Extended dynamic mode decomposition for two paradigms of non-linear dynamical systems

Abstract

We apply the Koopman operator theory and Extended Dynamic Mode Decomposition in two non-linear dynamical systems. The first one is the 3x+1-system which follows by the so-called Collatz conjecture and it is discrete. The second one is the SIR-model which is continuous and is used to describe the evolution of a disease through some population. Koopman operator captures the dynamics of a non-linear system, however it is infinite dimensional. In this study, we approximate the aforementioned systems with finite dimensional linear systems which are defined on some augmented state space. In the case of the Collatz system, this approach is expected to give some more insight to the 3x+1-problem. In the case of SIR-model, using some suitable dictionary of observables, we obtain a linear system which approximates the trajectories of the system and can be used for prediction and control purposes.