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Singularity of information flow at the Hopf bifurcation point

Kenshin Matsumoto* and Shin-ichi Sasa

  • *Contact author: matsumoto.kenshin.73z@st.kyoto-u.ac.jp
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Phys. Rev. Research 8, 033324 – Published 16 September, 2026

DOI: https://doi.org/10.1103/ljc9-72gq

Abstract

We investigate the singular behavior of information flow near the Hopf bifurcation point by analyzing the learning rate as a specific measure of information flow. We study the Brusselator, a model system exhibiting the Hopf bifurcation. We first numerically compute the learning rate in the stationary regime and find that it remains finite even in the deterministic limit, suggesting that the learning rate can be quantified in deterministic dynamics through probabilistic descriptions. Linear analysis accurately reproduces the numerical results in the stationary regime but fails near the bifurcation point. To overcome this limitation, we employ the singular perturbation method, well known in deterministic bifurcation theory, and carry out the corresponding calculation explicitly for a stochastic system described by a Langevin equation. This allows us to evaluate the learning rate near the bifurcation point. We then theoretically derive its nonsmooth behavior in the deterministic limit. Our results demonstrate that changes in dynamical behavior are reflected in the learning rate and provide a basis for analyzing information processing in biochemical oscillations.

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References (44)

  1. K. Sekimoto, Concept of heat on mesoscopic scales, in Stochastic Energetics (Springer, Berlin, 2010), pp. 135–174.
  2. J. M. R. Parrondo, J. M. Horowitz, and T. Sagawa, Thermodynamics of information, Nat. Phys. 11, 131 (2015).
  3. T. Sagawa, Second law, entropy production, and reversibility in thermodynamics of information, in Energy Limits in Computation: A Review of Landauer’s Principle, Theory and Experiments, edited by C. S. Lent, A. O. Orlov, W. Porod, and G. L. Snider (Springer, Cham, 2019), pp. 101–139.
  4. C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27, 379 (1948).
  5. T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. (John Wiley & Sons, Hoboken, NJ, 2006).
  6. L. Szilard, On the reduction of entropy in a thermodynamic system by the interference of an intelligent being, Z. Phys. 53, 840 (1929).
  7. R. Landauer, Irreversibility and heat generation in the computing process, IBM J. Res. Dev. 5, 183 (1961).
  8. T. Sagawa and M. Ueda, Nonequilibrium thermodynamics of feedback control, Phys. Rev. E 85, 021104 (2012).
  9. T. Sagawa and M. Ueda, Generalized Jarzynski equality under nonequilibrium feedback control, Phys. Rev. Lett. 104, 090602 (2010).
  10. A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  11. T. Tanogami, T. V. Vu, and K. Saito, Universal bounds on the performance of information-thermodynamic engine, Phys. Rev. Res. 5, 043280 (2023).
  12. A. E. Allahverdyan, D. Janzing, and G. Mahler, Thermodynamic efficiency of information and heat flow, J. Stat. Mech. (2009) P09011.
  13. J. M. Horowitz and M. Esposito, Thermodynamics with continuous information flow, Phys. Rev. X 4, 031015 (2014).
  14. J. M. Horowitz, Multipartite information flow for multiple Maxwell demons, J. Stat. Mech. (2015) P03006.
  15. A. C. Barato, D. Hartich, and U. Seifert, Efficiency of cellular information processing, New J. Phys. 16, 103024 (2014).
  16. D. Hartich, A. C. Barato, and U. Seifert, Stochastic thermodynamics of bipartite systems: Transfer entropy inequalities and a Maxwell’s demon interpretation, J. Stat. Mech. (2014) P02016.
  17. S. Ito and T. Sagawa, Maxwell’s demon in biochemical signal transduction with feedback loop, Nat. Commun. 6, 7498 (2015).
  18. D. Hartich, A. C. Barato, and U. Seifert, Sensory capacity: An information theoretical measure of the performance of a sensor, Phys. Rev. E 93, 022116 (2016).
  19. T. Matsumoto and T. Sagawa, Role of sufficient statistics in stochastic thermodynamics and its implication to sensory adaptation, Phys. Rev. E 97, 042103 (2018).
  20. M. P. Leighton, J. Ehrich, and D. A. Sivak, Information arbitrage in bipartite heat engines, Phys. Rev. X 14, 041038 (2024).
  21. I. Prigogine and G. Nicolis, Biological order, structure and instabilities, Q. Rev. Biophys. 4, 107 (1971).
  22. B. Novák and J. J. Tyson, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell Biol. 9, 981 (2008).
  23. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour (Cambridge University Press, Cambridge, 1996).
  24. M. Nakajima, K. Imai, H. Ito, T. Nishiwaki, Y. Murayama, H. Iwasaki, T. Oyama, and T. Kondo, Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro, Science 308, 414 (2005).
  25. J. E. Ferrell, Jr., T. Yu-Chen Tsai, and Q. Yang, Modeling the cell cycle: Why do certain circuits oscillate? Cell 144, 874 (2011).
  26. B. Nguyen, U. Seifert, and A. C. Barato, Phase transition in thermodynamically consistent biochemical oscillators, J. Chem. Phys. 149, 045101 (2018).
  27. A. C. Barato and U. Seifert, Coherence of biochemical oscillations is bounded by driving force and network topology, Phys. Rev. E 95, 062409 (2017).
  28. Z. Cao, H. Jiang, and Z. Hou, Design principles for biochemical oscillations with limited energy resources, Phys. Rev. Res. 2, 043331 (2020).
  29. C. del Junco and S. Vaikuntanathan, Robust oscillations in multi-cyclic Markov state models of biochemical clocks, J. Chem. Phys. 152, 055101 (2020).
  30. T. Xiao, Z. Hou, and H. Xin, Stochastic thermodynamics in mesoscopic chemical oscillation systems, J. Phys. Chem. B 113, 9316 (2009).
  31. Y. Cao, H. Wang, Q. Ouyang, and Y. Tu, The free-energy cost of accurate biochemical oscillations, Nat. Phys. 11, 772 (2015).
  32. D. Zhang, Y. Cao, Q. Ouyang, and Y. Tu, The energy cost and optimal design for synchronization of coupled molecular oscillators, Nat. Phys. 16, 95 (2020).
  33. Z. Cao and Z. Hou, Improved estimation for energy dissipation in biochemical oscillations, J. Chem. Phys. 157, 025102 (2022).
  34. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Westview Press, Cambridge, MA, 2000).
  35. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984).
  36. F. Baras, M. Malek Mansour, and C. Van den Broeck, Asymptotic properties of coupled nonlinear Langevin equations in the limit of weak noise. II: Transition to a limit cycle, J. Stat. Phys. 28, 577 (1982).
  37. E. Knobloch and K. A. Wiesenfeld, Bifurcations in fluctuating systems: The center-manifold approach, J. Stat. Phys. 33, 611 (1983).
  38. L. Arnold, N. Sri Namachchivaya, and K. R. Schenk-Hoppé, Toward an understanding of stochastic Hopf bifurcation, Int. J. Bifurcation Chaos 06, 1947 (1996).
  39. D. T. Gillespie, The chemical Langevin equation, J. Chem. Phys. 113, 297 (2000).
  40. C. W. Gardiner, Handbook of Stochastic Methods (Springer, Berlin, 1985).
  41. R. Chétrite, M. L. Rosinberg, T. Sagawa, and G. Tarjus, Information thermodynamics for interacting stochastic systems without bipartite structure, J. Stat. Mech. (2019) 114002.
  42. N. Freitas and M. Esposito, Information flows in macroscopic Maxwell’s demons, Phys. Rev. E 107, 014136 (2023).
  43. K. Matsumoto, Processed datasets and Julia scripts for singularity of information flow at the Hopf bifurcation point, GitHub repository, https://github.com/Ken-Matsumoto-code/hopf-learning-rate (2026).
  44. K. Tomita and H. Tomita, Irreversible circulation of fluctuation, Prog. Theor. Phys. 51, 1731 (1974).

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Phys. Rev. Research 8, 033324– Published 16 September, 2026
  • Received 3 March 2026
  • Accepted 8 August 2026
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