Dynamic System Mapping: Methods & Applications

• Dynamic system mapping is a set of mathematical and computational techniques that reconstruct and analyze evolving systems from data and models.
• It integrates geometric, algebraic, and statistical tools to enable applications in robotics, biological modeling, and network science.
• Practical implementations include state-space reconstruction, graph fibrations, category-theoretic frameworks, and model-free dynamic phase mapping to capture regime transitions.

Dynamic system mapping refers to a broad class of mathematical, algorithmic, and computational techniques for reconstructing, analyzing, and visualizing the structure and evolution of dynamical systems from data or models. The domain spans direct system-theoretic mapping between state spaces, parametric phase transitioning (bifurcation) studies, graph-based and category-theoretic system decompositions, and causal inference from multivariate time series. In modern computational practice, dynamic mapping is also critical in robotics (for online structural and semantic scene mapping), biological modeling (regime discovery, attractor mapping), network science (mapping systems of systems), and data-driven science (model-free phase diagram extraction). Approaches unify geometric, algebraic, and statistical principles to capture temporal evolution, topological structure, and inter-object/variable relations.

1. Mathematical Foundations and State-Space Reconstruction

Fundamental dynamic system mapping constructions originate in state-space theory. For continuous-time ordinary or partial differential equations, describing flows on manifolds or function spaces, system mapping is often formalized by specifying a tuple $(M, X)$, where $M$ is a state space manifold and $X : M \to TM$ a vector field. Modern networked system theory encodes such systems as networks of manifolds, with phase-space assignments to graph nodes, and their interconnections determined by the directed graph's wiring (DeVille et al., 2012, DeVille et al., 2013).

For data-driven cases, dynamic mapping begins with time series (trajectories), aiming to reconstruct the governing system. Takens’ theorem provides the basis for state-space reconstruction: delay-coordinate embeddings are constructed from scalar observations to topologically reconstruct the attractor. For multivariate data, joint or stacked embeddings are leveraged to reconstruct cross-variable manifolds for coupled systems (Zhang et al., 6 Feb 2025).

Mathematically, mappings between dynamical systems may be (i) direct (smooth) maps intertwining vector fields (conjugacies or semi-conjugacies), (ii) network-level combinatorial maps (such as graph fibrations specifying subsystem projections or synchrony constraints), or (iii) computational, learned mappings capturing regime boundaries in latent space (Romeo et al., 30 Jan 2025).

2. Mapping in Networked and Modular Dynamical Systems

Dynamical systems composed of interacting subsystems are mathematically modeled via directed graphs where nodes represent local phase spaces (manifolds) and edges encode couplings (DeVille et al., 2012, DeVille et al., 2013). The total system phase space is a product of local manifolds, and system interconnection is realized via "open systems"—parameterized vector fields on these products.

A map between such networked systems is induced by a graph fibration: a graph morphism that preserves the input-tree structure of each node. Fibrations yield two canonical mapping modalities:

• Surjective graph fibrations induce synchrony subspaces (polydiagonals) in the phase space; their images are invariant subsystems where certain variables are synchronized. The induced map is an embedding (DeVille et al., 2012, DeVille et al., 2013).
• Injective graph fibrations yield projections onto subsystems; the induced map is a surjective submersion, identifying "abstracted" or aggregated dynamics in the full system.

Operadic and double-category frameworks systematically capture how open-system interconnections and morphisms map between large, modular, or recursive system structures (Lerman, 2017), linking graph combinatorics to vector-field dynamics.

3. Data-Driven and Category-Theoretic Frameworks

When only discrete time-series data are available, dynamic system mapping may be framed as category-theoretic functorial reconstruction. "Dynamics, data and reconstruction" (Das et al., 2024) formalizes this: dynamical systems (as functor categories), observed/measured systems (comma categories), and data (functors from time-indexing categories to finite sequence categories) are related via a chain of functors. Proper dynamic system reconstruction from data is characterized as a functor $R: \mathrm{Data} \to \mathrm{Dyn}$, with universal properties articulated via Kan extensions.

The reconstruction is unique up to canonical equivalence when the data are "genuine" (i.e., arise from an observed system), and the Kan extension characterizes optimal reconstructions even in the presence of ambiguities due to indistinguishability or subsystem entanglement.

4. Dynamic Mapping in Robotic Perception and SLAM

In robotics, mapping dynamic environments involves distinguishing and tracking static and dynamic scene elements in real time. Recent systems realize dynamic mapping at several levels:

• Category- and instance-level SLAM: Systems like DyOb-SLAM implement parallel static and dynamic mapping branches. Instance masks (e.g., via Mask-RCNN), dense optical flow, and motion segmentation allow per-frame association of dynamic object IDs, separate mapping of static features, and continuous estimation of object trajectories and speeds. Mapping backend fuses keypoint triangulation (static) and trajectory optimization (dynamic) using bundle adjustment and motion smoothing (Wadud et al., 2022).
• Persistent object management: Bayesian filters and long-term data association (as in Changing-SLAM) maintain per-object persistence beliefs, enabling robust handling of object addition, removal, and relocation. Extended Kalman filters estimate object velocities, and fast keypoint filtering preserves feature density even in scenes with frequent occlusions (Soares et al., 2022).
• Dense volumetric mapping: Pipelines such as DynSLAM (Bârsan et al., 2019) use per-instance semantic segmentation and scene flow to build separate volumetric reconstructions of static backgrounds, moving objects, and potentially moving objects (e.g., parked vehicles). Each object is tracked, and map pruning via fixed-lag garbage collection ensures scalability.
• Geometric dynamic awareness: DUFOMap exploits void-region detection via ray casting—after a region is observed empty, subsequent occupancy marks are interpreted as dynamic, providing efficient, parameter-agnostic online dynamic/static labeling (Duberg et al., 2024).
• Open-vocabulary, dual-map frameworks: DualMap unifies semantic anchor-based global mapping and local, up-to-date concrete maps, supporting online open-vocabulary queries and efficient dynamic scene updates (Jiang et al., 2 Jun 2025).
• Unified optimization: DynoSAM constructs joint static/dynamic factor-graph optimization, simultaneously estimating camera poses, static and dynamic object trajectories, and latent structure. Motion factors are formulated to support both rigid and articulated objects, and flexible factorization enables downstream applications such as dynamic reconstruction and trajectory prediction (Morris et al., 21 Jan 2025).

5. Model-Free Dynamic Phase Mapping and Causal Inference

Modern model-free pipelines use contrastive or geometric learning to construct "dynamic maps" or "phase diagrams" from raw trajectory data without explicit model equations.

• Contrastive Cartography: Contrastive learning with invariance to invertible linear transforms enables the construction of a low-dimensional latent embedding that clusters trajectories by their qualitative dynamical regime. Clustering the embedding space yields empirical dynamical phase diagrams, recovering bifurcation boundaries and regime structure directly from data, robust even to moderate noise and high-dimensional systems (Romeo et al., 30 Jan 2025).
• Causal Structure in Dynamical Systems: Approaches like MXMap apply convergent cross mapping (CCM) and its multivariate extension (multiPCM) on observed time series to reconstruct the directed causal graph of the underlying dynamical system. This model-free framework reliably distinguishes direct and indirect causality even in strongly nonlinear, cyclic, or chaotic systems. Cross mapping correlation statistics (Pearson and partial correlations) are used to prune spurious links, producing interpretable and accurate dynamic system maps (Zhang et al., 6 Feb 2025).

6. Dynamic Relationship Mapping in Data Visualization

In multivariate data visualization, dynamic mapping generalizes multidimensional scaling (MDS), Sammon mapping, and t-SNE to sequences of objects whose relationships evolve over time:

• EvoMap and the evomap package: The EvoMap framework estimates time-indexed low-dimensional embeddings $\{X_t\}$ from a sequence of dissimilarity matrices $\{D_t\}$, jointly optimizing static fit at each $t$ and temporal coherence via smoothing penalties. This enables interpretation of object trajectories in embedding space, capturing temporal relationship dynamics and change-point structure. The evomap toolbox implements the framework for practical deployment, supporting data preprocessing, visualization, and quantitative temporal coherence diagnostics (Matthe, 6 Nov 2025).

7. Analytical and Numerical Regime Mapping

Dynamic system mapping in analytical models focuses on regime identification, topological transitions, and bifurcations:

• Parametric transition analysis: Explicitly parametrized ODE systems (e.g., the Alpha Group ODEs) are mapped by sweeping a system parameter (e.g., rotation $\theta$ in the structuring matrix) and solving for eigenvalue transitions. The resulting phase portraits (fixed points, periodic orbits, attractors at infinity) and bifurcation points directly define the regime mapping structure (Corrêa et al., 24 Jul 2025).
• Numerical tools: Fourth-order Runge–Kutta integration over parameter grids supports high-resolution mapping of critical transitions, attractors, and phase topologies.

---

Dynamic system mapping, as established across these traditions, unifies the geometric, algebraic, categorical, and statistical characterization of the structure, evolution, and interactions of dynamical systems. Whether via explicit model analysis, category-theoretic reconstruction, data-driven causal discovery, or robotic scene mapping, the field provides a multifaceted arsenal for extracting and interpreting the multi-scale, time-dependent structure of complex systems.