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  • Letter
  • Open Access

Weak chaos, anomalous diffusion, and weak ergodicity breaking in systems with delay

Tony Albers1,*, Lukas Hille1,†, David Müller-Bender1,‡, and Günter Radons1,2,§

  • *Contact author: tony.albers@physik.tu-chemnitz.de
  • †Contact author: hille.lukas@proton.me
  • ‡Contact author: david.mueller-bender@mailbox.org
  • §Deceased July 20, 2024
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Phys. Rev. E 112, L042201 – Published 14 October, 2025

DOI: https://doi.org/10.1103/qqcs-7pll

Abstract

We demonstrate that standard delay systems with a linear instantaneous and a delayed nonlinear term show weak chaos, asymptotically subdiffusive behavior, and weak ergodicity breaking if the nonlinearity is chosen from a specific class of functions. In the limit of large constant delay times, anomalous behavior may not be observable due to exponentially large crossover times. A periodic modulation of the delay causes a strong reduction of the effective dimension of the chaotic phases, leads to hitherto unknown types of solutions, and the occurrence of anomalous diffusion already at short times. The observed anomalous behavior is caused by nonhyperbolic fixed points in function space.

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Phys. Rev. E 112, L042201– Published 14 October, 2025
  • Received 12 July 2024
  • Revised 13 June 2025
  • Accepted 3 September 2025
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