Sparse Map of Dynamics (MoD)

• Sparse Map of Dynamics (MoD) is a framework that reconstructs the qualitative structure of continuous dynamical systems from limited, possibly noisy time series data.
• It discretizes phase space into cubical grids and constructs a sparse, multivalued transition graph to capture and certify invariant dynamics using Conley index theory.
• The method is computationally scalable and robust, ensuring persistence of invariant sets under noise and parameter variations.

A Sparse Map of Dynamics (MoD) is a computational framework that reconstructs the qualitative and topological features of an underlying continuous dynamical system from sparse and possibly noisy experimental or simulated time-series data. The approach achieves this by discretizing phase space, extracting combinatorial dynamics through multivalued transition graphs, and applying Conley index theory tailored for discrete multivalued maps. This construction rigorously isolates recurrent or invariant structures (such as fixed points, periodic orbits, and symbolic dynamics) and certifies their existence in the underlying true system, even when only limited data is available.

1. Discretization and Multivalued Map Construction

Let $\{x_k\in\mathbb{R}^d\mid k=0,1,\dots,N\}$ be a time series generated by sampling an unknown map $f : X \subset \mathbb{R}^d \rightarrow X$, possibly with noise. The phase space $X$ is covered with a regular cubical grid of mesh size $\delta>0$, yielding cubes $$Q_i = \prod_{\ell=1}^d [a_{i,\ell},a_{i,\ell}+\delta]$$ for $i\in I$. For each adjacent sample pair $(x_k,x_{k+1})$, the containing cubes $Q_i\ni x_k$, $Q_j\ni x_{k+1}$ are determined. To allow for noise and uncertainties, $x_{k+1}$ is allowed to fall within an expanded neighborhood $B(Q_j,\rho)$ of $Q_j$, where $\rho$ is a chosen covering radius.

A multivalued map $F:Q\rightrightarrows Q$ is then constructed as

$$F(Q_i) = \left\{ Q_j \in Q \;\bigg|\; \exists\,k : x_k \in Q_i,\; x_{k+1} \in B(Q_j, \rho) \right\}.$$

This map is efficiently represented as a directed graph on $I$, with a directed edge $i \rightarrow j$ whenever $Q_j \in F(Q_i)$. The resulting graph is sparse, as typically only a small subset of cube pairs are ever connected by observed transitions.

2. Isolated Invariant Sets and the Weak Index Pair

The search for meaningful dynamic behavior proceeds through identification of isolated invariant sets. For a multivalued map, a set $S \subset \bigcup Q$ is an isolated invariant set if there exists a cubical neighborhood $N$ such that

$$\operatorname{Inv}(N;F) = \{x\in N\mid \exists \text{ bi-infinite } F\text{-trajectory in }N\} = S, \quad S \subset \operatorname{int}(N).$$

A weak index pair $(N,L)$—with $L\subset N$—satisfies three conditions ensuring $N\setminus L$ is an isolating neighborhood for $S$ and that boundary effects are correctly handled for the purposes of Conley index computation.

The precise conditions are:

• $$F(N)\cap\overline{N\setminus L}\subset \operatorname{int}(N)$$,
• $$F(L)\cap\overline{N\setminus L} \subset \operatorname{int}(L)$$,
• $$\operatorname{Inv}(N\setminus L;F)\subset \operatorname{int}(N\setminus L)$$.

3. Conley Index for Multivalued Dynamics

Given a weak index pair, the (discrete) Conley index $h(S,F)$ is defined on the relative homology $H_*(N,L)$, with an associated index map $F_*: H_*(N,L)\rightarrow H_*(N,L)$ induced by the pairwise inclusion $(N,L)\hookrightarrow (F(N),F(L))$. The topological index, which includes both the homology and the induced map, is shift-equivalent across all weak index pairs isolating $S$ and thus provides a robust descriptor of the underlying dynamics.

A principal theoretical guarantee is the existence, for small enough $\delta$, of a continuous single-valued selector $\widetilde f$ with $\mathrm{graph}\, \widetilde f$ contained in an arbitrarily small neighborhood of $\mathrm{graph}\,F$, such that all isolated invariant sets and their Conley indices persist under $\widetilde f$ (Batko et al., 2019). Therefore, the computed indices are true certificates for existence and type of invariant sets in the original, possibly unknown, dynamics.

4. Algorithms and Computational Complexity

The essential computational pipeline consists of:

• Construction of $F$ (BuildF): For $N$ data points and $M$ cubes, locating the cell for each $x_k$ uses e.g., a kd-tree in $O(\log M)$ time. The entire sparse transition graph can be constructed in $O(N\log M)$ and compressed to a size proportional to the number of observed transitions $E$.
• Strongly Connected Components and Isolation: Tarjan’s or Kosaraju’s algorithm identifies strongly connected components (SCCs) in $O(M+E)$ time. Candidate invariant sets correspond to SCCs of the transition graph, from which isolating neighborhoods and boundary sets are extracted.
• Homology and Index Map: Cubical or simplicial homology computations on $(N,L)$ pairs use standard matrix reduction methods, typically scaling $O(K^3)$ with $K$ the number of cells in $N\setminus L$; $K$ is typically small in sparse data regimes. The index map $F_*$ is built by tracking correspondences under $F$.

This algorithmic structure is scalable. Sparse visitation of phase space (common in high-dimensional or short data sets) ensures that $E\ll M^2$ and SCCs are locally restricted, enabling practical analysis in dimensions up to at least $d=4$.

5. Empirical Illustrations and Robustness

Numerical studies illustrate the method's utility and robustness:

• For the Hénon map with parameter $(a,b)=(1.4,0.3)$, using $N=2000$ samples and a $100\times 100$ grid, only $\sim200$ cells are visited. SCC extraction detects two principal recurrent regions: one corresponding to the strange attractor (with $H_0\simeq\mathbb{Z}$, $H_1\simeq\mathbb{Z}^2$, hyperbolic index) and one to an attracting fixed point.
• In the presence of Gaussian observation noise ($\sigma=10^{-2}$), enlarging the covering radius $\rho$ maintains identification of these invariants and their indices.
• In cases where topological horseshoes or symbolic dynamics exist, the shifted equivalence class of $F_*$ in homology can evidence positive entropy and complex recurrence.

6. Parameter Tuning and Theoretical Guarantees

Selection of mesh size $\delta$ and covering radius $\rho$ is critical. Excessively coarse $\delta$ merges distinct phenomena; overly fine $\delta$ leads to unvisited cubes and fragmentation. Empirical guidance is to choose $\delta$ so that the fraction of visited cubes is $1$–$5\%$ of $M$. The covering radius $\rho$ should exceed the maximal noise and local dynamical expansion on cells, mitigating both noise and under-sampling. Stability of index computation under small parameter changes is used for cross-validation.

The topological underpinnings are secured by the upper-semicontinuity and acyclic-value properties of $F$: any continuous selector within a neighborhood of $\mathrm{graph} F$ preserves isolated invariants and indices. As such, the MoD approach yields provable lower bounds on the number and types of true invariant sets in the underlying continuous system, regardless of data sparsity and moderate noise.

7. Scope, Limitations, and Extensions

Sparse MoD produces a combinatorial skeleton of the true dynamics—certifying fixed points, periodic orbits, symbolic shifts, and their type—directly from data. The method’s performance depends on appropriate discretization and covering, and the resolution of highly localized dynamical phenomena may be limited by data density. However, by systematically leveraging sparsity, multivalued representations, and modern discrete Conley theory, the approach enables rigorous dynamical analysis in regimes previously inaccessible to classical method-of-images or density-based reconstruction, providing foundations for topological time-series validation and hypothesis testing in applied dynamics (Batko et al., 2019).