Related papers: Tracking Dynamical Features via Continuation and P…

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector…

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical…

We introduce a persistence-type invariant for finite weighted graphs based on combinatorial multivector dynamics. For each threshold parameter, a relation matrix determines a graph multivector field, whose induced directed dynamics admits a…

We investigate combinatorial dynamical systems on simplicial complexes considered as {\em finite topological spaces}. Such systems arise in a natural way from sampling dynamics and may be used to reconstruct some features of the dynamics…

This work presents new tools for studying reachability and set invariance for continuous-time mixed-monotone dynamical systems subject to a disturbance input. The vector field of a mixed-monotone system is decomposable via a decomposition…

The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…

In this paper, we introduce a novel persistence framework for Morse decompositions in Markov chains using combinatorial multivector fields. Our approach provides a structured method to analyze recurrence and stability in finite-state…

A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…

With the tracking condition, the stability of quintessence solutions are examined. It is found that there is only one physically relevant fixed point for the system generically. Two specific examples of quintessence potentials are worked…

Analyzing singular patterns in vector fields is a fundamental problem in theoretical and practical domains due to the ability of such patterns to detect the intrinsic characteristics of vector fields. In this study, we propose an approach…

Autonomous vehicles often perceive the environment by feeding sensor data to a learned detector algorithm, then feeding detections to a multi-object tracker that models object motions over time. Probabilistic models of multi-object trackers…

This note proposes a general control approach, called vector-field guided constraint-following control, to solve the dynamics control problem of geometric path-following for a class of uncertain mechanical systems. More specifically, it…

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove…

Continuation of algebraic structures in families of dynamical systems is described using category theory, sheaves, and lattice algebras. Well-known concepts in dynamics, such as attractors or invariant sets, are formulated as functors on…

The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…

We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets,…

In this paper, we consider topological featurizations of data defined over simplicial complexes, like images and labeled graphs, obtained by convolving this data with various filters before computing persistence. Viewing a convolution…

In this paper, we investigate distributed multi-agent tracking of a convex set specified by multiple moving leaders with unmeasurable velocities. Various jointly-connected interaction topologies of the follower agents with uncertainties are…

In this paper, we consider the problem of invariant set computation for black-box switched linear systems using merely a finite set of observations of system trajectories. In particular, this paper focuses on polyhedral invariant sets. We…

Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard…