B01
Complexification of Discrete Time

This project deals with the study of complexified time in ordinary differential equations and other systems that continuously propagate in time. Structural aspects (underlying Riemann surfaces, monodromy, singularities) as well as algorithmic and simulation aspects (complexified integrators, problem of path jumping, discretization, superconvergence effects) will be addressed. One of the main aims of the project, is the development of new generic integration methods that deal with singular situations (or nearly singular situations) by circumventing them in the complex plane rather than applying standard methods like decreasing the stepsize. This kind of complex integrators also allows for a deeper study of the solution structure of a differential equation by providing information on the underlying Riemann surface. In synergy with the project C1 the project also aims for a general algorithmic framework that applies to the treatment of related problems.

One starting point of the project is the experience with complex detour techniques in the context of situations in dynamic geometry . Dynamic geometry deals with geometric configurations (defined by a collection of constraints) and the modeling of consistent and continuous (i.e. constraint-stable) movements in their configuration space. A typical problem in this context arises if the configuration approaches a singular point of the configuration space, there an intrinsic ambiguity may cause unnatural behavior in the algorithmic simulation of such a configuration. The problem in this context can be completely resolved by embedding the original configuration into a ambient complex space and avoid the singularities by performing complex detours . Besides interesting theoretical insights in the structure of geometric configuration spaces these techniques allow for a satisfactory algorithmic treatment of such problems.

The long term aim of the presented research project is to transfer as much as possible from this approach to the situation of ordinary differential equations. In the situation of ODEs one usually assumes a continuous and real propagation of time. Singular or nearly singular situations may make the numerical simulation of an ODE hard or even impossible (like a simulated particle that is moving directly towards the center of an attracting singular force field). Considering the time as a complex variable introduces new freedom that allows for a resolution of the singularity. Singular situations may by this be understood in a broader framework and for nearly singular situations more stable discretization schemes may be found by this approach. Although this approach seems natural and well motivated, there seems to be only few literature on this topic.

Several fundamental difficulties arise that at first sight form fundamental obstacles in such a kind of approach. Resolving these difficulties will be a major research goal in this project. Here we list a few corner stones:

• Avoidance of path jumping: Typical numerical integrates for ODEs (like for instance Runge Kutta methods) involve many evaluations of the right side of the ODE y’(t) = f (g(t)). If the function f is allowed to branch then also the simulation method has to be adapted to this potential branching situation, otherwise path jumping of the solution may occur.
• Understanding monodromy: The complex interpretation of a singular situation may introduce new monodromy into the problem. This monodromy must be understood as part of the structure inherent to a specific ODE and the numerical simulation has to be adapted to the existence of this monodromy.
• Analyzing examples; algorithmic complexity: As soon as reliable algorithmic methods in the above sense are available we will investigate specific interesting examples. We also aim for intrinsic algorithmic bounds for these examples. The extraction of crucial examples that can serve as benchmarks for the developed methods is also of relevance.
• Structure preserving discretization: In particular (and in relation with project A1) we aim for the development for structure preserving or symplectic discretization techniques that still work well for ambient complex spaces of an ODE.