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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - COEXIST (Coexistence on the boundary of chaos)

Teaser

Summary

\"The project concerns renormalization phenomena in dynamical systems. They are important to understand because they describe how physical systems change from stable to unstable, or chaotic, behavior.

The potential impacts this research can have on society are potentially great as it is fundamental research. However, it is not the point of the project to have an immediate impact, but rather to lay the groundwork for future investigations.

The overall objectives are to prove understand the limit set of renormalization operators.

The conclusion of the action is that renormalization phenomena can be much more intricate than previously thought. A striking example of this was published in the article \"\"The Lorenz Renormalization Conjecture\"\". The action also investigated renormalization in cricial circle mappings but this is still ongoing research.\"

Work performed

\"Work performed mostly focused on: (1) renormalization of multicriticical circle maps, (2) renormalization of Lorenz maps, and (3) the dynamics of algorithms in reinforcement learning.

1. This work is in collaboration with Prof. Pablo Guarino at UFF, Brazil. Current methods for analyzing critical circle maps rely on complex methods that do not easily extend to arbitrary numbers of critical points and arbitrary exponents. Our research investigated how the methods of M. Martens for unimodal maps could be adapted to the multicritical circle case. This is still work in progress.

2. To investigate the possible renormalization behavior for Lorenz map a numerical experiment was carried out. Current methods are not able to rigorously prove the behavior of these systems so a conjecture was formulated based on the result of the numerics. This is published in the article \"\"The Lorenz Renormalization Conjecture\"\". It was found that the behavior of these operators is much more intricate than previously thought. This research was presented at Warwick University, UK, UNSW, Australia, and Inishmore, Ireland.

3. Reinformcement learning algorithms seek to train an agent to learn about its environment by maximizing future expected returns. Together with S. van Strien the learning problem for two agents playing an iterated prisoners dilemma game was investigated. Previous research indicate that these algorithms may exhibit chaotic behavior but this was never proven rigorously. We started applying dynamical systems techniques to understand whether this is truly the case or not. An article is under preparation.\"

Final results

The research into Lorenz renormalization is completely original. It was previously thought that these systems would behave like unimodal renormalization but our research shows that this is not the case.

The potential impact of this project is hard to estimate since it is about fundamental research. Understanding the real world phenomena that motivate this problems are far away, instead here a focus is upon providing a solid foundation for potential future research.