Spatiotemporal Patterns and Bifurcation Analysis With Degenerate Cases in a Charge Transfer Model

A priori estimates and the nonexistence of nonconstant positive steady-state solutions of a charge transfer model are established. Steady-state and Hopf bifurcations are carried out in detail, especially under the degenerate conditions such as invalid simple eigenvalue condition or crossing condition. It is shown that the steady-state solutions under the violated simple eigenvalue are characterized by a coupling of two eigenfunctions, and the bifurcation direction from simple eigenvalue is supercritical or subcritical. Meanwhile, there is no nonconstant steady-state solution under the degenerate case of violated crossing condition. In addition, whether the transversality condition in Hopf bifurcation is true or not, instead of the center manifold theory, we apply the Lyapunov-Schmidt reduction method and the singularity theory to illustrate the existence and stability of periodic solutions as well as the bifurcation diagrams. But it is proved that the usual pitchfork Hopf bifurcation is changed to the transcritical bifurcation through the invalid transversality condition. Finally, some numerical simulations allow us to identify the influence of external current on the predicted dynamics. The novel results on the effect of degeneracy obtained in this paper shall complement classical theories and enrich numerical examples in the existing work.