The zero set of an equation depending on one or more parameters may have a complicated geometric structure. In the one-parameter case, it is well known than pseudo-arclength continuation can be used to cross over folds. Less well-known is that in the multi-parameter case, the natural analogue of pseudo-arclength continuation permits the continuation of $n$-dimensional manifolds of zeros. However, when the ambient space is infinite-dimensional, controlling the error of a numerical implementation of this algorithm becomes more subtle. This makes it difficult to establish a posteriori error bounds between the numerical zero set and that of the function itself.

In this talk, we first overview validated two-parameter continuation of zeroes of a sufficiently smooth map $F:X\times\mathbb{R}^2\rightarrow Y$, for Banach spaces $X$ and $Y$, possibly infinite-dimensional. The first step of the validation procedure is to compute a simplical triangulation of the manifold of solutions in a suitable finite-dimensional projection of $X$. To validate a simplex --- abstractly denoted $\Delta\subset X\times\mathbb{R}^2$ --- of this triangulation, means to identify a radius $r_0>0$ such that every element of $\Delta$ is at most a distance $r_0$ away from a zero of $F$. Validating multiple adjacent simplices produces a global, validated triangulation of the solution manifold. The validation procedure is done using a theorem of Newton-Kantorovich type, applied to a bordered version of the map $F$.

As an application, we use a desingularization approach to rigorously compute two-dimensional manifolds of periodic orbits across regular and degenerate Hopf bifurcations in delay differential equations. The latter functionality has been incorporated into an existing MATLAB library for validated numerics of periodic orbits. The code is general-purpose and can be largely treated as a black box, computing all necessary technical bounds to complete validated continuation of periodic orbits in a given delay differential equation. We will conclude the talk with a live demo.

This is joint work with Elena Queirolo.