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.
Similar content being viewed by others
References
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).
J. C. Sprott, Elegant Chaos. Algebraically Simple Chaotic Flows, World Scientific, Singapore (2010).
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).
S. Haykin, Neural Networks. A Comprehensive Foundation, Prentice Hall, Upper Saddle River (1999).
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).
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).
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).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 1, pp. 82–102, January, 2013.
Rights 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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0767-9