Summary

Proceedings of the 2012 International Symposium on Nonlinear Theory and its Applications

2012

Session Number:D3L-D

Session:

Number:915

Ginzburg-Landau Equations Reduced from Coupled Delay Differential Equations

Ikuhiro Yamaguchi,  Yutaro Ogawa,  Hiroya Nakao,  Yasuhiko Jimbo,  Kiyoshi Kotani,  

pp.915-918

Publication Date:

Online ISSN:2188-5079

DOI:10.15248/proc.1.915

PDF download (811.9KB)

A coupled system of delay differential equations motivated through the corticothalamic dynamics is systematically reduced to the complex and the real Ginzburg-Landau equations. Two analytically solvable problems are discussed; (1) relaxation process from any initial state to the corresponding attractor (limit cycles or attractive fixed point) and (2) amplitude death of coupled two oscillators due to ”negative average bifurcation parameter” as well as due to large frequency difference. Projection from infinite dimensional phase space to the center subspace (subspace spanned by eigenfunctions belonging to zero or pure imaginary eigenvalues) plays mathematically essential role through this study and is clearly illustrated in (1). Physiologically, on the other hand, (2) provides unique insights into observed EEG activities such as occurrence of epileptic seizure.

[1] Mackey MC and Glass L (1977) Oscillation and chaos in physiological control systems. Science 197, 287.

[2] Hale J (1977) Theory of Functional Differential Equations. Springer.

[3] Guckenheimer J and Holmes P (1990) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer.

[4] Yamaguchi I, Ogawa Y, Jimbo Y, Nakao H, and Kotani K (2011) Reduction Theories Elucidate the Origins of Complex Biological Rhythms Generated by Interacting Delay-Induced Oscillations. PLoS ONE 6,e26497.

[5] Kuramoto (2003) Y Chemical oscillations, waves, and turbulence. Dover Pubns.

[6] Kim J, Robinson PA (2007) Compact dynamical model of brain activity. Phys.Rev. E 75:031907.