Enhancing the visualization of characteristic structures in dynamical systems (original) (raw)

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.

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.

Topologically relevant stream surfaces for flow visualization

Proceedings of the 2009 Spring Conference on Computer Graphics - SCCG '09, 2009

: Our algorithm applied to the Lorenz dynamical system. Stable manifold of the critical point at the origin, colored by geodesic distance (left), unstable manifold of a spiral saddle critical point, colored by node index (middle), and 2D manifolds of all three critical points (right).

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.

Visualizing the geometry of state space in plane Couette flow

arXiv preprint arXiv:0705.3957, 2007

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10 5 -dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Re turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.

TRAJECTORY-AUGMENTED VISUALIZATION OF LAGRANGIAN COHERENT STRUCTURES IN UNSTEADY FLOW

The finite-time Lyapunov exponent (FTLE) field can be used for many purposes, from the analysis of the predictability in dynamical systems to the topological analysis of timedependent vector fields. In the topological context, the topic of this work, FTLE ridges represent Lagrangian coherent structures (LCS), a counterpart to separatrices in vector field topology. Since the explicit vector field behavior cannot be deduced from these representations, they may be augmented by line integral convolution patterns, a computational flow visualization counterpart to the surface oil flow method. This is, however, strictly meaningful only in stationary vector fields. Here, we propose an augmentation that visualizes the LCS-inducing flow behavior by means of complete trajectories but avoids occlusion and visual clutter. For this we exploit the FTLE for both the selection of significant trajectories as well as their individual representation. This results in 3D line representations for 2D vector fields by treating 2D time-dependent vector fields in 3D space-time. We present two variants of the approach, one easing the choice of the finite advection time for FTLE analysis and one for investigating the flow once the time is chosen.

Flow visualization in science and mathematics

Journal of Physics: Conference Series, 2009

We present several methods for visualizing motion, vector fields, and flows, including polygonal surface advection, visibility driven transfer functions, feature extraction and tracking, and motion frequency analysis and enhancement. They are applied to chaotic attractors, turbulent vortices, supernovae, and seismic data.

Cartography of high-dimensional flows: A visual guide to sections and slices

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2012

Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a "template." Together these charts provide an atlas of the symmetry-reduced "slice" of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits. Visualizations of these solutions and their unstable manifolds reveal their interrelations and the role they play in organizing turbulence/chaos.