Introduction

Poincaré sections are an important tool for the investigation of dynamical systems in theory as well as in applications. They are used for models (usually in 3D) that exhibit periodic or quasi-periodic behavior. In addition to mathematical descriptions--there is a lot of theory about dynamical systems [4,5,32,92]--periodic or quasi-periodic dynamical systems can be found in many fields, e.g., in physics, chemistry, biology, ecology. Especially chaotic systems are often examined by the use of Poincaré sections [80,83,92].

A 2D Poincaré section through a periodic 3D flow is a planar cross-section transverse to the flow such that a periodic orbit intersects it at its center. The corresponding Poincaré map is defined as a map correlating consecutive intersections of flow trajectories with the Poincaré section. The Poincaré map is a discrete dynamical system of one dimension less than the continuous flow which it is constructed from. As many of the most important flow properties are inherited by the Poincaré map and its analysis is usually more simple due to its reduced dimensionality, it is often used for analysis instead of the 3D flow. See Sect. 5.2 for a more detailed discussion of some basics on Poincaré sections and Poincaré maps.

Since their introduction to dynamical system analysis by Henri Poincaré in 1899 [66] the visual representation of Poincaré maps has been always a very important part of this research technique. Hand-drawn sketches of the Poincaré map were used for a long time to guide or illustrate the mathematical analysis [1]. Due to the ability of integrating a dynamical system numerically by the use of a computer, visualization techniques that are based on the numerical approximation of the Poincaré map have become popular [63,83]. There are a number of programs that calculate Poincaré maps [22]. See Sect. 5.3 for a brief review of previous work in this field.

Many visualization techniques for Poincaré maps suffer from rather severe limitations. One problem is, that with the use of 2D visualization techniques the context of the 3D flow is lost. Certain features, e.g., the number of windings of a Möbius band, can not be derived from the 2D Poincaré map alone. Another problem with these techniques is that the temporal correlation between points of the Poincaré map is not encoded within the 2D image. Most limitations of the 2D techniques in this field are usually not due to a weakness of the software or method, but rather due to an inherent difficulty with dimension reduction approaches.

We therefore propose a set of advances within this rather untouched field of visualization. First, we suggest to adapt some well-known visualization techniques as, e.g., spot noise, to Poincaré maps to improve the visual representation of the 2D map. See Sect. 5.3 for a discussion of these ideas. Furthermore we present an embedding of these techniques within a 3D visualization of the underlying flow. This approach allows to significantly reduce some limitations of previously known techniques. Refer to Sect. 5.6 for a description of this approach.

mailto:helwig@cg.tuwien.ac.at.