Embedding the dynamics of a single delay system into a feed-forward ring

Vladimir Klinshov, Dmitry Shchapin, Serhiy Yanchuk, Matthias Wolfrum, Otti D'Huys, Vladimir Neokorkin

Research output: Contribution to journalArticlepeer-review


We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example we demonstrate how the complex bifurcation scenario of simultaneously emerging multi-jittering solutions can be transferred from a single oscillator with delayed pulse feedback to multi-jittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.
Original languageEnglish
Article number042217
Number of pages9
JournalPhysical Review E
Issue number4
Early online date27 Oct 2017
Publication statusPublished - 27 Oct 2017

Bibliographical note

Copyright: American Physical Society.

Funding: Russian Scientific Foundation (project 16-42-01043 for the Institute of Applied Physics) and the German Research Foundation (project SCHO 307/15-1 and
YA 225/3-1 for TU Berlin).


  • coupled oscillators
  • delay systems


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