# Conley index approach to sampled dynamics

## TL;DR

This paper enhances topological methods for reconstructing dynamics from time series data by using multivalued maps and a new Conley index theory, improving applicability to sparse and expanding data.

## Contribution

It introduces a multivalued map approach combined with a novel Conley index theory for better dynamical reconstruction from limited data.

## Key findings

- Effective in sparse data scenarios
- Handles strong expansion in dynamics
- Numerical validation with Henon maps

## Abstract

The topological method for the reconstruction of dynamics from time series [K. Mischaikow, M. Mrozek, J. Reiss, A. Szymczak. Construction of Symbolic Dynamics from Experimental Time Series, Physical Review Letters, 82 (1999), 1144-1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley index theory for multivalued maps [B. Batko and M. Mrozek. Weak index pairs and the Conley index for discrete multivalued dynamical systems, SIAM J. Applied Dynamical Systems 15 (2016), 1143-1162], [B.Batko. Weak index pairs and the Conley index for discrete multivalued dynamical systems. Part II: properties of the Index, SIAM J. Applied Dynamical Systems 16 (2017), 1587-1617] is used in computations. The existence of a continuous, single-valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of H\'enon type maps are presented.