Workshop Details

Combinatorial representations of dynamical systems have been used to extract rigorous statements about global dynamics computationally. Recent results have addressed how robustly these representations may capture the underlying dynamical structure. Further development of these methods and algorithms relies on understanding the natural order structures in global dynamics related to attractors and Morse decompositions.

In this talk we present combinatorial order-theoretic models for global dynamics. We give computational examples that illustrate the theory for both maps and flows as well as applications to implicitly / imprecisely defined systems and systems measured from data. 

Video

We discuss combinatorial methods based on algebraic topology to extract global dynamics from a discretized dynamical system. We represent the discretized dynamical system as a map on a cell complex and use this map and the associated cell complex to extract the regions of interest and to compute algebraic topological indices that provide information about the dynamics. The main focus of this talk will be on software packages available to perform  these computations. We will also discuss some of the algorithms used and present examples for ODEs and maps.

Video

Models for evolutionary processes like physical systems are conceptualized via continuous  dynamical systems. However, in general, the model is unknown and only finite data is observed, hence, it is a significant challenge to learn a continuous function that describes the desired dynamical system robustly. To address this, data-driven dynamics are being increasingly employed in order to identify the underlying dynamics based on finite data.

To overcome the gap between the complexity of a continuous system and the  description based on finite data, we use a Gaussian process as a surrogate model together with combinatorial dynamics to capture the global behavior of the underlying function that generates the dynamics. After introducing the main ideas, I will show some applications to robotic systems.

Video