Multiple Bifurcation Problems of Codimension Two

The term bifurcation in this paper refers to changes in the qualitative structure of any solutions to a system of ordinary differential equations with a varying parameter. This paper is about multiple bifurcations for which there is a multiple degeneracy in some feature of the system and a multi-dimensional parameter in its definition. The most immediate motivation for studying these problems is that they occur in the mathematical descriptions of many natural phenomena, but their importance extends beyond the fact that they can be identified in applications. Multiple bifurcations provide both a means of organizing knowledge about simple bifurcations and a powerful analytical tool for locating complicated dynamical behavior in some models. An intuitive reason for these features is that near some multiple bifurcations, the effect of nonlinear interactions is analytically accessible.