In this lecture we will consider the zero set of nonlinear systems of

equations that also depend on one or more parameters. In this case

the zero set is typically not just a single point but a whole curve

(or manifold). Moreover, on these curves typically appear bifurcation

points. These are points, where for example the number of roots

locally changes.

In the important case, that the zero set is in fact the set of steady

states of an ODE, it may well happen that along such a curve the

qualitative behavior of the solution changes: For example a stable

equilibrium may become unstable or a stationary solution becomes a

whole periodic orbit...

We will discuss techniques on how to approximate these zero sets and

how to detect bifurcation points.

In the exercises the algorithms will also be implemented and tested.

Note that these techniques are also implemented in the software

packages *auto* or *Matcont*.