Symmetric periodic solutions of delay-coupled optoelectronic oscillators

Advances in Difference Equations

Hongpeng Li¹  Chunrui Zhang² 

[1] College of Mechanical and Electronic Engineering, Northeast Forestry University, Harbin, P.R. China;Department of Mathematics, Northeast Forestry University, Harbin, P.R. China

关键词: optoelectronic oscillators;    symmetric bifurcation;    stability;    spatio-temporal patterns;    delay;   

DOI  :  10.1186/s13662-016-0755-0

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

Delay-coupled optoelectronic oscillators are considered. These structures are based on mutually coupled oscillators which oscillate at the same frequency. By taking the time delay as a bifurcation parameter, the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay are investigated, and then stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Using the symmetric functional differential equation theories combined with the representation theory of Lie groups, the multiple Hopf bifurcations of the equilibrium are demonstrated. In particular, we find that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling. The existence of multiple branches of bifurcating periodic solutions and their spatio-temporal patterns are obtained. Some numerical simulations are used to illustrate the effectiveness of the obtained results.

【 授权许可】

CC BY