Skip to main content
Log in

New methods for the investigation of periodic solutions in ring systems of unidirectionally coupled oscillators

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider special systems of ordinary differential equations, namely, the so-called ring chains of unidirectionally coupled oscillators. A new method is developed for the investigation of the problems of existence and stability of periodic solutions for this class of systems. As a specific feature of this approach, we can mention the use of certain auxiliary delay systems for the determination of cycles and the analysis of their properties. The proposed method is illustrated by a specific example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. D. Glyzin, A.Yu. Kolesov, and N. Kh. Rozov, “On the phenomena of chaos in a ring formed by three unidirectionally coupled generators,” Zh. Vychisl. Mat. Mat. Fiz., 46, No. 10, 1809–1821 (2006).

    MathSciNet  Google Scholar 

  2. J. C. Sprott, Elegant Chaos. Algebraically Simple Chaotic Flows, World Scientific, Singapore (2010).

    Book  MATH  Google Scholar 

  3. J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proc. Nat. Acad. Sci. USA, 81, 3088–3092 (1984).

    Article  Google Scholar 

  4. S. Haykin, Neural Networks. A Comprehensive Foundation, Prentice Hall, Upper Saddle River (1999).

    MATH  Google Scholar 

  5. A.Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “Relay with delay and its C 1-approximation,” in: Trudy MIAN im. Steklova, 216, 126–153 (1997).

  6. A.Yu. Kolesov, E. F. Mishchenko, and N. Kh. Rozov, “On a modification of the Hutchinson equation,” Zh. Vychisl. Mat. Mat. Fiz., 50, No. 12, 2099–2112 (2010).

    MathSciNet  MATH  Google Scholar 

  7. S. D. Glyzin, A.Yu. Kolesov, and N. Kh. Rozov, “Relaxation self-oscillations in neural networks. I,” Differents. Uravn., 47, No. 7, 919–932 (2011).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 1, pp. 82–102, January, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kolesov, A.Y., Rozov, N.K. New methods for the investigation of periodic solutions in ring systems of unidirectionally coupled oscillators. Ukr Math J 65, 91–113 (2013). https://doi.org/10.1007/s11253-013-0767-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-013-0767-9

Keywords

Navigation