Many processes (aka dynamical systems) in nature or technology exhibit complicated behavior. In particular, it is often inherently impossible to reliably predict future states of the process over long time spans. Reliable predictions, however, can be made on statistical or otherwise coarsened properties of the system. In this project, we construct coarse models of the behavior of a system which, in contrast to existing approaches, incorporate information about cycling motion, generalizing the classic notion of periodic or quasiperiodic dynamics.
We combine cohomological information about the invariant set with dynamical information (either from an underlying model or in form of time series data) in order to extract subsets on which the system performs cycling motion. Algorithmically, we first discretize the invariant set (e.g., based on a subcomplex of a cubical grid covering a trajectory) in order to infer its first cohomology, which can be identified with homotopy classes of maps to the circle. We then extract cohomology classes which identify cycling motion and use these to construct the coarse model. In particular, we will address the following challenges:
- Dynamically relevant cohomology classes: We need to reliably extract those classes with respect to which the system exhibits cycling motion. In particular, this identification shall be robust to discretization or (in the case of time series data) sampling artifacts.
- Stability of classification: Our method relies on classifying elementary cubical domains according to (locally) cycling and transient behavior, reflecting the transition between regions with different cycling behavior. We aim to develop methods that achieve a stable classification.
- Continuous limit: Since the algorithmic construction of the coarse model necessarily yields a discrete model, it is not clear a priori what the continuous object is that we approximate as the discretization becomes finer. We will begin to answer this question by starting from a continuous system with well-understood dynamics, like a pendulum system.
- Time series data: We aim for a method which is applicable in a model-free setting, relying on time series data only. This raises questions of how to balance the spatial discretization with the temporal resolution, and of stability of the method with regard to perturbations of the input. In order to address this, we plan to investigate the use of persistent cohomology groups with integer coefficients, i.e., the image of the homomorphism induced by the inclusion between two nested subcomplexes approximating the invariant set at different scales.
- Sampling conditions for cubical approximations: Under which conditions does some cubical covering have the same homology as the underlying invariant set? We actually plan to first answer this question for a (“static”) submanifold.