Steady-state and Hopf bifurcations in the Langford ODE and PDE systems

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• 作者：Gaihui Guo^a ; ^{guogaihui@sust.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Xiaoni Wang^a ; ^{wangxiaoni@163.com" class="auth_mail" title="E-mail the corresponding author} ; Xiaolin Lin^a ; ^{linxl@sust.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Meihua Wei^b ; ^{wei_meihua@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Langford system ; Steady-state bifurcation ; Hopf bifurcation ; Lyapunov&ndash ; Schmidt method ; Center manifold theory
• 刊名：Nonlinear Analysis: Real World Applications
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：34
• 期：Complete
• 页码：343-362
• 全文大小：630 K

文摘

This paper is concerned with the Langford ODE and PDE systems. For the Langford ODE system, the existence of steady-state solutions is firstly obtained by Lyapunov–Schmidt method, and the stability and bifurcation direction of periodic solutions are established. Then for the Langford PDE system, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalue. Finally, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions for the PDE system are investigated.