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This paper concerns the computation and identification of the (homological) Conley index over the integers, in the context of discrete dynamical systems generated by continuous maps. We discuss the significance with respect to nonlinear dynamics of using integer, as opposed to field, coefficients. We translate the problem into the language of commutative ring theory. More precisely, we relate shift equivalence in the category of finitely generated abelian groups to the classification of $\mathbb{Z}[t]$-modules whose underlying abelian group is given. We provide tools to handle the classification problem, but also highlight the associated computational challenges.

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to $S^1$ (resp., to a bouquet of circles).

This paper proposes an integration of surrogate modeling and topology to significantly reduce the amount of data required to describe the underlying global dynamics of robot controllers, including closed-box ones. A Gaussian Process (GP), trained with randomized short trajectories over the state-space, acts as a surrogate model for the underlying dynamical system. Then, a combinatorial representation is built and used to describe the dynamics in the form of a directed acyclic graph, known as {\it Morse graph}. The Morse graph is able to describe the system's attractors and their corresponding regions of attraction (\roa). Furthermore, a pointwise confidence level of the global dynamics estimation over the entire state space is provided. In contrast to alternatives, the framework does not require estimation of Lyapunov functions, alleviating the need for high prediction accuracy of the GP. The framework is suitable for data-driven controllers that do not expose an analytical model as long as Lipschitz-continuity is satisfied. The method is compared against established analytical and recent machine learning alternatives for estimating \roa s, outperforming them in data efficiency without sacrificing accuracy. Link to code: https://go.rutgers.edu/49hy35en

We study a Weiner process that is conditioned to pass through a finite set of points and consider the dynamics generated by iterating a sample path from this process. Using topological techniques we are able to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, we compute the probability that this characterization is correct. This work is probabilistic in nature and intended to provide a theoretical foundation for the statistical analysis of dynamical systems which can only be queried via finite samples., 18 pages, 3 figures

In this paper we describe a combined combinatorial/numerical approach to studying equilibria and bifurcations in network models arising in Systems Biology. ODE models of the dynamics suffer from high dimensional parameters which presents a significant obstruction to studying the global dynamics via numerical methods. The main point of this paper is to demonstrate that combining classical techniques with recently developed combinatorial methods provides a richer picture of the global dynamics despite the high parameter dimension. Given a network topology describing state variables which regulate one another via monotone and bounded functions, we first use the Dynamic Signatures Generated by Regulatory Networks (DSGRN) software to obtain a combinatorial summary of the dynamics. This summary is coarse but global and we use this information as a first pass to identify "interesting'' subsets of parameters in which to focus. We construct an associated ODE model with high parameter dimension using our {\em Network Dynamics Modeling and Analysis} (NDMA) Python library. We introduce algorithms for efficiently investigating the dynamics in these ODE models restricted to these parameter subsets. Finally, we perform a statistical validation of the method and several interesting dynamical applications including finding saddle-node bifurcations in a 54 parameter model.

Understanding the global dynamics of a robot controller, such as identifying attractors and their regions of attraction (RoA), is important for safe deployment and synthesizing more effective hybrid controllers. This paper proposes a topological framework to analyze the global dynamics of robot controllers, even data-driven ones, in an effective and explainable way. It builds a combinatorial representation representing the underlying system's state space and non-linear dynamics, which is summarized in a directed acyclic graph, the Morse graph. The approach only probes the dynamics locally by forward propagating short trajectories over a state-space discretization, which needs to be a Lipschitz-continuous function. The framework is evaluated given either numerical or data-driven controllers for classical robotic benchmarks. It is compared against established analytical and recent machine learning alternatives for estimating the RoAs of such controllers. It is shown to outperform them in accuracy and efficiency. It also provides deeper insights as it describes the global dynamics up to the discretization's resolution. This allows to use the Morse graph to identify how to synthesize controllers to form improved hybrid solutions or how to identify the physical limitations of a robotic system.

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 behavior 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 using algebraic topological invariants (Conley index). The GP predictive distribution provides a lower bound on the confidence that these topological invariants, and hence the characterized dynamics, apply to the unknown dynamical system (a sample path of the GP). The focus of this paper is on explaining the ideas, thus we restrict our examples to one-dimensional systems and show how to capture the existence of fixed points, periodic orbits, connecting orbits, bistability, and chaotic dynamics.

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 which produce inconsistent predictions of the same test data points. This calls for techniques to be able to empirically quantify the difference in the trajectories and highlight problematic regions. While much focus is placed on discovering what models learn, the question of how a model learns is less studied beyond theoretical landscape characterizations and local geometric approximations near optimal conditions. Here, we present a toolkit for the Dynamical Organization Of Deep Learning Loss Landscapes, or DOODL3. DOODL3 formulates the training of neural networks as a dynamical system, analyzes the learning process, and presents an interpretable global view of trajectories in the loss landscape. Our approach uses the coarseness of topology to capture the granularity of geometry to mitigate against states of instability or elongated training. Overall, our analysis presents an empirical framework to extract the global dynamics of a model and to use that information to guide the training of neural networks., Comment: 9 pages, 3 figures, Supplementary

We introduce the notion of a template for discrete Morse theory. Templates provide a memory efficient approach to the computation of homological invariants (e.g., homology, persistent homology, Conley complexes) of cell complexes. We demonstrate the effectiveness of templates in two settings: first, by computing the homology of certain cubical complexes which are homotopy equivalent to $\mathbb{S}^d$ for $1\leq d \leq 20$, and second, by computing Conley complexes and connection matrices for a collection of examples arising from a Conley-Morse theory on spaces of braids diagrams., Comment: Updated timing information for improved version of PyCHomP

We describe a computational method for constructing a coarse combinatorial model of some dynamical system in which the macroscopic states are given by elementary cycling motions of the system. Our method is in particular applicable to time series data. We illustrate the construction by a perturbed double well Hamiltonian as well as the Lorenz system., Replacement of the accidentally submitted v2

To make progress in understanding the issue of memory loss and history dependence in evolving complex systems, we consider the mixing rate that specifies how fast the future states become independent of the initial condition. We propose a simple measure for assessing the mixing rate that can be directly applied to experimental data observed in any metric space $X$. For a compact phase space $X \subset R^M$, we prove the following statement. If the underlying dynamical system has a unique physical measure and its dynamics is strongly mixing with respect to this measure, then our method provides an upper bound of the mixing rate. We employ our method to analyze memory loss for the system of slowly sheared granular particles with a small inertial number $I$. The shear is induced by the moving walls as well as by the linear motion of the support surface that ensures approximately linear shear throughout the sample. We show that even if $I$ is kept fixed, the rate of memory loss (considered at the time scale given by the inverse shear rate) depends erratically on the shear rate. Our study suggests a presence of bifurcations at which the rate of memory loss increases with the shear rate while it decreases away from these points. We also find that the memory loss is not a smooth process. Its rate is closely related to frequency of the sudden transitions of the force network. The loss of memory, quantified by observing evolution of force networks, is found to be correlated with the loss of correlation of shear stress measured on the system scale. Thus, we have established a direct link between the evolution of force networks and macroscopic properties of the considered system.

We explore the chaotic dynamics of Rayleigh-B\'enard convection using large-scale, parallel numerical simulations for experimentally accessible conditions. We quantify the connections between the spatiotemporal dynamics of the leading-order Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the spatiotemporal features of the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. Using precision-recall curves, we quantify the complex relationship between the pattern diagnostics and the regions where the magnitude of the leading-order Lyapunov vector is significant.

We explore the chaotic dynamics of Rayleigh-B��nard convection using large-scale, parallel numerical simulations for experimentally accessible conditions. We quantify the connections between the spatiotemporal dynamics of the leading-order Lyapunov vector and different measures of the flow field pattern's topology and dynamics. We use a range of pattern diagnostics to describe the spatiotemporal features of the flow field structures which includes many of the traditional diagnostics used to describe convection as well as some diagnostics tailored to capture the dynamics of the patterns. Using precision-recall curves, we quantify the complex relationship between the pattern diagnostics and the regions where the magnitude of the leading-order Lyapunov vector is significant.

We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network (DSGRN) database. The motivation stems from urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and decades of parameters. The input to the database computations is a regulatory network, i.e.\ a directed graph with edges indicating up or down regulation, from which a computational model based on switching networks is generated. The phase space dimension equals the number of nodes. The associated parameter space consists of one parameter for each node (a decay rate), and three parameters for each edge (low and high levels of expression, and a threshold at which expression levels change). Since the nonlinearities of switching systems are piece-wise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamic signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5-nodes and a 29-dimensional parameter space.

We use persistent homology to build a quantitative understanding of large complex systems that are driven far-from-equilibrium; in particular, we analyze image time series of flow field patterns from numerical simulations of two important problems in fluid dynamics: Kolmogorov flow and Rayleigh-B��nard convection. For each image we compute a persistence diagram to yield a reduced description of the flow field; by applying different metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding flow patterns. We also examine the dynamics of the flow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an effective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.

The aim of this paper is to provide a short introduction to computational homology based on cubical complexes. The discussed topics include cubical complexes, a reduction algorithm for computing homology of finitely generated chain complexes, and an algorithmic construction of homology of continuous maps via multivalued acyclic representations.

Two implementations of a homology algorithm based on the Forman’s discrete Morse theory combined with the coreduction method are presented. Their efficiency is compared with other implementations of homology algorithms.

We use a quantitative topological characterization of complex dynamics to measure geometric structures. This approach is used to analyze the weakly turbulent state of spiral defect chaos in experiments on Rayleigh-Benard convection. Different attractors of spiral defect chaos are distinguished by their homology. The technique reveals pattern asymmetries that are not revealed using statistical measures. In addition we observe global stochastic ergodicity for system parameter values where locally chaotic dynamics has been observed previously., 5 pages, 5 figures

Algebraic topology (homology) is used to analyze the state of spiral defect chaos in both laboratory experiments and numerical simulations of Rayleigh-Bénard convection. The analysis reveals topological asymmetries that arise when non-Boussinesq effects are present. The asymmetries are found in different flow fields in the simulations and are robust to substantial alterations to flow visualization conditions in the experiment. However, the asymmetries are not observable using conventional statistical measures. These results suggest homology may provide a new and general approach for connecting spatiotemporal observations of chaotic or turbulent patterns to theoretical models.

We use Conley index theory to solve the singular boundary value problem $\varepsilon^2D u_{xx} + f(u,\varepsilon u_x,x) = 0$ on an interval $[-1,1]$, where $u \in \mathbb R^n$ and $D$ is a diagonal matrix, with separated boundary conditions. Since we use topological methods the assumptions we need are weaker then the standard set of assumptions. The Conley index theory is used here not for detection of an invariant set, but for tracking certain cohomological information, which guarantees existence of a solution to the boundary value problem.

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPAN

Die Akbulut-Kirby Sphären gibt eine unendliche Familie von (glatten) 4-Mannigfaltigkeiten, die diffeomorph zur Standardsphäre sind; sie haben jeweils eine handlebody Zersetzung, die durch eine bestimmte nicht-triviale Vorstellung der trivialen Gruppe beschrieben wird. Diese Sphären wurden als Kandidaten Gegenbeispiele zum reibungslosen Poincaré-Vermutung in Dimension 4. Die glatte Poincaré-Vermutung in Dimension 4 vorgeschlagen wird, ist noch offen heute, sondern die ganze Familie von Kugeln sind nicht mehr exotisch Kandidaten. In Dimension 4 die Kategorien PL und DIFF zusammenfallen, so PL Triangulierungen der Akbulut-Kirby Kugeln sind Standard PL 4-Sphären. Wir beschreiben drei verschiedene Möglichkeiten, explizite Triangulierungen der Akbulut-Kirby Sphären zu konstruieren. Aber um zu zeigen, dass diese PL Triangulierungen sind in der Tat PL 4-Sphären ist keine einfache Aufgabe. Das Problem der PL Kugel Anerkennung in Dimension 4 ist auch noch offen. In Dimension 3 ist Sphäre Erkennung bekannt, in der Komplexitätsklasse NP sein. In höheren Dimensionen sind PL Sphären nicht wiederzuerkennen. Es gibt jedoch heuristische Algorithmen, die Kugeln effektiv auf vielen Beispielen erkennen kann. Wir werden einige dieser Algorithmen und ihre Grenzen zu diskutieren. Unsere Experimente zeigen, dass die Heuristik - selbst wenn sie in Kombination verwendet werden - nicht in der Lage, den Großteil der Akbulut-Kirby erkennen trianguliert PL 4-Sphären. Wir werden auch kurz auf unsere Bau einer Triangulation von Mazur 4-Verteiler, die viele der Werkzeuge, die wir entwickelt, um die Dreiecks Akbulut-Kirby Kugeln bauen verwendet. Mazur 4-Verteilers ist eine zusammenziehbare 4-Verteiler, der eine Grenze, die eine Homologie Kugel, die nicht die Standard-3-Kugel hat. Wir verwenden diese Triangulation, eine zusammenklappbare nicht-PL 5-Ball und eine nicht-PL 5-Sphäre zu bauen. The Akbulut-Kirby spheres are an infinite family of (smooth) 4-manifolds that are diffeomorphic to the standard sphere; they each have a handlebody decomposition that is described by a certain nontrivial presentation of the trivial group. These spheres were proposed as candidate counterexamples to the smooth Poincare conjecture in dimension 4. The smooth Poincare conjecture in dimension 4 is still open today, but the entire family of spheres are no longer exotic candidates. In dimension 4, the categories PL and DIFF coincide, so PL triangulations of the Akbulut-Kirby spheres are standard PL 4-spheres. We describe three different ways to construct explicit triangulations of the Akbulut-Kirby spheres. However, to show that these PL triangulations are indeed PL 4-spheres is no simple task. The problem of PL sphere recognition in dimension 4 is also still open. In dimension 3, sphere recognition is known to be in the complexity class NP. In higher dimensions, PL spheres are unrecognizable. There are, however, heuristic algorithms that can recognize spheres effectively on many examples. We will discuss some of these algorithms and their limitations. Our experiments show that the heuristics—-even when used in combination—-were not able to recognize the majority of the Akbulut-Kirby triangulated PL 4-spheres. We will also briefly discuss our construction of a triangulation of Mazur's 4-manifold, which used many of the tools we designed to build the triangulated Akbulut-Kirby spheres. Mazur's 4-manifold is a contractible 4-manifold that has a boundary that is a homology sphere that is not the standard 3-sphere. We use this triangulation to build a collapsible non-PL 5-ball and a non-PL 5-sphere.

Switching networks are a common model for biological systems, especiallyfor genetic transcription networks. Stuart Kaufman originally proposedthe usefulness of the Boolean framework, but much of the dynamical features there are not realizable in a continuous analogue. We introduce the notionof braid-like dynamics as a bridge between Boolean and continuous dynamics andstudy its importance in the local dynamics of ring and ring-like networks. We discussa near-theorem on the global dynamics of general feedback networks, and in the finalchapter study the main ideas of this thesis in models of a yeast cell transcription network.