# Convergent Chaos

## TL;DR

This paper refines the understanding of chaos by demonstrating that trajectories can exhibit strong convergence over long periods, challenging the traditional view of chaos as purely divergent, with implications for various fields.

## Contribution

It introduces the concept of 'convergent chaos', showing that chaotic systems can have long-term strong convergence, supported by statistical analysis and modeling.

## Key findings

- Trajectories can be strongly convergent over long times
- Clustering in phase space is intense and widespread
- Power-law distributions characterize particle clustering

## Abstract

Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we demonstrate that this criterion should be refined. Despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods, as revealed by our investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow). We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions. Power laws, indicative of scale-free features, characterise the distribution of particles in the system. We use large-deviation and extreme-value statistics to explain the effect. Our results show that the interpretation of the 'butterfly effect' needs to be carefully qualified. We argue that the combination of mixing and clustering processes makes our specific model relevant to understanding the evolution of simple organisms. Lastly, this notion of 'convergent chaos', which implies the existence of conditions for which uncertainties are unexpectedly small, may also be relevant to the valuation of insurance and futures contracts.