Conley Index Approach to Sampled Dynamics

Abstract

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.

Keywords: 37B30; 37M05; 37M10; 54C60; 54H20; Conley index; chaos; dynamical system; nonlinear dynamics; topological data analysis; topological semiconjugacy.

Figure 1.1.

Construction of an upper semicontinuous acyclic multivalued map F covering points representing the data.

Figure 1.2.

Domain of sunflower enclosure for $g_{\overline{x}}$ consisting of 1184 2-dimensional cubes, an isolating neighborhood (in dark sea green), its weak index pair (in blue violet), and the graph of transitions between components of an isolating neighborhood.

Figure 1.3.

Domain of sunflower enclosure for $g_{\overline{x}}$ consisting of 1029 3-dimensional cubes, an isolating neighborhood (in dark cyan), its weak index pair (in orange), and the graph of transitions between components of an isolating neighborhood.

Figure 1.4.

Domain of sunflower enclosure for $g_{\overline{x}}$ consisting of 106 2-dimensional cubes, an isolating neighborhood (in dark sea green), its weak index pair (P₁ in yellow, P₂ in green), and the graph of transitions between components of an isolating neighborhood. Lower dimensional cubes are enlarged to 2-dimensional cubes.