A guided tour to wonderland: Visualizing the slow-fast dynamics of an analytical dynamical system (original) (raw)

Visualization of dynamical systems

Future Generation Computer Systems - FGCS, 1999

The visualization of analytically de#ned dynamical systems is important for a thoroughunderstanding of the underlying system behavior. An introduction to analyticallyde#ned dynamical systems is given. Various visualization techniques fordynamical systems are discussed. Several current research directions concerning thevisualization of dynamical systems are treated in more detail. These are: texturebased techniques, visualization of high-dimensional dynamical systems, advancedstreamsurface ...

Visualizing the behaviour of higher dimensional dynamical systems

Proceedings. Visualization '97 (Cat. No. 97CB36155), 1997

In recent years scientific visualization has been driven by the need to visualize high-dimensional data sets within high-dimensional spaces. However most visualization methods are designed to show only some statistical features of the data set. This paper deals with the visualization of trajectories of high-dimensional dynamical systems which form a L n n data set of a smooth n-dimensional flow.

Depicting time evolving flow with illustrative visualization techniques

Arts and Technology, 2010

Visualization has become an indispensable tool for scientists to extract knowledge from large amounts of data and convey that knowledge to others. Visualization may be exploratory or illustrative. Exploratory visualization generally provides multiple views of the data at different levels of abstraction and should be highly interactive, whereas illustrative visualization is often made offline at high quality with sufficient knowledge about the data and features of interest. Techniques used by professional illustrators may be borrowed to enhance the clarity and aesthetics of the visualization. This paper presents a set of visualization techniques for presenting the evolution of 3D flow. While the spatial features of the data is rendered in 3D space, the temporal behaviors of the flow are depicted using image-based methods. We demonstrate visualization results generated using three data sets obtained from simulations.

Stream arrows: enhancing the use of stream surfaces for the visualization of dynamical systems

The Visual Computer, 1997

We present a new approach to deal with stream surfaces that occlude major parts of the system representation. While analyzing mixed-mode oscillations, we came across geometrically complex stream surfaces with a curly shape. Certain regions of these surfaces may occlude major parts of the model. We adapted visualization techniques to deal with this. Stream arrows make portions of the stream surface semitransparent and this diminish the problem of occlusion. Cross-sections and removing portions of the model also improve visual perception. Choosing the shape of an arrow for segmentation helps visualize the flow direction. An anisotropic spot noise texture emphasizes the flow within a stream surface. Animation techniques facilitate the interpretation of dynamical systems with complex stream surfaces.

Enhancing the visualization of characteristic structures in dynamical systems

1998

Abstract. We present a thread of streamlets as a new technique to visualize dynamical systems in three-space. A trade-off is made between solely visualizing a mathematical abstraction through lower-dimensional manifolds, ie, characteristic structures such as fixed point, separatrices, etc., and directly encoding the flow through stream lines or stream surfaces. Bundlers of streamlets are selectively placed near characteristic trajectories. An over-population of phase space with occlusion problems as a consequence is omitted.

Hierarchical streamarrows for the visualization of dynamical systems

1997

Abstract. Streamarrows are a technique to enhance the use of streamsurfaces by separating arrow-shaped portions from the remaining streamsurface. We present a hierarchical streamarrows algorithm as an extension to this technique: Streamarrows are locally chosen from a stack of scaled streamarrows textures to avoid too big or small streamarrows in the rendered image.

Analysing spatially extended high-dimensional dynamics by recurrence plots

Recurrence plot based measures of complexity are capable tools for characterizing complex dynamics. In this letter we show the potential of selected recurrence plot measures for the investigation of even high-dimensional dynamics. We apply this method on spatially extended chaos, such as derived from the Lorenz96 model and show that the recurrence plot based measures can qualitatively characterize typical dynamical properties such as chaotic or periodic dynamics. Moreover, we demonstrate its power by analysing satellite image time series of vegetation cover with contrasting dynamics as a spatially extended and potentially high-dimensional example from the real world.

Analysis of population dynamics with interactive multi-dimensional graphics

Landscape and Urban Planning, 1992

Increased access to supercomputers permits entomologists and ecologists to simuiate complex models consisting of many state variables representing one or more populations. To explain the dynamics of these populatizs over time and space, more than the ordinary summary data and two-dimensional graphs are needed. We have developed a set of graphics programs to address the typical concerns.

EGAP-Helix: A Novel Geometric Approach to Visualize Economic Growth Dynamics

This paper introduces a groundbreaking graphical approach, termed the Economic Growth Accumulation Process Helix (EGAP-Helix), to visually represent the total output (GDP) of a country. The EGAP-Helix offers a unique geometrical perspective, allowing for a comprehensive understanding of economic growth patterns. This research advocates the utilization of a 3-Dimensional geometrical approach and proposes 3D printing for the materialization of EG-Spheres and EGAP-Helix, enhancing its methodological and educational applications (Tufte, 2001) (Ware, 2008). The study employs real GDP growth rates of the U.S. from 2005 to 2022.

A Visual Analytics Approach for Ecosystem Dynamics based on Empirical Dynamic Modeling

IEEE Transactions on Visualization and Computer Graphics, 2021

Fig. 1. Schematic of visual analytics to study nonlinear interactions with empirical dynamic modeling (EDM). A dynamic graph of changing interaction coefficients is first constructed using the (A) measured time series data and (B) interaction coefficients extracted via EDM techniques. (C) Our proposed visual analytics system enables the detection and interpretation of system states using dimension reduction and brush-link visualization techniques. (D) Using the process of annotation and summarization, the state transition graph can be obtained for interpretation. Abstract-An important approach for scientific inquiry across many disciplines involves using observational time series data to understand the relationships between key variables to gain mechanistic insights into the underlying rules that govern the given system. In real systems, such as those found in ecology, the relationships between time series variables are generally not static; instead, these relationships are dynamical and change in a nonlinear or state-dependent manner. To further understand such systems, we investigate integrating methods that appropriately characterize these dynamics (i.e., methods that measure interactions as they change with time-varying system states) with visualization techniques that can help analyze the behavior of the system. Here, we focus on empirical dynamic modeling (EDM) as a state-of-the-art method that specifically identifies causal variables and measures changing state-dependent relationships between time series variables. Instead of using approaches centered on parametric equations, EDM is an equation-free approach that studies systems based on their dynamic attractors. We propose a visual analytics system to support the identification and mechanistic interpretation of system states using an EDM-constructed dynamic graph. This work, as detailed in four analysis tasks and demonstrated with a GUI, provides a novel synthesis of EDM and visualization techniques such as brush-link visualization and visual summarization to interpret dynamic graphs representing ecosystem dynamics. We applied our proposed system to ecological simulation data and real data from a marine mesocosm study as two key use cases. Our case studies show that our visual analytics tools support the identification and interpretation of the system state by the user, and enable us to discover both confirmatory and new findings in ecosystem dynamics. Overall, we demonstrated that our system can facilitate an understanding of how systems function beyond the intuitive analysis of high-dimensional information based on specific domain knowledge.