Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh-Rose neurons

作者：Guo, Yuxiao; Jiang, Weihua^*; Niu, Ben

来源：Nonlinear Dynamics, 2013, 71(3): 515-529.

DOI：10.1007/s11071-012-0678-6

• Networks
• Delay differential equations
• Multiple scale
• Equivariant bifurcation
• Codimension two bifurcation
• Synchronization

We study the appearance and stability of spatiotemporal periodic patterns like phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or antiphase oscillations, and coexistence of multiple patterns, in a ring of bidirectionally delay coupled oscillators. Hopf bifurcation, Hopf-Hopf bifurcation, and the equivariant Hopf bifurcation are studied in the viewpoint of normal forms obtained by using the method of multiple scales which is a kind of perturbation technique, thus a clear bifurcation scenario is depicted. We find time delay significantly affects the dynamics and induces rich spatiotemporal patterns. With the help of the unfolding system near Hopf-Hopf bifurcation, it is confirmed in some regions two kinds of stable oscillations may coexist. These phenomena are shown for the delay coupled limit cycle oscillators as well as for the delay coupled chaotic Hindmarsh-Rose neurons.

• 出版日期2013-2
• 单位哈尔滨学院; 哈尔滨工业大学