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A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set XK has n-simplices given by the simplicial For a fixed N, we analyze the space of all sequences z = (z¹, . . ., zN), 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 S¹. We also consider the space of functions on Y -shaped (resp., starshaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to S¹ (resp., to a bouquet of circles). [ABSTRACT FROM AUTHOR]

The algebraic structure of the attractors in a dynamical system determines much of its global dynamics. The collection of all attractors has a natural lattice structure, and this structure can be detected through attracting neighborhoods, which can in principle be computed. Indeed, there has been much recent work on developing and implementing general computational algorithms for global dynamics, which are capable of computing attracting neighborhoods efficiently. Here we address the question of whether all of the algebraic structure of attractors can be captured by these methods. [ABSTRACT FROM AUTHOR]

We provide explicit and efficient reduction algorithms based on discrete Morse theory to simplify homology computation for a very general class of complexes. A set-valued map of top-dimensional cells between such complexes is a natural discrete approximation of an underlying (and possibly unknown) continuous function, especially when the evaluation of that function is subject to measurement errors. We introduce a new Morse theoretic preprocessing framework for deriving chain maps from such set-valued maps, and hence provide an effective scheme for computing the morphism induced on homology by the approximated continuous function. [ABSTRACT FROM AUTHOR]

We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations. [ABSTRACT FROM AUTHOR]

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six. [ABSTRACT FROM AUTHOR]

The concept of activation in transcriptional regulation is based on the assumption that product mRNA increases monotonically as a function of regulator concentration. We analyze the Shea-Ackers model of transcription and find this assumption to be correct only for the simplest of promoters. We define a new regulatory constant that is a nonlinear combination of association and transcription initiation constants characterizing activation and repression for more complicated promoters. Our results can guide the synthesis of new promoters and lead to a deeper understanding of the constraints guiding the natural promoters evolution. [ABSTRACT FROM AUTHOR]

We describe the fine structure of the global attractor of the Cahn–Hilliard equation on two-dimensional square domains. This is accomplished by combining recent numerical results on the set of equilibrium solutions due to [Maier-Paape & Miller, 2002] with algebraic Conley index techniques. Using the information on the set of equilibria as assumption, we build Morse decompositions and connection matrices. The latter imply existence of heteroclinic connections between the equilibria inside the attractor. While path-following of the parameter range of Cahn–Hilliard, we find more and more complicated dynamical behavior. One of our main results describes the fine structure of the attractor for mean mass zero with four stable cosine structured equilibria and eight other stable equilibria that have a quarter circle nodal line. Besides that, we also study the attractor in symmetry fixed point spaces where we, for example, find nonunique connection matrices and saddle–saddle connections of Morse sets. [ABSTRACT FROM AUTHOR]

In this paper, we propose a rigorous computational method for detecting homoclinic tangencies and structurally unstable connecting orbits. It is a combination of several tools and algorithms, including the interval arithmetic, the subdivision algorithm, the Conley index theory, and the computational homology theory. As an example, we prove the existence of generic homoclinic tangencies in the Hénon family. [ABSTRACT FROM AUTHOR]

We introduce an efficient algorithm to compute the homomorphism induced in (relative) homology by a continous map. The algorithm is based on a cubical approximation of the map and the theory of multivalued maps. A software implementation of the algorithms introduced in this paper is available at [27]. [ABSTRACT FROM AUTHOR]

Abstract: Phase separation processes in compound materials can produce intriguing and complicated patterns. Yet, characterizing the geometry of these patterns quantitatively can be quite challenging. In this paper we propose the use of computational algebraic topology to obtain such a characterization. Our method is illustrated for the complex microstructures observed during spinodal decomposition and early coarsening in both the deterministic Cahn–Hilliard theory, as well as in the stochastic Cahn–Hilliard–Cook model. While both models produce microstructures that are qualitatively similar to the ones observed experimentally, our topological characterization points to significant differences. One particular aspect of our method is its ability to quantify boundary effects in finite size systems. [Copyright &y& Elsevier]

Considers a computational homology based on cubical complexes. Reduction algorithm for computing homology of finitely generated chain complexes; Algorithmic construction of homology of continuous maps.

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 (assumed to be 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. [ABSTRACT FROM AUTHOR]

History dependence of the evolution of complex systems plays an important role in forecasting. The precision of the predictions declines as the memory of the systems is lost. We propose a simple method for assessing the rate of memory loss that can be applied to experimental data observed in any metric space X. This rate indicates how fast the future states become independent of the initial condition. Under certain regularity conditions on the invariant measure of the dynamical system, we prove that our method provides an upper bound on the mixing rate of the system. This rate can be used to infer the longest time scale on which the predictions are still meaningful. We employ our method to analyze the memory loss of a slowly sheared granular system with a small inertial number I. We show that, even if I is kept fixed, the rate of memory loss depends erratically on the shear rate. Our study suggests the 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 rate of memory loss is closely related to the frequency of the sudden transitions of the force network. Moreover, the rate of memory loss is also well correlated with the loss of correlation of shear stress measured at the system scale. Thus, we have established a direct link between the evolution of force networks and the macroscopic properties of the considered system. [ABSTRACT FROM AUTHOR]

In this paper, we prove that Wright's equation y ′ ( t ) = − α y ( t − 1 ) { 1 + y ( t ) } has a unique slowly oscillating periodic solution (SOPS) for all parameter values α ∈ [ 1.9 , 6.0 ] , up to time translation. Our proof is based on the same strategy employed earlier by Xie [27] ; show that every SOPS is asymptotically stable. We first introduce a branch and bound algorithm to control all SOPS using bounding functions at all parameter values α ∈ [ 1.9 , 6.0 ] . Once the bounding functions are constructed, we then control the Floquet multipliers of all possible SOPS by solving rigorously an eigenvalue problem, again using a formulation introduced by Xie. Using these two main steps, we prove that all SOPS of Wright's equation are asymptotically stable for α ∈ [ 1.9 , 6.0 ] , and the proof follows. This result is a step toward the proof of the Jones' Conjecture formulated in 1962. [ABSTRACT FROM AUTHOR]

Large programs of dynamic gene expression, like cell cyles and circadian rhythms, are controlled by a relatively small "core" network of transcription factors and post-translational modifiers, working in concerted mutual regulation. Recent work suggests that system-independent, quantitative features of the dynamics of gene expression can be used to identify core regulators. We introduce an approach of iterative network hypothesis reduction from time-series data in which increasingly complex features of the dynamic expression of individual, pairs, and entire collections of genes are used to infer functional network models that can produce the observed transcriptional program. The culmination of our work is a computational pipeline, Iterative Network Hypothesis Reduction from Temporal Dynamics (Inherent dynamics pipeline), that provides a priority listing of targets for genetic perturbation to experimentally infer network structure. We demonstrate the capability of this integrated computational pipeline on synthetic and yeast cell-cycle data. Author summary: In this work we discuss a method for identifying promising experimental targets for genetic network inference by leveraging different features of time series gene expression data along a chained set of previously published software tools. We aim to locate small networks that control oscillations in the genome-wide expression profile in biological functions such as the circadian rhythm and the cell cycle. We infer the most promising targets for further experimentation, emphasizing that modeling and experimentation are an essential feedback loop for confident predictions of core network structure. Our major offering is the reduction of experimental time and expense by providing targeted guidance from computational methods for the inference of oscillating core networks, particularly in novel organisms. [ABSTRACT FROM AUTHOR]

We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points. [ABSTRACT FROM AUTHOR]

Understanding how the structure of within-system interactions affects the dynamics of the system is important in many areas of science. We extend a network dynamics modeling platform DSGRN, which combinatorializes both dynamics and parameter space to construct finite but accurate summaries of network dynamics, to new types of interactions. While the standard DSGRN assumes that each network edge controls the rate of abundance of the target node, the new edges may control either activity level or a decay rate of its target. While motivated by processes of post-transcriptional modification and ubiquitination in systems biology, our extension is applicable to the dynamics of any signed directed network. [ABSTRACT FROM AUTHOR]

Abstract: The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated. [Copyright &y& Elsevier]

One of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor-corrector continuation techniques. In the case of partial differential equations this procedure must be applied to some finite-dimensional approximation, which of course raises the question of the validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite-dimensional partial differential equation. Using the Cahn-Hilliard and Swift-Hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone. [ABSTRACT FROM AUTHOR]

Simple point detection is an important task for several problems in discrete geometry, such as topology preserving thinning in image processing to compute discrete skeletons. In this paper, the approach to simple point detection is based on techniques from cubical homology, a framework ideally suited for problems in image processing. A (d-dimensional) unitary cube (for a d-dimensional digital image) is associated with every discrete picture element, instead of a point in ϵd (the d-dimensional Euclidean space) as has been done previously. A simple point in this setting then refers to the removal of a unitary cube without changing the topology of the cubical complex induced by the digital image. The main result is a characterization of a simple point p (i.e., simple unitary cube) in terms of the homology groups of the (3d - 1) neighborhood of p for arbitrary, finite dimensions [ABSTRACT FROM AUTHOR]

Abstract: We use the Conley index theory to develop a general method to prove existence of periodic and heteroclinic orbits in a singularly perturbed system of ODEs. This is a continuation of the authors'' earlier work [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast–slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427–470] which is now extended to systems with multidimensional slow variables. The key new idea is the observation that the Conley index in fast–slow systems has a cohomological product structure. The factors in this product are the slow index, which captures information about the flow in the slow direction transverse to the slow flow, and the fast index, which is analogous to the Conley index for fast–slow systems with one-dimensional slow flow [T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, J. Reineck, The Conley index for fast–slow systems I: One-dimensional slow variable, J. Dynam. Differential Equations 11 (1999) 427–470]. [Copyright &y& Elsevier]

This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semiconjugacy between an attractor for the Swift--Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of computed branches as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index, also computed during this verification procedure, is then used to build a model for the attractor consisting of stationary solutions and connecting orbits. [ABSTRACT FROM AUTHOR]

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 represents either down- or up-regulation, and is modeled by a sigmoidal nonlinear function. We show that the existence and stability of equilibria of a sigmoidal system are determined by a combinatorial analysis of the limiting switching system with piecewise constant nonlinearities. In addition, we describe a local decomposition of a switching system into a product of simpler cyclic feedback systems, where the cycles in each decomposition correspond to a particular subset of network loops. [ABSTRACT FROM AUTHOR]

We demonstrate a modeling and computational framework that allows for rapid screening of thousands of potential network designs for particular dynamic behavior. To illustrate this capability we consider the problem of hysteresis, a prerequisite for construction of robust bistable switches and hence a cornerstone for construction of more complex synthetic circuits. We evaluate and rank most three node networks according to their ability to robustly exhibit hysteresis where robustness is measured with respect to parameters over multiple dynamic phenotypes. Focusing on the highest ranked networks, we demonstrate how additional robustness and design constraints can be applied. We compare our results to more traditional methods based on specific parameterization of ordinary differential equation models and demonstrate a strong qualitative match at a small fraction of the computational cost. Author summary: A major challenge in the domains of systems and synthetic biology is an inability to efficiently predict function(s) of complex networks. This work demonstrates a modeling and computational framework that allows for a mathematically justifiable rigorous screening of thousands of potential network designs for a wide variety of dynamical behavior. We screen all 3-node genetic networks and rank them based on their ability to act as an inducible bistable switch. Our results are summarized in a searchable database that can be used to construct robust switches. The ability to quickly screen thousands of designs significantly reduces the set of viable designs and allows synthetic biologists to focus their experimental and more traditional modeling tools to this much smaller set. [ABSTRACT FROM AUTHOR]

Since the seminal 1961 paper of Monod and Jacob, mathematical models of biomolecular circuits have guided our understanding of cell regulation. Model-based exploration of the functional capabilities of any given circuit requires systematic mapping of multidimensional spaces of model parameters. Despite significant advances in computational dynamical systems approaches, this analysis remains a nontrivial task. Here, we use a nonlinear system of ordinary differential equations to model oocyte selection in Drosophila, a robust symmetry-breaking event that relies on autoregulatory localization of oocyte-specification factors. By applying an algorithmic approach that implements symbolic computation and topological methods, we enumerate all phase portraits of stable steady states in the limit when nonlinear regulatory interactions become discrete switches. Leveraging this initial exact partitioning and further using numerical exploration, we locate parameter regions that are dense in purely asymmetric steady states when the nonlinearities are not infinitely sharp, enabling systematic identification of parameter regions that correspond to robust oocyte selection. This framework can be generalized to map the full parameter spaces in a broad class of models involving biological switches. Author summary: Identification of qualitatively different regimes in models of biomolecular switches is essential for understanding dynamics of complex biological processes, including symmetry breaking in cells and cell networks. We demonstrate how topological methods, symbolic computation, and numerical simulations can be combined for systematic mapping of symmetry-broken states in a mathematical model of oocyte specification in Drosophila, a leading experimental system of animal oogenesis. Our algorithmic framework reveals global connectedness of parameter domains corresponding to robust oocyte specification and enables systematic navigation through multidimensional parameter spaces in a large class of biomolecular switches. [ABSTRACT FROM AUTHOR]

We present a new modeling and computational tool that computes rigorous summaries of network dynamics over large sets of parameter values. These summaries, organized in a database, can be searched for observed dynamics, e.g., bistability and hysteresis, to discover parameter regimes over which they are supported. We illustrate our approach on several networks underlying the restriction point of the cell cycle in humans and yeast. We rank networks by how robustly they support hysteresis, which is the observed phenotype. We find that the best 6-node human network and the yeast network share similar topology and robustness of hysteresis, in spite of having no homology between the corresponding nodes of the network. Our approach provides a new tool linking network structure and dynamics. [ABSTRACT FROM AUTHOR]

Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid. [ABSTRACT FROM AUTHOR]

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications. [ABSTRACT FROM AUTHOR]

This paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. We study switching models of regulatory networks. To each switching network we associate a Morse graph, a computable object that describes a Morse decomposition of the dynamics. In this paper we show that all smooth perturbations of the switching system share the same Morse graph and we compute explicit bounds on the size of the allowable perturbation. This shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities. [ABSTRACT FROM AUTHOR]

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–Bénard 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. [ABSTRACT FROM AUTHOR]

We describe the theoretical and computational framework for the Dynamic Signatures Generated by Regulatory Networks (DSGRN) database. The motivation stems from an 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 have 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. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate) and three parameters for each edge (low level of expression, high level of expression, and threshold at which expression levels change). Since the nonlinearities of switching systems are piecewise 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 a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. 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 dynamical 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 of 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. [ABSTRACT FROM AUTHOR]

We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group ( capd.ii.uj.edu.pl ), the other based on the COSY Infinity project ( cosyinfinity.org ). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time- T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T , and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. [ABSTRACT FROM AUTHOR]

In this paper we showcase the technique of Conley-Morse databases for studying a parameterized family of dynamical systems. The dynamical system of interest arises from considering the limiting behavior of Newton's root-finding method applied to functions f : R² - R² when the iterates converge to the origin. Considering the progression of angular orientations gives rise to a self- map of the unit circle we call the angular dynamics map. We demonstrate how the technique of Conley{Morse dynamical databases allows us to quickly survey and prove theorems about the global dynamics of the parameterized family of angular dynamics maps. [ABSTRACT FROM AUTHOR]

We discuss sets of measures with the goal of describing dynamical properties of force networks in dense particulate systems. The presented approach is based on persistent homology and allows for extracting precise, quantitative measures that describe the evolution of geometric features of the interparticle forces, without necessarily considering the details related to individual contacts between particles. The networks considered emerge from discrete element simulations of two-dimensional particulate systems consisting of compressible frictional circular disks. We quantify the evolution of the networks for slowly compressed systems undergoing jamming transition. The main findings include uncovering significant but localized changes of force networks for unjammed systems, global (systemwide) changes as the systems evolve through jamming, to be followed by significantly less dramatic evolution for the jammed states. We consider both connected components, related in a loose sense to force chains, and loops and find that both measures provide a significant insight into the evolution of force networks. In addition to normal, we consider also tangential forces between the particles and find that they evolve in the consistent manner. Consideration of both frictional and frictionless systems leads us to the conclusion that friction plays a significant role in determining the dynamical properties of the considered networks. We find that the proposed approach describes the considered networks in a precise yet tractable manner, making it possible to identify features which could be difficult or impossible to describe using other approaches. [ABSTRACT FROM AUTHOR]

We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes of these distributions: digital, position, and interaction, motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between the particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to the different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks. [ABSTRACT FROM AUTHOR]

We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conley's topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications. [ABSTRACT FROM AUTHOR]

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation. [ABSTRACT FROM AUTHOR]

A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decompositions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the existence of certain types of dynamics. The power of the developed method is illustrated with an application to the two-dimensional density-dependent Leslie population model. An interactive visualization of the results of computations discussed in the paper can be accessed at the Web site http://chomp.rutgers.edu/database/, and the source code of the software used to obtain these results has also been made freely available. [ABSTRACT FROM AUTHOR]

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ3. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ3, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We propose a simple method for reconstructing vascular trees from 3D images. Our algorithm extracts persistent maxima of the intensity on all axis-aligned 2D slices of the input image. The maxima concentrate along 1D intensity ridges, in particular along blood vessels. We build a forest connecting the persistent maxima with short edges. The forest tends to approximate the blood vessels present in the image, but also contains numerous spurious features and often fails to connect segments belonging to one vessel in low contrast areas. We improve the forest by applying simple geometric filters that trim short branches, fill gaps in blood vessels and remove spurious branches from the vascular tree to be extracted. Experiments show that our technique can be applied to extract coronary trees from heart CT scans. [Copyright &y& Elsevier]

By using current biological understanding, a conceptually simple, but mathematically complex, model is proposed for the dynamics of the gene circuit responsible for regulating nitrogen catabolite repression (NCR) in yeast. A variety of mathematical "structure" theorems are described that allow one to determine the asymptotic dynamics of complicated systems under very weak hypotheses. It is shown that these theorems apply to several subcircuits of the full NCR circuit most importantly to the URE2-.61N3 subcircuit that is independent of the other constituents but governs the switching behavior of the full NCR circuit under changes in nitrogen source. Under hypotheses that are fully consistent with biological data, it is proven that the dynamics of this subcircuit is simple periodic behavior in synchrony with the cell cycle. Although the current mathematical structure theorems do not apply to the full NCR circuit extensive simulations suggest that the dynamics is constrained in much the same way as that of the URE2-GLN3 subcircuit. This finding leads to the proposal that mathematicians study genetic circuits to find new geometries for which structure theorems may exist. [ABSTRACT FROM AUTHOR]

We carry out a direct comparison of experimental and numerical realizations of the exact same granular system as it undergoes shear jamming. We adjust the numerical methods used to optimally represent the experimental settings and outcomes up to microscopic contact force dynamics. Measures presented here range from microscopic through mesoscopic to systemwide characteristics of the system. Topological properties of the mesoscopic force networks provide a key link between microscales and macroscales. We report two main findings: (1) The number of particles in the packing that have at least two contacts is a good predictor for the mechanical state of the system, regardless of strain history and packing density. All measures explored in both experiments and numerics, including stress-tensor-derived measures and contact numbers depend in a universal manner on the fraction of nonrattler particles, fNR. (2) The force network topology also tends to show this universality, yet the shape of the master curve depends much more on the details of the numerical simulations. In particular we show that adding force noise to the numerical data set can significantly alter the topological features in the data. We conclude that both fNR and topological metrics are useful measures to consider when quantifying the state of a granular system. [ABSTRACT FROM AUTHOR]