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  • Open Access

Inference of a time delay in stochastic systems

Robin A. Kopp*, Sabine H. L. Klapp†, and Deepak Gupta‡

  • *Contact author: r.kopp@tu-berlin.de
  • †Contact author: sabine.klapp@tu-berlin.de
  • ‡Contact author: phydeepak.gupta@gmail.com
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Phys. Rev. Research 8, 013129 – Published 5 February, 2026

DOI: https://doi.org/10.1103/7hsc-mmzc

Abstract

Time delay is ubiquitous in many experimental and real-world situations. However, from measurable data it is often unclear whether time delay plays a significant role in the dynamics, and if it does, how long the time lag really is. This would be invaluable knowledge when analyzing and modeling such systems. Hitherto, no universal method is available by which the time delay can be inferred. To address this problem, we propose and demonstrate two different methods to infer time delay in overdamped Langevin systems with delayed feedback. In the first part, we focus on the power spectral density based on the positional data and use a characteristic signature of the time delay to infer the delay time. In limiting cases, we establish a direct relation between the observations made for nonlinear time-delayed feedback forces and analytical results obtained for the linear system. In other situations, despite the absence of this direct relation, the characteristic signature remains and can be exploited by a semiautomatic method to infer the delay time. Furthermore, it may not always desirable or possible to observe a system for a long time to infer dependencies and parameters. Thus, in the second part, we propose a probing method combined with a neural network to infer the delay time, which requires only short observation time series. These proposed methods for inferring time delays in stochastic systems can be valuable tools for gaining deeper insight into the role of delay across a wide range of applications—from the behavior of individual colloidal particles under feedback control to emergent collective phenomena such as flocking and swarming.

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References (103)

  1. E. Schöll, S. H. L. Klapp, and P. Hövel, Control of Self- Organizing Nonlinear Systems (Springer, Cham, 2016).
  2. K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A 170, 421 (1992).
  3. K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A 206, 323 (1995).
  4. A. P. Tsui and A. J. Jones, The control of higher dimensional chaos: Comparative results for the chaotic satellite attitude control problem, Physica D 135, 41 (2000).
  5. M. Gavrilov and J. Bechhoefer, Feedback traps for virtual potentials, Philos. Trans. R. Soc. A 375, 20160217 (2017).
  6. A. Kumar and J. Bechhoefer, Nanoscale virtual potentials using optical tweezers, Appl. Phys. Lett. 113, 183702 (2018).
  7. M. C. R. Bell-Davies, A. Curran, Y. Liu, and R. P. A. Dullens, Dynamics of a colloidal particle driven by continuous time-delayed feedback, Phys. Rev. E 107, 064601 (2023).
  8. K. Lichtner and S. H. L. Klapp, Feedback-controlled transport in an interacting colloidal system, Europhys. Lett. 92, 40007 (2010).
  9. B. J. Lopez, N. J. Kuwada, E. M. Craig, B. R. Long, and H. Linke, Realization of a feedback controlled flashing ratchet, Phys. Rev. Lett. 101, 220601 (2008).
  10. S. A. M. Loos and S. H. L. Klapp, Heat flow due to time-delayed feedback, Sci. Rep. 9, 2491 (2019).
  11. R. A. Kopp and S. H. L. Klapp, Heat production in a stochastic system with nonlinear time-delayed feedback, Phys. Rev. E 110, 054126 (2024).
  12. T. Munakata, S. Iwama, and M. Kimizuka, Linear stochastic system with delay: Energy balance and entropy production, Phys. Rev. E 79, 031104 (2009).
  13. T. Munakata and M. L. Rosinberg, Entropy production and fluctuation theorems for Langevin processes under continuous non-Markovian feedback control, Phys. Rev. Lett. 112, 180601 (2014).
  14. M. Debiossac, D. Grass, J. J. Alonso, E. Lutz, and N. Kiesel, Thermodynamics of continuous non-Markovian feedback control, Nat. Commun. 11, 1360 (2020).
  15. M. Debiossac, M. L. Rosinberg, E. Lutz, and N. Kiesel, Non-Markovian feedback control and acausality: An experimental study, Phys. Rev. Lett. 128, 200601 (2022).
  16. M. Fränzl, S. Muiños-Landin, V. Holubec, and F. Cichos, Fully steerable symmetric thermoplasmonic microswimmers, ACS Nano 15, 3434 (2021).
  17. J. Hindes, K. Szwaykowska, and I. B. Schwartz, Hybrid dynamics in delay-coupled swarms with mothership networks, Phys. Rev. E 94, 032306 (2016).
  18. G. Stoica, A stochastic delay financial model, Proc. Am. Math. Soc. 133, 1837 (2005).
  19. J. L. Callen, M. Khan, and H. Lu, Accounting quality, stock price delay, and future stock returns, Contemp. Accoun. Res. 30, 269 (2013).
  20. M. Mijalkov, A. McDaniel, J. Wehr, and G. Volpe, Engineering sensorial delay to control phototaxis and emergent collective behaviors, Phys. Rev. X 6, 011008 (2016).
  21. R. Sipahi, F. M. Atay, and S.-I. Niculescu, Stability of traffic flow behavior with distributed delays modeling the memory effects of the drivers, SIAM J. Appl. Math. 68, 738 (2008).
  22. L. Yu, T. Li, and Z.-K. Shi, Density waves in a traffic flow model with reaction-time delay, Physica A 389, 2607 (2010).
  23. V. Holubec, D. Geiss, S. A. M. Loos, K. Kroy, and F. Cichos, Finite-size scaling at the edge of disorder in a time-delay vicsek model, Phys. Rev. Lett. 127, 258001 (2021).
  24. F. Pakpour and T. Vicsek, Delay-induced phase transitions in active matter, Physica A 634, 129453 (2024).
  25. R. A. Kopp and S. H. L. Klapp, Spontaneous velocity alignment of Brownian particles with feedback-induced propulsion, Europhys. Lett. 143, 17002 (2023).
  26. S. Tarama, S. U. Egelhaaf, and H. Löwen, Traveling band formation in feedback-driven colloids, Phys. Rev. E 100, 022609 (2019).
  27. S. Tarama, Engineering living worms and active crystals with colloids propelled by attractive time-delayed feedback, J. Chem. Phys. 163, 044903 (2025).
  28. E. Pigani, D. Sgarbossa, S. Suweis, A. Maritan, and S. Azaele, Delay effects on the stability of large ecosystems, Proc. Natl. Acad. Sci. USA 119, e2211449119 (2022).
  29. M. G. Earl and S. H. Strogatz, Synchronization in oscillator networks with delayed coupling: A stability criterion, Phys. Rev. E 67, 036204 (2003).
  30. T. Dahms, J. Lehnert, and E. Schöll, Cluster and group synchronization in delay-coupled networks, Phys. Rev. E 86, 016202 (2012).
  31. P. Siegle, I. Goychuk, P. Talkner, and P. Hänggi, Markovian embedding of non-Markovian superdiffusion, Phys. Rev. E 81, 011136 (2010).
  32. S. A. M. Loos and S. H. L. Klapp, Fokker–Planck equations for time-delayed systems via Markovian embedding, J. Stat. Phys. 177, 95 (2019).
  33. T. J. Doerries, S. A. M. Loos, and S. H. L. Klapp, Correlation functions of non-Markovian systems out of equilibrium: Analytical expressions beyond single-exponential memory, J. Stat. Mech. (2021) 033202.
  34. M. Fränzl and F. Cichos, Active particle feedback control with a single-shot detection convolutional neural network, Sci. Rep. 10, 12571 (2020).
  35. S. A. M. Loos, S. Hermann, and S. H. L. Klapp, Medium entropy reduction and instability in stochastic systems with distributed delay, Entropy 23, 696 (2021).
  36. C. Otto, K. Lüdge, and E. Schöll, Modeling quantum dot lasers with optical feedback: Sensitivity of bifurcation scenarios, Phys. Stat. Sol. (B) 247, 829 (2010).
  37. C. Otto, B. Globisch, K. Lüdge, E. Schöll, and T. Erneux, Complex dynamics of semiconductor quantum dot lasers subject to delayed optical feedback, Int. J. Bifurcation Chaos 22, 1250246 (2012).
  38. U. Dorner and P. Zoller, Laser-driven atoms in half-cavities, Phys. Rev. A 66, 023816 (2002).
  39. A. Carmele, J. Kabuss, F. Schulze, S. Reitzenstein, and A. Knorr, Single photon delayed feedback: A way to stabilize intrinsic quantum cavity electrodynamics, Phys. Rev. Lett. 110, 013601 (2013).
  40. Complex Time-Delay Systems: Theory and Applications, edited by F. M. Atay, Understanding Complex Systems (Springer, Berlin, Heidelberg, 2010).
  41. U. von Toussaint, Bayesian inference in physics, Rev. Mod. Phys. 83, 943 (2011).
  42. S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113, 3932 (2016).
  43. Z. Chen, Y. Liu, and H. Sun, Physics-informed learning of governing equations from scarce data, Nat. Commun. 12, 6136 (2021).
  44. Á. T. Köpeczi-Bócz, H. Sykora, and D. Takács, Data-driven delay identification with SINDy, in Advances in Nonlinear Dynamics, edited by W. Lacarbonara, NODYCON Conference Proceedings Series, Vol. III (Springer, Cham, Switzerland, 2024), pp. 481–491.
  45. G. Leylaz, S. Wang, and J.-Q. Sun, Identification of nonlinear dynamical systems with time delay, Int. J. Dyn. Control 10, 13 (2022).
  46. Y. Zhao and S. W. K. Wong, Inference for delay differential equations using manifold-constrained Gaussian processes, arXiv:2406.15170.
  47. U. Küchler and M. Sørensen, Statistical inference for discrete-time samples from affine stochastic delay differential equations, Bernoulli 19, 409 (2013).
  48. H. Voss and J. Kurths, Reconstruction of non-linear time delay models from data by the use of optimal transformations, Phys. Lett. A 234, 336 (1997).
  49. W. Horbelt, J. Timmer, and H. Voss, Parameter estimation in nonlinear delayed feedback systems from noisy data, Phys. Lett. A 299, 513 (2002).
  50. M. J. Bünner, M. Popp, Th. Meyer, A. Kittel, and J. Parisi, Tool to recover scalar time-delay systems from experimental time series, Phys. Rev. E 54, R3082(R) (1996).
  51. T. M. Muenker, G. Knotz, M. Krüger, and T. Betz, Accessing activity and viscoelastic properties of artificial and living systems from passive measurement, Nat. Mater. 23, 1283 (2024).
  52. K. Berg-Sørensen and H. Flyvbjerg, Power spectrum analysis for optical tweezers, Rev. Sci. Instrum. 75, 594 (2004).
  53. K. Berg-Sørensen, E. J. G. Peterman, T. Weber, C. F. Schmidt, and H. Flyvbjerg, Power spectrum analysis for optical tweezers. II: Laser wavelength dependence of parasitic filtering, and how to achieve high bandwidth, Rev. Sci. Instrum. 77, 063106 (2006).
  54. V. Sposini, D. Krapf, E. Marinari, R. Sunyer, F. Ritort, F. Taheri, C. Selhuber-Unkel, R. Benelli, M. Weiss, R. Metzler, and G. Oshanin, Towards a robust criterion of anomalous diffusion, Commun. Phys. 5, 305 (2022).
  55. S. L. Brunton and J. N. Kutz, Data-Driven Science and Eng- ineering: Machine Learning, Dynamical Systems, and Control, 2nd ed. (Cambridge University Press, Cambridge, 2022).
  56. H. I. Fawaz, G. Forestier, J. Weber, L. Idoumghar, and P.-A. Muller, Deep learning for time series classification: A review, Data Min. Knowl. Discov. 33, 917 (2019).
  57. U. Küchler and B. Mensch, Langevins stochastic differential equation extended by a time-delayed term, Stoch. Stoch. Rep. 40, 23 (1992).
  58. S. Yanchuk and P. Perlikowski, Delay and periodicity, Phys. Rev. E 79, 046221 (2009).
  59. T. D. Frank and P. J. Beek, Stationary solutions of linear stochastic delay differential equations: Applications to biological systems, Phys. Rev. E 64, 021917 (2001).
  60. S. Yanchuk, M. Wolfrum, P. Hövel, and E. Schöll, Control of unstable steady states by long delay feedback, Phys. Rev. E 74, 026201 (2006).
  61. D. Saha, S. Tarama, H. Löwen, and S. U. Egelhaaf, Cybloids—Creation and control of cybernetic colloids, Soft Matter 20, 8112 (2024).
  62. R. A. Kopp and S. H. L. Klapp, Persistent motion of a Brownian particle subject to repulsive feedback with time delay, Phys. Rev. E 107, 024611 (2023).
  63. T. Lequy and A. M. Menzel, Stochastic motion under nonlinear friction representing shear thinning, Phys. Rev. E 108, 064606 (2023).
  64. S. Guillouzic, I. L’Heureux, and A. Longtin, Small delay approximation of stochastic delay differential equations, Phys. Rev. E 59, 3970 (1999).
  65. S. Guillouzic, I. L’Heureux, and A. Longtin, Rate processes in a delayed, stochastically driven, and overdamped system, Phys. Rev. E 61, 4906 (2000).
  66. M. S. Bartlett, Periodogram analysis and continuous spectra, Biometrika 37, 1 (1950).
  67. R. B. Blackman and J. W. Tukey, The measurement of power spectra from the point of view of communications engineering—Part I, Bell Syst. Tech. J. 37, 185 (1958).
  68. A. Squarcini, A. Solon, and G. Oshanin, Spectral density of individual trajectories of an active Brownian particle, New J. Phys. 24, 013018 (2022).
  69. D. Krapf, N. Lukat, E. Marinari, R. Metzler, G. Oshanin, C. Selhuber-Unkel, A. Squarcini, L. Stadler, M. Weiss, and X. Xu, Spectral content of a single non-Brownian trajectory, Phys. Rev. X 9, 011019 (2019).
  70. D. Krapf, E. Marinari, R. Metzler, G. Oshanin, X. Xu, and A. Squarcini, Power spectral density of a single Brownian trajectory: What one can and cannot learn from it, New J. Phys. 20, 023029 (2018).
  71. R. Metzler, Brownian motion and beyond: First-passage, power spectrum, non-Gaussianity, and anomalous diffusion, J. Stat. Mech. (2019) 114003.
  72. S. A. McKinley and H. D. Nguyen, Anomalous diffusion and the generalized Langevin equation, SIAM J. Math. Anal. 50, 5119 (2018).
  73. L. He and B. Feng, Fundamentals of Measurement and Signal Analysis (Springer, Singapore, 2022).
  74. P. Welch, The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Trans. Audio Electroacoust. 15, 70 (1967).
  75. J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19, 297 (1965).
  76. M. L. Rosinberg, T. Munakata, and G. Tarjus, Stochastic thermodynamics of Langevin systems under time-delayed feedback control: Second-law-like inequalities, Phys. Rev. E 91, 042114 (2015).
  77. M. L. Rosinberg, G. Tarjus, and T. Munakata, Stochastic thermodynamics of Langevin systems under time-delayed feedback control. II. Nonequilibrium steady-state fluctuations, Phys. Rev. E 95, 022123 (2017).
  78. T. Erneux, Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences (Springer, New York, NY, 2009).
  79. S. Ciliberto, Experiments in stochastic thermodynamics: Short history and perspectives, Phys. Rev. X 7, 021051 (2017).
  80. S. Dago, J. Pereda, S. Ciliberto, and L. Bellon, Virtual double-well potential for an underdamped oscillator created by a feedback loop, J. Stat. Mech. (2022) 053209.
  81. S. Toyabe, H.-R. Jiang, T. Nakamura, Y. Murayama, and M. Sano, Experimental test of a new equality: Measuring heat dissipation in an optically driven colloidal system, Phys. Rev. E 75, 011122 (2007).
  82. M. A. Fernandez-Rodriguez, F. Grillo, L. Alvarez, M. Rathlef, I. Buttinoni, G. Volpe, and L. Isa, Feedback-controlled active Brownian colloids with space-dependent rotational dynamics, Nat. Commun. 11, 4223 (2020).
  83. Y. Jun and J. Bechhoefer, Virtual potentials for feedback traps, Phys. Rev. E 86, 061106 (2012).
  84. B. Ertel, J. Van Der Meer, and U. Seifert, Waiting time distributions in hybrid models of motor–bead assays: A concept and tool for inference, Int. J. Mol. Sci. 24, 7610 (2023).
  85. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, edited by M. Cardona, P. Fulde, K. Von Klitzing, H.-J. Queisser, and H. K. V. Lotsch, Springer Series in Solid-State Sciences (Springer, Berlin, Heidelberg, 1991), Vol. 31.
  86. U. Marconi, A. Puglisi, L. Rondoni, and A. Vulpiani, Fluctuation–dissipation: Response theory in statistical physics, Phys. Rep. 461, 111 (2008).
  87. M. S. Gomes-Filho, L. C. Lapas, E. Gudowska-Nowak, and F. A. Oliveira, The fluctuation–dissipation relations: Growth, diffusion, and beyond, Phys. Rep. 1141, 1 (2025).
  88. A. V. Chechkin and R. Klages, Fluctuation relations for anomalous dynamics, J. Stat. Mech. (2009) L03002.
  89. X. Wang, P.-C. Chen, K. Kroy, V. Holubec, and F. Cichos, Spontaneous vortex formation by microswimmers with retarded attractions, Nat. Commun. 14, 56 (2023).
  90. T. Bäuerle, R. C. Löffler, and C. Bechinger, Formation of stable and responsive collective states in suspensions of active colloids, Nat. Commun. 11, 2547 (2020).
  91. N. Miller and R. Gerlai, From schooling to shoaling: Patterns of collective motion in zebrafish (Danio rerio), PLoS ONE 7, e48865 (2012).
  92. J. D. Partridge, Surveying a swarm: Experimental techniques to establish and examine bacterial collective motion, Appl. Environ. Microbiol. 88, e01853-21 (2022).
  93. B. Koger, A. Deshpande, J. T. Kerby, J. M. Graving, B. R. Costelloe, and I. D. Couzin, Quantifying the movement, behaviour and environmental context of group-living animals using drones and computer vision, J. Anim. Ecol. 92, 1357 (2023).
  94. S. L. Miller and D. Childers, Power spectral density, in Probability and Random Processes (Elsevier, Amsterdam, 2012), pp. 429–471.
  95. C. R. Harris, et al., Array programming with NumPy, Nature (London) 585, 357 (2020).
  96. P. Virtanen, et al., SciPy 1.0: Fundamental algorithms for scientific computing in Python, Nat. Methods 17, 261 (2020).
  97. J. D. Hunter, Matplotlib: A 2D graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  98. F. Chollet, et al., Keras, 2015, https://keras.io.
  99. M. Abadi, et al., TENSORFLOW: Large-scale machine learning on heterogeneous systems, 2015, https://www.tensorflow.org/.
  100. G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo 4, 48 (1955).
  101. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Stochastic Modelling and Applied Probability (Springer, Berlin, Heidelberg, 1992).
  102. R. H. Shumway and D. S. Stoffer, Time Series Analysis and Its Applications Book Subtitle: With R Examples, Springer Texts in Statistics (Springer, Cham, 2025).
  103. P. Hänggi and P. Jung, Colored noise in dynamical systems, Adv. Chem. Phys. 89, 239 (1994).

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Phys. Rev. Research 8, 013129– Published 5 February, 2026
  • Received 1 August 2025
  • Accepted 5 December 2025
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