Flow‐Induced Inertial Steady Vector Field Topology

Traditionally, vector field visualization is concerned with 2D and 3D flows. Yet, many concepts can be extended to general dynamical systems, including the higher‐dimensional problem of modeling the motion of finite‐sized objects in fluids. In the steady case, the trajectories of these so‐called inertial particles appear as tangent curves of a 4D or 6D vector field. These higher‐dimensional flows are difficult to map to lower‐dimensional spaces, which makes their visualization a challenging problem. We focus on vector field topology, which allows scientists to study asymptotic particle behavior. As recent work on the 2D case has shown, both extraction and classification of isolated critical points depend on the underlying particle model. In this paper, we aim for a model‐independent classification technique, which we apply to two different particle models in not only 2D, but also 3D cases. We show that the classification can be done by performing an eigenanalysis of the spatial derivatives' velocity subspace of the higher‐dimensional 4D or 6D flow. We construct glyphs that depict not only the types of critical points, but also encode the directional information given by the eigenvectors. We show that the eigenvalues and eigenvectors of the inertial phase space have sufficient symmetries and structure so that they can be depicted in 2D or 3D, instead of 4D or 6D.     

Visualizing the Phase Space of Heterogeneous Inertial Particles in 2D Flows

In many scientific disciplines, the motion of finite‐sized objects in fluid flows plays an important role, such as in brownout engineering, sediment transport, oceanology or meteorology. These finite‐sized objects are called inertial particles and, in contrast to traditional tracer particles, their motion depends on their current position, their own particle velocity, the time and their size. Thus, the visualization of their motion becomes a high‐dimensional problem that entails computational and perceptual challenges. So far, no visualization explored and visualized the particle trajectories under variation of all seeding parameters. In this paper, we propose three coordinated views that visualize the different aspects of the high‐dimensional space in which the particles live. We visualize the evolution of particles over time, showing that particles travel different distances in the same time, depending on their size. The second view provides a clear illustration of the trajectories of different particle sizes and allows the user to easily identify differences due to particle size. Finally, we embed the trajectories in the space‐velocity domain and visualize their distance to an attracting manifold using ribbons. In all views, we support interactive linking and brushing, and provide abstraction through density volumes that are shown by direct volume rendering and isosurface slabs. Using our method, users gain deeper insights into the dynamics of inertial particles in 2D fluids, including size‐dependent separation, preferential clustering and attraction. We demonstrate the effectiveness of our method in multiple steady and unsteady 2D flows.     

Phase Space Projection of Dynamical Systems

Dynamical systems are commonly used to describe the state of time-dependent systems. In many engineering and control problems, the state space is high-dimensional making it difficult to analyze and visualize the behavior of the system for varying input conditions. We present a novel dimensionality reduction technique that is tailored to high-dimensional dynamical systems. In contrast to standard general purpose dimensionality reduction algorithms, we use energy minimization to preserve properties of the flow in the high-dimensional space. Once the projection operator is optimized, further high-dimensional trajectories are projected easily. Our 3D projection maintains a number of useful flow properties, such as critical points and flow maps, and is optimized to match geometric characteristics of the high-dimensional input, as well as optional user constraints. We apply our method to trajectories traced in the phase spaces of second-order dynamical systems, including finite-sized objects in fluids, the circular restricted three-body problem and a damped double pendulum. We compare the projections with standard visualization techniques, such as PCA, t-SNE and UMAP, and visualize the dynamical systems with multiple coordinated views interactively, featuring a spatial embedding, projection to subspaces, our dimensionality reduction and a seed point exploration tool.     

An Interactive Visualization System for Large Sets of Phase Space Trajectories

We introduce a visual analysis system with GPU acceleration techniques for large sets of trajectories from complex dynamical systems. The approach is based on an interactive Boolean combination of subsets into a Focus+Context phase‐space visualization. We achieve high performance through efficient bitwise algorithms utilizing runtime generated GPU shaders and kernels. This enables a higher level of interactivity for visualizing the large multivariate trajectory data. We explain how our design meets a set of carefully considered analysis requirements, provide performance results, and demonstrate utility through case studies with many‐particle simulation data from two application areas.     

Fractals in neurodynamics: From hybrid dynamical systems point of view

Recently, hybrid dynamical systems have attracted considerable attention in the automatic control domain. In this article, a theory for recurrent neural networks is presented from a hybrid dynamical systems point of view. The hybrid dynamical system is defined by a continuous dynamical system discretely switched by external temporal inputs. The theory suggests that the dynamics of continuous-time recurrent neural networks, which are stochastically excited by external temporal inputs, are generally characterized by a set of continuous trajectories with a fractal-like structure in hyper-cylindrical phase space. This work was presented, in part, at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

Route‐Aware Edge Bundling for Visualizing Origin‐Destination Trails in Urban Traffic

Origin‐destination (OD) trails describe movements across space. Typical visualizations thereof use either straight lines or plot the actual trajectories. To reduce clutter inherent to visualizing large OD datasets, bundling methods can be used. Yet, bundling OD trails in urban traffic data remains challenging. Two specific reasons hereof are the constraints implied by the underlying road network and the difficulty of finding good bundling settings. To cope with these issues, we propose a new approach called Route Aware Edge Bundling (RAEB). To handle road constraints, we first generate a hierarchical model of the road‐and‐trajectory data. Next, we derive optimal bundling parameters, including kernel size and number of iterations, for a user‐selected level of detail of this model, thereby allowing users to explicitly trade off simplification vs accuracy. We demonstrate the added value of RAEB compared to state‐of‐the‐art trail bundling methods on both synthetic and real‐world traffic data for tasks that include the preservation of road network topology and the support of multiscale exploration.     

Visual Analysis of protein‐ligand interactions

The analysis of protein‐ligand interactions is complex because of the many factors at play. Most current methods for visual analysis provide this information in the form of simple 2D plots, which, besides being quite space hungry, often encode a low number of different properties. In this paper we present a system for compact 2D visualization of molecular simulations. It purposely omits most spatial information and presents physical information associated to single molecular components and their pairwise interactions through a set of 2D InfoVis tools with coordinated views, suitable interaction, and focus+context techniques to analyze large amounts of data. The system provides a wide range of motifs for elements such as protein secondary structures or hydrogen bond networks, and a set of tools for their interactive inspection, both for a single simulation and for comparing two different simulations. As a result, the analysis of protein‐ligand interactions of Molecular Simulation trajectories is greatly facilitated.     

Computational dynamics: Modeling and visualizing trajectory flows in phase space

This paper describes a computational technique for modeling and visualizing dynamical behaviors of complex systems in phase space. The technique employs a novel idea offlow pipes to model trajectory bundles that exhibit the same qualitative features. It parses a continuous phase space of a dynamical system, consisting of an infinite number of individual trajectories, into a manageable discrete collection of flow pipes that a computer can efficiently reason about. The technique provides a computational way for both machines and humans to visualize and manipulate dynamics of a physical system. The flow-pipe modeling technique is implemented in a program called MAPS. The technique has been applied to the automatic control synthesis in which programs automatically analyze and design high-performance, global controllers.     

ℋ︁∞ and l2–l∞ filtering for two‐dimensional linear parameter‐varying systems

In this paper, the ??_∞ and l₂–l_∞ filtering problem is investigated for two‐dimensional (2‐D) discrete‐time linear parameter‐varying (LPV) systems. Based on the well‐known Fornasini–Marchesini local state‐space (FMLSS) model, the mathematical model of 2‐D systems under consideration is established by incorporating the parameter‐varying phenomenon. The purpose of the problem addressed is to design full‐order ??_∞ and l₂–l_∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in ??_∞ and l₂–l_∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive architectures for control of uncertain dynamical systems with actuator and unmodeled dynamics

In adaptive control of uncertain dynamical systems, it is well known that the presence of actuator and/or unmodeled dynamics in feedback loops can yield to unstable closed‐loop system trajectories. Motivated by this standpoint, this paper focuses on the analysis and synthesis of multiple adaptive architectures for control of uncertain dynamical systems with both actuator and unmodeled dynamics. Specifically, we first analyze model reference adaptive control architectures with standard, hedging‐based, and expanded reference models for this class of uncertain dynamical systems and develop sufficient stability conditions. We then synthesize a robustifying term for the latter architecture and analytically show that this term can allow for a relaxed sufficient stability condition. The proposed theoretical treatments involve Lyapunov stability theory, linear matrix inequalities, and matrix mathematics. Finally, we compare the resulting sufficient stability conditions of the considered adaptive control architectures on a benchmark mechanical system subject to actuator and unmodeled dynamics.     

Analyzing Residue Surface Proximity to Interpret Molecular Dynamics

The surface of a molecule holds important information about the interaction behavior with other molecules. In dynamic folding or docking processes, residues of amino acids with different properties change their position within the molecule over time. The atoms of the residues that are accessible to the solvent can directly contribute to binding interactions, while residues buried within the molecular structure contribute to the stability of the molecule. Understanding patterns and causality of structural changes is important for experts in the pharmaceutical domain, e.g., in the process of drug design. We apply an iterative computation of the Solvent Accessible Surface in order to extract virtual layers of a molecule. The extraction allows to track the movement of residues in the body of the molecule, with respect to the distance of the residue to the surface or the core during dynamics simulations. We visualize the obtained layer information for the complete time span of the molecular dynamics simulation as a 2D‐map and for individual time‐steps as a 3D‐representation of the molecule. The data acquisition has been implemented alongside with further analysis functionality in a prototypical application, which is available to the public domain. We underline the feasibility of our approach with a study from the pharmaceutical domain, where our approach has been used for novel insights into the folding behavior of μ‐conotoxins.     

HyperMoVal: Interactive Visual Validation of Regression Models for Real‐Time Simulation

During the development of car engines, regression models that are based on machine learning techniques are increasingly important for tasks which require a prediction of results in real‐time. While the validation of a model is a key part of its identification process, existing computation‐ or visualization‐based techniques do not adequately support all aspects of model validation. The main contribution of this paper is an interactive approach called HyperMoVal that is designed to support multiple tasks related to model validation: 1) comparing known and predicted results, 2) analyzing regions with a bad fit, 3) assessing the physical plausibility of models also outside regions covered by validation data, and 4) comparing multiple models. The key idea is to visually relate one or more n‐dimensional scalar functions to known validation data within a combined visualization. HyperMoVal lays out multiple 2D and 3D sub‐projections of the n‐dimensional function space around a focal point. We describe how linking HyperMoVal to other views further extends the possibilities for model validation. Based on this integration, we discuss steps towards supporting the entire workflow of identifying regression models. An evaluation illustrates a typical workflow in the application context of car‐engine design and reports general feedback of domain experts and users of our approach. These results indicate that our approach significantly accelerates the identification of regression models and increases the confidence in the overall engineering process.     

Multi‐tasking optimal control of networked control systems: A delta operator approach

In this paper, the multi‐tasking optimal control problem is addressed in the delta‐domain for a class of networked control systems (NCSs) with external disturbances. A delta‐domain model is proposed to describe the NCSs with random packet dropouts and long‐time delay, and the so‐called ε‐Nash equilibrium is employed to quantify the impacts from the disturbances on the underlying NCSs. The multi‐tasking optimal control strategies are developed and the upper bound for the ε‐Nash equilibrium is provided explicitly. Some simulation results on the two‐area load frequency control system are given to show the validity and applicability of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

An analytical workbench for system level multibody dynamics

There has been considerable focus in the research community on developing accurate models, as well as on fast algorithms for solving the equations of motion of multibody systems required for simulating the dynamics of such systems. This paper focuses on the less explored complementary topic of evaluating system level dynamics properties of multibody systems. Examples of such dynamics properties are the system mass matrix, Jacobians and sensitivities of these quantities. These system level quantities manifest the dynamical properties of the system and are important for design, optimization and control. While such system level quantities are often used in theory to describe the underlying mathematical physics of the systems, due to their complexity there is a lack of a systematic methods for computing them when desired. In this paper we describe a computational workbench framework that provides a bridge between theory and the computation of such system level quantities. This workbench framework builds upon the Spatial Operator Algebra (SOA) that has been used for analysis and algorithm development for multibody dynamics. Mathematical operator expressions from the SOA can be transcribed literally to the workbench command line to allow the easy evaluation of complex dynamics quantities. We use a specific Python/C++ implementation of a workbench called PyCraft to illustrate the structure and use of such a workbench. Several examples illustrating operator-based analysis and corresponding PyCraft-based computation are included.     

E&F Chaos: A User Friendly Software Package for Nonlinear Economic Dynamics

The use of nonlinear dynamic models in economics and finance has expanded rapidly in the last two decades. Numerical simulation is crucial in the investigation of nonlinear systems. E&F Chaos is an easy-to-use and freely available software package for simulation of nonlinear dynamic models to investigate stability of steady states and the presence of periodic orbits and chaos by standard numerical simulation techniques such as time series, phase plots, bifurcation diagrams, Lyapunov exponent plots, basin boundary plots and graphical analysis. The package contains many well-known nonlinear models, including applications in economics and finance, and is easy to use for non-specialists. New models and extensions or variations are easy to implement within the software package without the use of a compiler or other software. The software is demonstrated by investigating the dynamical behavior of some simple examples of the familiar cobweb model, including an extension with heterogeneous agents and asynchronous updating of strategies. Simulations with the E&F Chaos software quickly provide information about local and global dynamics and easily lead to challenging questions for further mathematical analysis.     

Constructing subsystems for nonlinear controlled systems

We consider the problem of constructing subsystems representing simplified models of nonlinear controlled dynamical systems. Subsystems define a part of phase trajectories of the original controlled system.     

Core Lines in 3D Second‐Order Tensor Fields

Vortices are important features in vector fields that show a swirling behavior around a common core. The concept of a vortex core line describes the center of this swirling behavior. In this work, we examine the extension of this concept to 3D second‐order tensor fields. Here, a behavior similar to vortices in vector fields can be observed for trajectories of the eigenvectors. Vortex core lines in vector fields were defined by Sujudi and Haimes to be the locations where stream lines are parallel to an eigenvector of the Jacobian. We show that a similar criterion applied to the eigenvector trajectories of a tensor field yields structurally stable lines that we call tensor core lines. We provide a formal definition of these structures and examine their mathematical properties. We also present a numerical algorithm for extracting tensor core lines in piecewise linear tensor fields. We find all intersections of tensor core lines with the faces of a dataset using a simple and robust root finding algorithm. Applying this algorithm to tensor fields obtained from structural mechanics simulations shows that it is able to effectively detect and visualize regions of rotational or hyperbolic behavior of eigenvector trajectories.     

SetCoLa: High‐Level Constraints for Graph Layout

Constraints enable flexible graph layout by combining the ease of automatic layout with customizations for a particular domain. However, constraint‐based layout often requires many individual constraints defined over specific nodes and node pairs. In addition to the effort of writing and maintaining a large number of similar constraints, such constraints are specific to the particular graph and thus cannot generalize to other graphs in the same domain. To facilitate the specification of customized and generalizable constraint layouts, we contribute SetCoLa: a domain‐specific language for specifying high‐level constraints relative to properties of the backing data. Users identify node sets based on data or graph properties and apply high‐level constraints within each set. Applying constraints to node sets rather than individual nodes reduces specification effort and facilitates reapplication of customized layouts across distinct graphs. We demonstrate the conciseness, generalizability, and expressiveness of SetCoLa on a series of real‐world examples from ecological networks, biological systems, and social networks.     

Reliable approximation of separatrix manifolds in competition models with safety niches

In dynamical systems saddle points partition the domain into basins of attractions of the remaining locally stable equilibria. This situation is rather common especially in population dynamics models, like prey–predator or competition systems. Focusing on squirrels population models with niche, in this paper we design algorithms for the detection and the refinement of points lying on the separatrix manifold partitioning the phase space. We consider both the two populations and the three populations cases. To reconstruct the separatrix curve and surface, we apply the Partition of Unity method, which makes use of Wendland's functions as local approximants.