Cycling signatures: Identifying cycling motions in time series using algebraic topology

Figure 1.  Cycling segments in the classical Lorenz attractor. The black points come from numerical integration of the system over a large time span. Three segments are highlighted. The orange and blue segments belong to the same cycling class which should be different from the cycling class of the green segment

Figure 2.  a) A periodic orbit is a topological circle. b) Almost periodic time series segment (black) with thickening (blue). Only the thickening has nontrivial $ H_1 $

Figure 3.  Periodic segment (black) with thickening (blue) sampled from a squished periodic orbit (gray)

Figure 4.  Periodic segment (black) with unit tangent vectors (blue) sampled from a squished periodic orbit

Figure 5.  Time series with (simplified) comparison space. The time series (black dots) is covered with boxes, the union of the boxes is the comparison space $ Y $. Three cycling segments are highlighted (dark blue, dark green dark purple) together with a thickening (light blue, light green, light purple). The comparison space classifies these into two different classes, one containing the blue, the other containing the green and purple segments

Figure 6.  Comparison space in the unit tangent bundle. We use the $ S^1_\infty $ unit tangent bundle instead of $ S^1_2 $ in this figure (and for computations), since $ S^1_\infty $ is easier to cover with boxes than $ S^1_2 $. Each cluster of boxes corresponds to the cover of $ Q\times S^1_\infty $ where $ Q $ is a box in Fig. 5. The comparison space is the union all the black boxes

Figure 7.  Segments with higher-dimensional cycling spaces

Figure 8.  Segment with different thickenings. The light green set in a has rank 0, and the one in b has rank 1

Figure 9.  Time series for the Lorenz system (left), a stochastically perturbed Hamiltonian system with a double well potential (center) and the Dadras system projected to the first three coordinates (right). The color of the $ x_i $ changes gradually with time $ t_i $ to visualize time evolution

Figure 10.  Distribution of cycling ranks for segments of different time spans. Each plot consists of stacked barplots (one for each value of $ \tau $) showing the distribution of cycling ranks for the sampled segments. In all three plots, rank zero dominates for short time spans while long segments eventually attain the maximal possible cycling rank of the system

Figure 11.  Distribution of rank 1 segments for different time spans. For the double well and Dadras systems, some infrequent signatures are omitted

Figure 12.  Rank 1 segments in the Lorenz system. The full time series $ \Gamma $ is shown in black. The red points in each plot highlight a segment with cycling space $ V_1 $, $ V_2 $ and $ V_3 $, respectively

Figure 13.  Distribution of rank 2 signatures in the example systems

Figure 14.  Inclusion graphs. In each plot, the nodes in the bottom and top rows correspond to frequent rank 1 and 2 signatures, respectively. Node colors correspond to the ones in Fig. 11 and Fig. 13. An edge indicates that the rank 1 signature is a subspace of the rank 2 signature

Figure 15.  Distribution of cycling ranks for different segment lengths and evaluation radii. Each column corresponds to a dimension, each row to one of the example systems, i.e. the top left heat map corresponds to rank 0 segments in Lorenz

Figure 16.  Distribution of the most frequent 1d cycling spaces for different segment lengths and filtration radii

Figure 17.  Illustration of the map $ f_r $

Figure 18.  Rank distribution versus segment time span for a variety of parameter choices. All cycling signatures are evaluated at $ r = 4 $

Figure 19.  Rank 1 signature distribution versus segment time span for a variety of parameter choices. All cycling signatures are evaluated at $ r = 4 $

Figure 20.  Illustration of $ \operatorname{Dec} $ for a cubical cover of $ S^1_\infty $

Figure 21.  a) Internal and external balls at $ p $ and $ q $ with radius $ r $. Since the external balls intersect while the internal balls do not, the two sets are not homotopy equivalent. b) Illustration for $ \pi $ and $ \psi $. The map $ \pi $ projects $ B(p, \theta(r)) $, indicated in blue, radially onto $ S^1 $. The map $ \psi $ projects projects $ B^S(p, r) $, indicated in orange, radially onto the pink sphere with radius $ \tfrac{2-r^2}{2} $. The resulting arc is a deformation retract of $ B(p, \theta(r)) $ via a straight-line homotopy