Conley Index Approach to Sampled Dynamics

The topological method for the reconstruction of dynamics from time series [K. Mischaikow et al., Phys. Rev. Lett., 82 (1999), pp. 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, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1587-1617; B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143-1162] 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 Hénon-type maps are presented.