // Phys. Rev. Lett. 1998. Vol. 81, no. 3. P. 558–561. 
  14. Bunner M.J., Ciofini M., Giaquinta A., Hegger R., Kantz H., Meucci R., Politi A. Reconstruction of systems with delayed feedback: I. Theory // The European Physical Journal D. 2000. Vol. 10, no. 2. P. 165–176. 
  15. Yu-Chu Tian, Furong Gao. Extraction of delay information from chaotic time series based on information entropy // Physica D: Nonlinear Phenomena. 1997. Vol. 108, no. 1. P. 113–118. 
  16. Bunner M.J., Meyer Th., Kittel A., Parisi J. Recovery of the time-evolution equation of time-delay systems from time series // Phys. Rev. E. 1997. Vol. 56, no. 5. P. 5083–5089. 
  17. Voss H., Kurths J. Reconstruction of non-linear time delay models from data by the use of optimal transformations // Physics Letters A. 1997. Vol. 234, no. 5. P. 336–344. 
  18. Ellner S.P., Kendall B.E., Wood S.N., McCauley E., Briggs Ch.J. Inferring mechanism from timeseries data: Delay-differential equations // Physica D: Nonlinear Phenomena. 1997. Vol. 110, no. 3. P. 182–194. 
  19. Prokhorov M.D., Ponomarenko V.I., Karavaev A.S., Bezruchko B.P. Reconstruction of time-delayed feedback systems from time series // Physica D: Nonlinear Phenomena. 2005. Vol. 203, no. 3. P. 209–223. 
  20. Prokhorov M.D., Ponomarenko V.I., Khorev V.S. Recovery of delay time from time series based on the nearest neighbor method // Physics Letters A. 2013. Vol. 377, no. 43. P. 3106–3111. 
  21. Udaltsov V.S., Larger L., Goedgebuer J.P., Locquet A., Citrin D.S. Time delay identification in chaotic cryptosystems ruled by delay-differential equations // J. Opt. Technol. 2005, no. 5. P. 373–377. 
  22. Zunino L., Soriano M.C., Fischer I., Rosso O.A., Mirasso C.R. Permutation-information-theory approach to unveil delay dynamics from time-series analysis // Phys. Rev. E. 2010. Vol. 82, no. 4. P. 046212. 
  23. Horbelt W., Timmer J., Voss H.U. Parameter estimation in nonlinear delayed feedback systems from noisy data // Physics Letters A. 2002. Vol. 299, no. 5. P. 513–521.
  24. Dai Chaohua, Chen Weirong, Li Lixiang, Zhu Yunfang, Yang Yixian. Seeker optimization algorithm for parameter estimation of time-delay chaotic systems // Phys. Rev. E. 2011. Vol. 83, no. 3. P. 036203.
  25. Sorrentino F. Identification of delays and discontinuity points of unknown systems by using synchronization of chaos // Phys. Rev. E. 2010. Vol. 81, no. 6. P. 066218. 
  26. Ma Huanfei, Xu Bing, Lin Wei, Feng Jianfeng. Adaptive identification of time delays in nonlinear dynamical models // Phys. Rev. E. 2010. Vol. 82, no. 6. P. 066210. 
  27. Siefert M. Practical criterion for delay estimation using random perturbations // Phys. Rev. E. 2007. Vol. 76, no. 2. P. 026215. 
  28. Yu Dongchuan, Frasca Mattia, Liu Fang. Control-based method to identify underlying delays of a nonlinear dynamical system // Phys. Rev. E. 2008. Vol. 78, no. 4. P. 046209. 
  29. Ponomarenko V.I., Prokhorov M.D. Recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes // Phys. Rev. E. 2008. Vol. 78, no. 6. P. 066207. 
  30. Prokhorov M.D., Ponomarenko V.I. Reconstruction of time-delay systems using small impulsive disturbances // Phys. Rev. E. 2009. Vol. 80, no. 6. P. 066206. 
  31. Prokhorov M.D., Ponomarenko V.I. Estimation of coupling between time-delay systems from time series // Phys. Rev. E. 2005. Vol. 72. P. 016210. 
  32. Afraimovich V.S., Nekorkin V.I., Osipov G.V., Shalfeev V.D. Stability, Structures and Chaos in Nonlinear Synchronization Networks. WORLD SCIENTIFIC, 1995.
  33. Pikovsky A., Rosenblum M. and Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. London: Cambridge University Press, 2001.
  34. Xiaoming Wu, Zhiyong Sun, Feng Liang, Changbin Yu. Online estimation of unknown delays and parameters in uncertain time delayed dynamical complex networks via adaptive observer // Nonlinear Dynamics. 2013. Vol. 73, no. 3. P. 1753–1768. 
  35. Wang W.X., Yang R., Lai Y.C., Kovanis V., Grebogi C. Predicting catastrophes in nonlinear dynamical systems by compressive sensing // Phys. Rev. Lett. 2011. Vol. 106. P. 154101. 
  36. Han X., Shen Z., Wang W.-X., Di Z. Robust Reconstruction of Complex Networks from Sparse Data // Phys. Rev. Lett. 2015. Vol. 114. P. 28701. 
  37. Brunton S.L., Proctor J.L., Kutz J.N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems // Proc. Natl. Acad. Sci. U.S.A. 2016. Vol. 113. P. 3932-7. 
  38. Mangan N.M., Brunton S.L., Proctor J.L., Kutz J.N. Inferring biological networks by sparse identification of nonlinear dynamics// IEEE Trans. Mol. Biol. Multi-Scale Commun. 2016. Vol. 2. P. 52–63. 
  39. Jose Casadiego, Mor Nitzan, Sarah Hallerberg, Marc Timme. Model-free inference of direct network interactions from nonlinear collective dynamics // Nature Communications. 2017. Vol. 8. P. 2192. 
  40. Sysoev I.V., Ponomarenko V.I., Prokhorov M.D., Bezruchko B.P. Reconstruction of ensembles of coupled time-delay system from time series // Phys. Rev. E. 2014. Vol. 89. P. 062911. 
  41. Sysoev I.V., Ponomarenko V.I., Kulminsky D.D., Prokhorov M.D. Recovery of couplings and parameters of elements in networks of time-delay systems from time series//Phys. Rev. E. 2016. Vol. 94. P. 052207. 
  42. Sysoev Ilya V., Ponomarenko Vladimir I. Reconstruction of the coupling matrix in the ensemble of identical neuron-like oscillators with time delay in coupling. Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 567–576.
  43. Sysoev I.V., Ponomarenko V.I., Pikovsky A. Reconstruction of coupling architecture of neural field networks from vector time series // Commun. Nonlinear Sci. Numer. Simulat. 2018. Vol. 57. P. 342–351. 
  44. Sysoev I.V., Ponomarenko V.I., Prokhorov M.D. Reconstruction of Ensembles of Oscillators with Nonlinear Time-Delay Feedbacks. Technical Physics Letters, 2018, vol. 44, no. 11, pp. 1024–1027.
  45. Sysoev I.V., Ponomarenko V.I., Prokhorov M.D. Reconstruction of ensembles of nonlinear neurooscillators with sigmoid coupling function // Nonlinear Dynamics. 2019. Vol. 95, no. 3. P. 2103–2116. 
  46. Sysoev I.V. Reconstruction of ensembles of generalized Van der Pol oscillators from vector time series // Physica D. 2018. Vol. 384–385, no. 1. P. 1–11.
  47. Savitzky A., Golay M.J.E. Smoothing and Differentiation of Data by Simplified Least Squares Procedures // Analytical Chemistry. 1964. Vol. 38, no. 8. P. 1627–1639. 
  48. Nelder J.A., Mead R. A simplex for function minimization // Computer Journal. 1965. Vol. 7. P. 308–313. 
  49. Kendall M., Stuart A. The Advanced Theory of Statistics. New York: MacMillan, 1979. 
  50. Johnson N.L., Kotz S., Balakrishnan N. Continuous Univariate Distributions. New York: Wiley, 1995. Vol. 2. 752 p. 
  51. Ikeda K. Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system // Optics Communications. 1979. Vol. 30, no. 2. P. 257–261. 
  52. Mackey M.C., Glass L. Oscillation and chaos in physiological control systems // Science. 1977. Vol. 197, no. 4300. P. 287–289. 
  53. Mandel I.D. Cluster Analysis. Moscow: Finansy i Statistika, 1988, 176 p. (in Russian). 
  54. Sompolinsky H., Crisanti A., Sommers H.E. Chaos in random neural networks // Phys. Rev. Lett. 1988. Vol. 61, no. 3. P. 259–262. 
  55. Levenberg K. A method for the solution of certain non-linear problems in least squares // Quarterly of Applied Mathematics. 1944. Vol. 2. P. 164–168. 
  56. Marquardt D. An algorithm for least-squares estimation of nonlinear parameters // SIAM Journal on Applied Mathematics. 1963. Vol. 11, no. 2. P. 431–441.
  57. Coleman T.F., Li Y. An interior trust region approach for nonlinear minimization subject to bounds // SIAM J. Opt. 1996. Vol. 6. P. 418–445. 
  58. Bezruchko B.P., Smirnov D.A. Constructing nonautonomous differential equations from experimental time series // Phys. Rev. E. Vol. 63, no. 1. P. 016207.
Received: 
22.07.2019
Accepted: 
26.08.2019
Published: 
26.08.2019
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