50 citations · 53 across the 13 of their papers we have counts for
16 papers
Computation of Attractor Lattices for a Class of Hybrid Systems
William Kalies, Bernardo Rivas, Tony Wehbe
We study the attractor lattice of a class of hybrid dynamical systems on compact metric spaces. Under the trapping guard condition, we prove that hybrid attractors form a bounded d…
Characterizing High-dimensional Dynamics by Combinatorial-Topological Methods on a Latent Space
Patrick Bailon, Marcio Gameiro, Brittany Gelb +5
Combinatorial-topological methods for characterizing dynamics are rigorous, generalizable, computable, and they only require approximations, but the dimension of the phase space is…
Compositionality of Global Dynamics in Product and Skew-Product Systems
William D. Kalies, Tony Wehbe
We study the compositionality of global dynamics through attractor lattices and order structures of recurrent dynamics in product and skew-product systems using Conley theory. For…
Conley Index Theory for Hybrid Systems
Bernardo Rivas, William Kalies
We define a homological Conley index for a class of hybrid dynamical systems. This is achieved by factoring through the hybrid suspension semiflow, which views a class of hybrid dy…
Topological Dynamics via Learned Hybrid Systems
Bernardo Rivas, Kaito Iwasaki, William Kalies +2
The analysis of global dynamics, particularly the identification and characterization of attractors and their regions of attraction, is essential for complex nonlinear and hybrid s…
Data-driven Identification of Attractors Using Machine Learning
Marcio Gameiro, Brittany Gelb, William Kalies +3
In this paper we explore challenges in developing a topological framework in which machine learning can be used to robustly characterize global dynamics. Specifically, we focus on…
Priestley duality and representations of recurrent dynamics
William Kalies, Robert Vandervorst
For an arbitrary dynamical system there is a strong relationship between global dynamics and the order structure of an appropriately constructed Priestley space. This connection pr…
Identifying Nonlinear Dynamics with High Confidence from Sparse Data
Bogdan Batko, Marcio Gameiro, Ying Hung +3
We introduce a novel procedure that, given sparse data generated from a stationary deterministic nonlinear dynamical system, can characterize specific local and/or global dynamic b…
Extracting Global Dynamics of Loss Landscape in Deep Learning Models
Mohammed Eslami, Hamed Eramian, Marcio Gameiro +2
Deep learning models evolve through training to learn the manifold in which the data exists to satisfy an objective. It is well known that evolution leads to different final states…
Equilibria and their Stability in Networks with Steep Sigmoidal Nonlinearities
William Duncan, Tomas Gedeon, Hiroshi Kokubu +2
In this paper we investigate equilibria of continuous differential equation models of network dynamics. The motivation comes from gene regulatory networks where each directed edge…
Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation
K. Dowling, W. D. Kalies, R. C. A. M. Vandervorst
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 at…
Lattice Structures for Attractors III
William D. Kalies, Konstantin Mischaikow, Robert C. A. M. Vandervorst
The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattice…
Topological portraits of multiscale coordination dynamics
Mengsen Zhang, William D. Kalies, J. A. Scott Kelso +1
Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaning…
Analytic continuation of local (un)stable manifolds with rigorous computer assisted error bounds
William D. Kalies, Shane Kepley, J. D. Mireles James
We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The proc…
Lattice Structures for Attractors II
William D. Kalies, Konstantin Mischaikow, Robert C. A. M. Vandervorst
The algebraic structure of the attractors in a dynamical system determine much of its global dynamics. The collection of all attractors has a natural lattice structure, and this st…
Lattice Structures for Attractors I
William D. Kalies, Konstantin Mischaikow, Robert C. A. M. Vandervorst
We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like…