  # B01Complexification of Discrete Time

## How Complex Time Helps to Make Algorithms and Simulations More Stable

Time may not always be considered to be a linear flow. If a time dependent problem is regular enough one may replace a one-dimensional (real) time variable by a two-dimensional (complex) time variable. By this it suddenly becomes possible to bypass critical singular or nearly-singular situations and to broaden the practical applicability of simulations. The research goal of B01 is to exploit these detours in time for problems in dynamic geometry and elementary particle mechanics.

#### Mission-

Many processes in geometry and dynamics can be considered as time dependent systems. For instance, dynamic geometry studies elementary geometric constructions in which the base elements of a construction may be moved and the dependent elements move accordingly and consistently. Such a movement may be considered as a process in time. In physics there are also time-dependent processes. One may think of a planet orbiting around a sun or an arrangement of atoms moving under mutual forces.
Both systems - moving geometric constructions and particle/force systems - suffer from the fundamental problem that singular situations, which occur at a specific moment in time, may prevent the system from being continued beyond this moment. For example, the orbit of a planet, directly heading into a sun, is stopped at the moment of the collision.
The philosophy in research project B01 expands† the concept of an ordinary linear time flow to a concept in two dimensions, where the time is represented by a variable over the complex numbers. Complex numbers are inherently two-dimensional objects, but in many aspects they behave like ordinary numbers. In particular all formulas governing geometric constructions or force/particle systems allow direct generalizations to complex numbers. Now, figuratively speaking, the two-dimensional nature of complex numbers allows us to bypass critical singular situations by so-called complex detours. At first, this seems to be a purely esoteric approach to geometry and physics. However, this approach, which is a central element in the research in B01, creates possibilities for new, more stable and consistent methods in dynamic geometry as well as for physics simulations.

#### Scientific Details+

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.

#### Publications+

##### Papers
• Stefan Kranich.
Generation of real algebraic loci via complex detours.
preprint, October 2015.
arXiv:1510.05464.
• Stefan Kranich.
GPU-based visualization of domain-coloured algebraic Riemann surfaces.
preprint, July 2015.
arXiv:1507.04571.
• Stefan Kranich.
An epsilon-delta bound for plane algebraic curves and its use for certified homotopy continuation of systems of plane algebraic curves.
preprint, May 2015.
arXiv:1505.03432.

#### Prof. Dr. Dr. Jürgen Richter-Gebert   +

Projects: CaP, Z02
University: TU München, Zentrum Mathematik, M10, 02.06.054
Address: Boltzmannstr. 3, D-85748 Garching bei München
Tel: +49 (89) 289 18354
E-Mail: richter[at]ma.tum.de
Website: http://www-m10.ma.tum.de/bin/view/Lehrstuhl/RichterGebert