Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Mixed-Mode Oscillations with Multiple Time Scales

Our starting point for developing the Studierstube system was the belief that augmented reality, the less obtrusive cousin of virtual reality, has a better chance of becoming a viable user interface for applications requiring manipulation of complex three-dimensional information as a daily routine. In essence, we are searching for a 3-D user interface metaphor as powerful as the desktop metaphor for 2-D. At the heart of the Studierstube system, collaborative augmented reality is used to embed computer-generated images into the real work environment. In the first part of this paper, we review the user interface of the initial Studierstube system, in particular the implementation of collaborative augmented reality, and the Personal Interaction Panel, a two-handed interface for interaction with the system. In the second part, an extended Studierstube system based on a heterogeneous distributed architecture is presented. This system allows the user to combine multiple approaches-augmented reality, projection displays, and ubiquitous computing--to the interface as needed. The environment is controlled by the Personal Iteraction Panel, a two-handed, pen-and-pad interface that has versatile uses for interacting with the virtual environment. Studierstube also borrows elements from the desktop, such as multi-tasking and multi-windowing. The resulting software architecture is a user interface management system for complex augmented reality applications. The presentation is complemented by selected application examples.

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The studierstube augmented reality project

Flow visualization has been a very attractive component of scientific visualization research for a long time. Usually very large multivariate datasets require processing. These datasets often consist of a large number of sample locations and several time steps. The steadily increasing performance of computers has recently become a driving factor for a reemergence in flow visualization research, especially in texture-based techniques. In this paper, dense, texture-based flow visualization techniques are discussed. This class of techniques attempts to provide a complete, dense representation of the flow field with high spatio-temporal coherency. An attempt of categorizing closely related solutions is incorporated and presented. Fundamentals are shortly addressed as well as advantages and disadvantages of the methods.

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The State of the Art in Flow Visualization : Dense and Texture-Based Techniques

We provide a research agenda for the International Cartographic Association's Commission on Visualization and Virtual Environment Working Group on Cognitive and Usability Issues in Geovisualization. Developments in hardware and software have led to (and will continue to stimulate) numerous novel methods for visualizing geospatial data. It is our belief that these novel methods will be of little use if they are not developed within a theoretical cognitive framework and iteratively tested using usability engineering principles. We argue that cognitive and usability issues should be considered in the context of six major research themes: 1) geospatial virtual environments (GeoVEs); 2) dynamic representations (including animated and interactive maps); 3) metaphors and schemata in user interface design; 4) individual and group differences; 5) collaborative geovisualization; and 6) evaluating the effectiveness of geovisualization methods. A key point underlying our use of theoretical cognitive principles is tha...

Cognitive and Usability Issues in Geovisualization

Flow visualization is a fascinating sub-branch of scientific visualization. With ever increasing computing power, it is possible to process ever more complex fluid simulations. However, a gap between data set sizes and our ability to visualize them remains. This is especially true for the field of flow visualization which deals with large, timedependent, multivariate simulation datasets. In this paper, geometry based flow visualization techniques form the focus of discussion. Geometric flow visualization methods place discrete objects in the vector field whose characteristics reflect the underlying properties of the flow. A great amount of progress has been made in this field over the last two decades. However, a number of challenges remain, including placement, speed of computation, and perception. In this survey, we review and classify geometric flow visualization literature according to the most important challenges when considering such a visualization, a central theme being the seeding object upon which they are based. This paper details our investigation into these techniques with discussions on their applicability and their relative merits and drawbacks. The result is an up-to-date overview of the current state-of-the-art that highlights both solved and unsolved problems in this rapidly evolving branch of research. It also serves as a concise introduction to the field of flow visualization research.

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Over Two Decades of Integration-Based, Geometric Flow Visualization

We present a geometric explanation of a basic mechanism generating mixed-mode oscillations in a prototypical simple model of a chemical oscillator. Our approach is based on geometric singular perturbation theory and canard solutions. We explain how the small oscillations are generated near a special point, which is classified as a folded saddle-node for the reduced problem. The canard solution passing through this point separates small oscillations from large relaxation type oscillations. This allows to define a one-dimensional return map in a natural way. This bimodal map is capable of explaining the observed bifurcation sequence convincingly.

Geometry of mixed-mode oscillations in the 3-d autocatalator

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. The paper deals with the visualization of trajectories of high-dimensional dynamical systems which form a L/sub n//sup n/ data set of a smooth n-dimensional flow. Three methods that are based on the idea of parallel coordinates are presented and discussed. Visualizations done with these new methods are shown and an interactive visualization tool for the exploration of high-dimensional dynamical systems is proposed.

https://www.cg.tuwien.ac.at/research/vis/dynsys/ndim/ndim_crc.pdf

Visualizing the behaviour of higher dimensional dynamical systems

The authors discuss Studierstube, a low-cost augmented reality system. The system features true stereoscopy, 3D interaction, individual viewpoints and customized views for multiple users, and unhindered natural collaboration.

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Collaborative visualization in augmented reality

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.

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

We present a set of advanced techniques for the visualization of 2D Poincare maps. Since 2D Poincare maps are a mathematical abstraction of periodic or quasi-periodic 3D flows, we propose to embed the 2D visualization with standard 3D techniques to improve the understanding of the Poincare maps. Methods to enhance the representation of the relation x ↔ P(x), e.g., the use of spot noise, are presented as well as techniques to visualize the repeated application of P, e.g., the approximation of P as a warp function. It is shown that animation can be very useful to further improve the visualization. For example, the animation of the construction of Poincare map P is a very intuitive visualization. During the paper we present a set of examples which demonstrate the usefulness of our techniques.

Helwig Löffelmann

Papers

Geometry of mixed-mode oscillations in the 3-d autocatalator

Visualizing the behaviour of higher dimensional dynamical systems

Collaborative visualization in augmented reality

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

Visualizing Poincaré Maps together with the Underlying Flow