Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems

Abstract

This chapter is devoted to generic bifurcations of equilibria in two-parameter systems of differential equations. First, we make a complete list of such bifurcations. Then, we derive a parameter-dependent normal form for each bifurcation in the minimal possible phase dimension and specify relevant nondegeneracy conditions. Next, we truncate higher-order terms and present the bifurcation diagrams of the resulting system. The analysis is completed by a discussion of the effect of the higher-order terms. In those cases where the higher-order terms do not qualitatively alter the bifurcation diagram, the truncated systems provide topological normal forms for the relevant bifurcations. The results of this chapter can be applied to n-dimensional systems by means of the parameter-dependent version of the Center Manifold Theorem (see Chapter 5).