arXiv research

The paper develops algorithms and topological invariants for distinguishing dynamic systems.

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…

The paper studies bifurcations in discrete dynamical systems on manifolds.

Constructs algorithms to recognize and classify 2D surfaces.

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…

The abstract applies waist inequality to dynamical systems and entropy.

Deep learning detects bifurcations in dynamical systems.

The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.

The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.

A Riemmanian foliated dynamical system of 3-dimension $(\mathrm{RFDS}^{3})$ is a closed Riemannian 3-manifold with additional structures: foliation, dynamical system. In the context of arithmetic topology, it is a geometric/analytic analogue of an arithmetic scheme with a conjectural dynamical system suggested by C. De…

Topological method detects Hopf bifurcations from time series.

This work develops agents to learn generalizable policies for dynamic network environments.

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.

GD-VAEs learn dynamics from observations using geometric and topological information.

Extends rigidity results to non-homogeneous manifolds.

Paper tackles dynamic behavior of variable topology mechanisms, presenting new transition conditions.

Bayesian method infers network topology and dynamics from noisy, sparse measurements.

In the present work we analyse the dynamics of indirect connections between insurance companies that result from market price channels. In our analysis we assume that the stock quotations of insurance companies reflect market sentiments which constitute a very important systemic risk factor. Interlinkages between insur…

AdaCGP learns dynamic graph topology from time series data, improving over existing methods.

This paper classifies links in 3D dynamical systems.

This paper studies the Yang--Mills ASD equation over the cylinder as a non-linear evolution equation. We consider a dynamical system consisting of bounded orbits of this evolution equation. This system contains many chaotic orbits, and moreover it becomes an infinite dimensional and infinite entropy system. We study th…

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…

This study uses persistent homology to analyze complex transitional networks from time series data.

In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited …

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…

Framework for analyzing dynamic topological changes in point clouds using persistent homology and dynamic optimal transport.

Researchers develop multi-agent systems for quadcopters to collaborate in missions.

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…

This paper is part of the research on the interlinkages between insurers and their contribution to systemic risk on the insurance market. Its main purpose is to present the results of the analysis of linkage dynamics and systemic risk in the European insurance sector which are obtained using correlation networks. These…

Study global geometry of dynamical systems with entire vector fields.

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…

We study topological structures of the sets $(0,1/2)^3 \cap Ω$ and $(0,1/2)^3 \setminus Ω$, where~$Ω$ is one special algebraic surface defined by a symmetric polynomial in variables $a_1,a_2,a_3$ of degree~$12$. These problems arise in studying of general properties of degenerate singular points of dynamical systems ob…

New method detects dynamical system changes in time series data.

Final version. To appear in Discrete and Continuous Dynamical Systems - A.

Classical knot theory deals with {\em diagrams} and {\em invariants}. By means of horizontal {\em trisecants}, we construct a new theory of classical braids with invariants valued in {\em pictures}. These pictures are closely related to diagrams of the initial object. The main tool is the notion of {\em free $k$-braid …

New method identifies network dynamics and noise structure.

This text is about geometric structures imposed by robust dynamical behaviour. We explain recent results towards the classification of partially hyperbolic systems in dimension 3 using the theory of foliations and its interaction with topology. We also present recent examples which introduce a challenge in the classifi…

The topological classification of gradient like Morse-Smale vector fields and diffeomorphisms on 3-manifolds was obtained.

The paper explores how topological methods can reveal insights into electric charge distributions on knots.

Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how traj…

The financial market is a complex dynamical system composed of a large variety of intricate relationships between several entities, such as banks, corporations and institutions. At the heart of the system lies the stock exchange mechanism, which establishes a time-evolving network of trades among companies and individu…

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…

Develops VAEs for learning complex physical systems from data.

We study the dynamics of the vector field on an open surface given by the gradient of a Green's function. This dynamical approach enables us to show that this field induces an invariant decomposition of the surface as the union of a disk and a 1-skeleton that encodes the topology of the surface. We analyze the structur…

A growing body of studies on systemic risk in financial markets has emphasized the key importance of taking into consideration the complex interconnections among financial institutions. Much effort has been put in modeling the contagion dynamics of financial shocks, and to assess the resilience of specific financial ma…

Graph neural controlled differential equations learn graph dynamics from vertex observations.

Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromo…