Geometric Approach to the Bifurcation at Infinity: A Case Study

In this paper, the reaction-diffusion equation in space 1D with negative power nonlinearity is presented as an example the stationary solutions of which are characterized by the bifurcation at infinity. The all dynamics, including to infinity, of the ordinary differential equation concerning with the stationary problem of the reaction-diffusion equation are obtained by the Poincaré-type compactification. The results of this paper give relationship between the bifurcation at infinity of the stationary solutions for the reaction-diffusion equation and the connecting orbits of the ordinary differential equations, especially, that connect the equilibria at infinity. Then, for the bifurcation at infinity, the question of what kind of bifurcation structures are present in this ODEs not only before the solutions go to infinity but also after they go to infinity is answered by a precise analysis. Moreover, this result also answers the question of what kind of solutions appear (or disappear) associated with the bifurcation. In addition, from the bifurcation theory view point, to clarify the bifurcation structures of the equilibria in the ordinary differential equation, the dynamics of it through the one-point compactification called the Bendixson compactification is also shown.