The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
Explains research on 3D dynamics and manifold topology.
problem Understanding obstructions for Anosov flows on 3-manifolds.
method Expository note on partially hyperbolic diffeomorphisms and Anosov flows.
result Margulis and Plante-Thurston's topological obstructions for Anosov flows.
Framework for analyzing dynamic topological changes in point clouds using persistent homology and dynamic optimal transport.
problem Analyzing transient structural reorganizations during dynamic phase transitions in time-evolutionary point clouds.
method Hierarchical dynamic evaluation framework driven by topological and hypergraph reconstruction strategy.
result Combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.
Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.
problem Dynamic behavior of mechanisms with changing kinematic topology.
method Presented new transition conditions for variable topology mechanisms using projected motion equations and Voronets equations.
result Results show the dynamic behavior of joint locking in 3R and 6DOF mechanisms.
We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…
In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathe…
New method identifies network dynamics and noise structure.
problem Estimating network and disturbance topologies in dynamic systems.
method Extended multi-step Sequential Linear Regression and Weighted Null Space Fitting methods.
result Consistent estimation of dynamic networks with reduced computational burden.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
Proposes a model for identifying edges in low-rank dynamical networks.
problem Inability of conventional methods to handle low-rank dynamical networks.
method Low rank dynamical network model with causal Wiener filtering.
result Consistent method for estimating all network edges.
The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.
problem Understanding the dynamics of period-doubling routes to chaos in complex systems.
method Introducing three topological invariants to describe the topology of period-doubling routes to chaos.
result Ascribed symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and dynamics of the Henon map.
Research predicts XRP price anomalies using graph topologies.
problem Forecasting extreme price movements in XRP cryptoasset.
method Analyzed topological features of XRP transaction graphs.
result Topological features indicate extreme price surges.
Topological surgery occurs in natural phenomena where two points are selected and attracting or repelling forces are applied. The two points are connected via an invisible `thread'. In order to model topologically such phenomena we introduce dynamics in 1-, 2- and 3-dimensional topological surgery, by means of attracti…
The paper studies dynamical properties in semigroups modulo ideals.
problem Analyzing shadowing, expansivity, and stability in semigroups with ideals.
method Investigates shadowing, expansivity, and stability properties in uniform transformation semigroups modulo an ideal.
result Establishes that if a semigroup exhibits shadowing and expansivity modulo an ideal, it is also topologically stable modulo that ideal.
Topological data analysis quantifies structural dynamics using persistent homology.
problem Analyzing the shape and topology of structural dynamics data.
method Topological Data Analysis (TDA) with persistent homology to quantify shape over scales.
result Persistent homology reveals significant changes in manifold shape due to damage, not temperature.
The paper proves metrizability and dynamics of Weil bundles.
problem Metrizability and dynamics of Weil bundles in differential geometry.
method Investigation of metrizability and dynamics of Weil bundles for smooth compact manifolds and Weil algebras.
result A canonical, complete, weighted metric \(\mathfrak{d}_w\) on \(M^\mathbf{A}\) that encodes geometry and deformations.
Study dynamics and topology of flows near non-saddle sets or W-sets.
problem Understanding the dynamics and topology of flows near specific invariant sets.
method Cohomological relations and global properties analysis.
result Dynamical classification of surfaces and robustness of non-saddle-sets.
The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.
problem Understanding the structure and dynamics of deep learning models.
method Topological dynamical systems, index theory, and computational homology.
result Neurons correspond to simplexes in a simplicial complex, and topological invariants can be computed.
We introduce dynamic asymptotic dimension, a notion of dimension for actions of discrete groups on locally compact spaces, and more generally for locally compact étale groupoids. We study our notion for minimal actions of the integer group, its relation with conditions used by Bartels, Lück, and Reich in the context of…
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
Topological method detects Hopf bifurcations from time series.
problem Detecting Hopf bifurcations in nonlinear systems from time series data.
method Persistent homology applied to Takens embedding for phase space reconstructions.
result A simple scalar topological functional identifies critical bifurcation points.
Study bounds topological entropy of toroidal attractors.
problem Bounding entropy of toroidal attractors.
method Analyzing topological properties of toroidal sets to bound entropy.
result Entropy of toroidal attractors is bounded from below.
This study uses persistent homology to analyze complex transitional networks from time series data.
problem Lack of effective tools to summarize complex topology in transitional networks.
method Persistent homology from topological data analysis applied to coarse-grained state-space networks (CGSSN).
result CGSSN improves dynamic state detection and noise robustness compared to other methods.
This work develops agents to learn generalizable policies for dynamic network environments.
problem Real-world network topologies change due to attackers, defenders, or system failures, leading to failures in adaptive ACD systems.
method Developing agents to learn generalizable policies across dynamic network environments.
result Agents can learn robust policies for dynamic network topologies and diverse attackers.
We survey results on compact Clifford-Klein forms of homogeneous spaces, with a focus on recent contributions and organized around approaches via topology, geometry and dynamics. In addition, we survey results on moduli spaces of compact forms.
Learning influence pathways of a network of dynamically related processes from observations is of considerable importance in many disciplines. In this article, influence networks of agents which interact dynamically via linear dependencies are considered. An algorithm for the reconstruction of the topology of interacti…
New example solves topological dynamics problem.
problem Embedding compact metric space into cubical shift.
method Borsuk-Ulam theorem, p-adic completions, equivariant Sullivan conjecture.
result Existence of a compact metric space not embeddable into a cubical shift.
The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting…
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
The theoretical explanation for deep neural network (DNN) is still an open problem. In this paper DNN is considered as a discrete-time dynamical system due to its layered structure. The complexity provided by the nonlinearity in the dynamics is analyzed in terms of topological entropy and chaos characterized by Lyapuno…
STS clarifies chaos and stochastic dynamics, linking algebraic topology and physics.
problem Chaos and stochastic dynamics in arbitrary form SDEs.
method Supersymmetric theory of stochastic dynamics (STS) using generalized transfer operator (GTO) and topological field theories (TFT).
result Positive 'pressure' in GTOs corresponds to spontaneous breakdown of topological supersymmetry, explaining 1/f noise.
Study of spectral invariants on CR contact manifolds with circle action.
problem Analytic torsion and eta-like invariants on CR contact manifolds.
method Interpret spectral series topologically and dynamically using Reeb flow.
result Spectral series can be interpreted both topologically and dynamically.
We address the issue of estimating the topology and dynamics of sparse linear dynamic networks in a hyperparameter-free setting. We propose a method to estimate the network dynamics in a computationally efficient and parameter tuning-free iterative framework known as SPICE (Sparse Iterative Covariance Estimation). The …
Contagions such as the spread of popular news stories, or infectious diseases, propagate in cascades over dynamic networks with unobservable topologies. However, "social signals" such as product purchase time, or blog entry timestamps are measurable, and implicitly depend on the underlying topology, making it possible …
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
Stable topological summary captures evolving dependency structure in dynamic Bayesian networks.
problem Missing larger-scale patterns in evolving dependency structures in dynamic Bayesian networks.
method Topological approach using Dynamic Bayesian Graphs and persistent homology.
result Stable topological summary (barcodes) captures evolving dependency structure in DBNs.
Proposes TSBP for matching topological signal distributions.
problem Matching signal distributions on topological domains.
method Topological Schrödinger Bridge (TSBP) with linear topology-aware stochastic dynamics.
result Derives closed-form topological SB (TSB) for Gaussian boundary distributions.
The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.
problem Understanding the dynamics and properties of composite symplectic Dehn twists.
method Analyzing the form of nonuniform hyperbolicity, growth of Floer cohomology, and classification of symplectic mapping classes.
result Composite symplectic Dehn twists exhibit positive topological entropy and exponential growth in Floer cohomology.
Deep learning detects bifurcations in dynamical systems.
problem Predicting catastrophic changes in dynamical systems across sciences.
method Data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries.
result Extracts topologically invariant features to detect bifurcation boundaries in unseen systems.
CMS formulation solves Poincare conjecture for all dimensions.
problem Poincaré Conjecture in higher dimensions.
method Calculus of moving surfaces (CMS) for evolving hypersurfaces.
result Compact simply connected hypersurfaces relax to constant mean curvature (CMC) manifolds.
Stable actions of hyperbolic groups on their boundaries.
problem Stability of group actions on boundaries.
method Dynamical coding and semi-conjugacy analysis.
result Topological stability of actions on hyperbolic group boundaries.
Graph neural controlled differential equations learn graph dynamics from vertex observations.
problem Predicting future states of dynamical systems on graphs with limited vertex data.
method Incorporates graph topology information into NCDE to predict graph dynamics.
result Informed NCDE requires fewer parameters and lower MAE compared to previous methods.
We present an empirical analysis of the network formed by the trade relationships between all world countries, or World Trade Web (WTW). Each (directed) link is weighted by the amount of wealth flowing between two countries, and each country is characterized by the value of its Gross Domestic Product (GDP). By analysin…
We consider the dynamics of a smart grid system characterized by widespread distributed generation and storage devices. We assume that agents are free to trade electric energy over the network and we focus on the emerging market dynamics. We consider three different models for the market dynamics for which we present a…
Paper tackles dynamic graph topology identification in time-varying graphs.
problem Dynamic graph topology identification in time-varying graphs.
method Proposes an online algorithm for time-varying optimization, with intrinsic temporal regularization.
result Demonstrates performance on Gaussian graphical model problem.
The paper deals with the problem of reconstructing the topological structure of a network of dynamical systems. A distance function is defined in order to evaluate the "closeness" of two processes and a few useful mathematical properties are derived. Theoretical results to guarantee the correctness of the identificatio…
Survey on computational models in dynamical systems, including new universality concepts.
problem Understanding the relationship between computational models and dynamical systems.
method Review of recent works on Turing universality, Topological Kleene Field Theories, and dynamical bordisms.
result Introduction of new perspectives on computability through dynamical systems.
EvoNet predicts the evolution of dynamic graphs using a graph neural network and recurrent architecture.
problem Predicting the evolution of dynamic graphs is challenging and underexplored.
method EvoNet uses a graph neural network and recurrent architecture to predict the evolution of dynamic graphs.
result EvoNet effectively predicts the evolution of dynamic graphs on both artificial and real-world datasets.
We survey the 19th century development of the signature of a quadratic form, and the applications in the 20th and 21st century to the topology of manifolds and dynamical systems. Version 2 is an expanded and corrected version of Version 1, including an Appendix by the second named author "Algebraic L-theory of rings wi…