19.3 Higher dimensional stability

The trace-determinant plane is a really use approach to analyze stability without computing the eigenvalues directly. It is also useful when you have a parameter in your Jacobian - the trace-determinant will allow you to characterize the stability of solutions as a function of a parameter. This leads to what is known as a bifurcation diagram, which we will study in the next section.

The conditions between the trace and determinant really describe the notion of eigenvalues as a function of the entries of a matrix. It may be natural to ask if similar conditions exist for larger dimensioned matrices. It turns out yes … to a point. The Routh-Hurwitz stability criterion is one such approach, but it gets tricky for larger matrices.