EXPOSÉS
Elizabeth Bradley (University of Colorado, Boulder, USA)
Computational topology techniques for characterizing time series data
Abstract
The first step in computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics using delay-coordinate embedding; the second is the construction of an appropriate simplicial complex from the results. The first of these steps involves a number of free parameters, though, and the computation of homology for a large number of simplices can be expensive. I will discuss some approaches for computing the homology efficiently and effectively, in the face of these challenges, using a range of examples from the classic Lorenz system through musical instruments and honeybee colonies to demonstrate the associated ideas.
Tamal Dey (Purdue University, USA)
Connection matrix: Computation, updates, and persistence
Abstract
Connection matrices, a generalization of Morse boundary operators from the classical Morse theory, capture the connection made by the flow among critical structures like attractors, repellers, and orbits in a vector field. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Recently, in the context of a combinatorial framework, an efficient persistence-like algorithm to compute connection matrices has been proposed in [DLMS24]. We show that, actually, the classical persistence algorithm in topological data analysis with exhaustive reduction indeed retrieves the connection matrix thus simplifying the algorithm of [DLMS24] even further. We also present an approach for updating a connection matrix under changes of a minimal Morse decomposition of the vector field in play. We conclude the talk by proposing a concept of persistence in the vector field derived from its connection matrix and presenting some preliminary experimental results.
[DLMS24] Computing Connection Matrices via Persistence-Like Reductions. Tamal K. Dey, Michal Lipinski, Marian Mrozek, and Ryan Slechta. SIAM J. Appl. Dyn. Syst., Vol. 23, No. 1, pages 81-97, 2024.
Bernardo Do Prado Rivas (Rutgers, USA)
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Marcio Gameiro (Rutgers, USA)
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Tomas Gedeon (Montana State, USA)
Combinatorial structure of continuous dynamics in gene regulatory networks
Abstract
We describe a close connection between two seemingly disparate types of gene regulatory network models: ordinary differential equations and monotone Boolean maps. We show that switching (Glass) models connect the world of continuous dynamics of ODEs and discrete dynamics of Boolean maps by describing connection between collection of all parameterizations of switching systems and collections of monotone Boolean maps compatible with the network structure. This connection allows combinatorial (i.e. finite) description of ODE dynamics over parameter space and allows conceptualization of bifurcation theory of Boolean maps.
We discuss an iterative construction of the set of all monotone Boolean maps with $k$ inputs that facilitates construction of parameter spaces of switching ODE systems
Oliver Junge (TUM, Germany)
Cycling signatures: Identifying cycling motions in time series using algebraic topology
Abstract
Recurrence is a fundamental characteristic of dynamical systems with complicated behavior. Understanding the inner structure of recurrence is challenging, especially if the system has many degrees of freedom and is subject to noise. We develop algebraic topological notions for identifying and classifying elementary recurrent motions – called cycling – and the transitions between those. Statistics on these cycling motions can be computed from sampled trajectories (time series data), providing coarse global information on the structure of the recurrent behavior. We demonstrate this through three examples; in particular, we identify and analyze six cycling motions in a four-dimensional system with a hyperchaotic attractor. We see this as a promising approach to reveal coarse-grained dynamical information on high-dimensional systems.
William Kalies (U. of Toledo, USA)
Attractor Sheaves: Computing Sheaf Cohomology for the Detection of Bifurcations
Abstract
Algebraic structures such as the lattices of attractors and Morse poset decompositions provide a computable description of global dynamics. For parameterized families of dynamical systems, these structures form a sheaf over the parameter space. Sheaf cohomology provides information about the global sections of the attractor sheaf and changes in this structure that detect bifurcations. Advancements in the theory and implementation of the Dynamic Signatures Generated by Regulatory Networks (DSGRN) framework have facilitated the computation of cellular sheaf cohomology across the parameter space. In particular, we can use attractor sheaves to discover paths that exhibit hysteresis, as well as higher-codimension bifurcations.
Rachel Kuske (Georgia Institute of Technology, USA)
Computer-assisted global analysis of non-smooth impact dynamics inspired by energy harvesting
Abstract
We discuss a novel return map approach for studying the global dynamics of a vibro-impact (VI) pair, that is, a ball moving in a harmonically forced capsule. Results are relevant for recent designs of VI-based energy harvesters and nonlinear energy transfer, and hold promise for other non-smooth systems. Computationally efficient short-time realizations are based on “words” that represent key impact sequences and divide the state space according to different dynamics. The resulting maps define surfaces that illuminate weak and strong attraction to certain periodic and complex states of interest for energy output. A small collection of reduced piece-wise polynomial approximations for the maps on these surfaces yields a composite map. They capture transients and focus the analysis on maps (surfaces) with attracting properties, confirming and complementing the bifurcation structure of the full system. Valuable for cobweb analysis, this framework inspires auxiliary maps based on the extreme bounds of the maps, yielding global dynamics of energetically favorable states. We apply this framework to understand how the surfaces explain stochastically enhanced grazing and bi-stability when the scales of noise and attraction compete.
Michal Lipinski (IST, Austria)
Perspectives on computing infinite dimensional (stable) manifolds in PDEs with validated numerics
Abstract
Dynamical systems offer a rich set of tools for analyzing the long-term behavior of solutions to evolutionary equations, and many finite dimensional techniques may be extended to analyze partial differential equations. However to develop a validated numerical approximation of stable manifolds, one is met with several difficulties: Proof techniques that work for finite dimensional problems or for discrete systems simply do not work for PDEs; and the stable manifold is an infinite dimensional object, so any sort of high-order approximation of it quickly encounters the curse of dimensionality. A way forward may be found with the Lyapunov-Perron method. While impractical to compute with, it allows one to set up a contraction mapping on the space of graphs over the stable eigenspace. A no-frills version of the theorem can be nicely obtained for a linear approximation of the stable manifold, but in order to take advantage of a higher order approximation one must leverage a detailed analysis of the semi-group operator in a transformed coordinate system. I will present how this approach may be applied to yield computer-assisted-proofs of stable manifolds in the Swift-Hohenberg equation, and discuss future directions and challenges.
Facundo Memoli (Ohio State U., USA)
Spatiotemporal persistent homology for dynamic metric spaces
Abstract
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in animals and social networks in the human sphere. A natural mathematical model for such collective behaviors is a dynamic point cloud, or more generally a dynamic metric space (DMS). We extend the Rips filtration stability result for (static) metric spaces to the setting of DMSs. We do this by devising a certain three-parameter “spatiotemporal” filtration of a DMS. Applying the homology functor to this filtration gives rise to multidimensional persistence module derived from the DMS. We show that this multidimensional module enjoys stability under a suitable generalization of the Gromov–Hausdorff distance which permits metrization of the collection of all DMSs.
On the other hand, it is recognized that, in general, comparing two multidimensional persistence modules leads to intractable computational problems. For the purpose of practical comparison of DMSs, we focus on both the rank invariant or the dimension function of the multidimensional persistence module that is derived from a DMS. We specifically propose to utilize a certain metric d for comparing these invariants: In our work this d is either (1) a certain generalization of the erosion distance by Patel, or (2) a specialized version of the well-known interleaving distance. In either case, the metric d can be computed in polynomial time.
Marian Mrozek (Jagiellonian U., Poland)
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Alessandro Pugliese (University of Bari, Italy)
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Luis Scoccola (Oxford, UK)
On the need for sparsity in topological inference
Abstract
In topological inference, one assumes that the input data approximates a topological object, with the goal of inferring topological properties of this object. As such, topological inference includes versions of the clustering problem (by interpreting clusters as connected components), density estimation (by considering the merge tree induced by suplevel sets), as well as manifold learning and dimensionality reduction. Global topological properties, such as number of components, loops or voids, are often quantified and computed using algebra, with the computational cost usually scaling poorly in the size and dimensionality of the input data. A main culprit for this is that most of the standard ways of building topological spaces (graphs or simplicial complexes) on data scale at best linearly in the data, and quite often quadratically or worse. I will overview some motivations for developing a theory of sparse simplicial complexes built on data, and I will describe an approach to sparsification based on approximate nerve theorems and topological optimization.
Robert Vandervorst (VU Amsterdam, Netherlands)
Priestley duality and representations of recurrent dynamics
Abstract
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 provides an order-theoretic framework for studying global dynamics. In the classical setting, the chain recurrent set, introduced by C. Conley, is an example of an ordered Stone space or Priestley space. Priestley duality can be applied in the setting of dynamics on arbitrary topological spaces and yields a notion of Hausdorff compactification of the (chain) recurrent set.
Ewerton Vieira (Rutgers University, USA)
Characterizing Global Dynamics from Sparse Data
Abstract
In this talk, we introduce a novel framework that, given sparse data generated by a stationary deterministic nonlinear dynamical system, can characterize global dynamic behaviors with rigorous probability guarantees. More precisely, the sparse data is used to construct a statistical surrogate model based on a Gaussian Process (GP). The dynamics of the surrogate model is interrogated using combinatorial methods and characterized through algebraic topological invariants (Conley index). The GP’s predictive distribution provides a lower bound on the confidence that these topological invariants, and therefore the characterized dynamics, apply to the unknown dynamical system. We apply this framework to significantly reduce the amount of data required to describe the underlying global dynamics of robot controllers.
Yanghong Yu (Tokyo Tech, Japan)
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