Vector Field Editing and Periodic Orbit Extraction Using Morse Decomposition

Abstract

Abstract-Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.

An image-space Morse decomposition for 2D vector fields

“…13 Theisel et al 14 proposed a grid-independent method for the location of periodic orbits. Chen et al 3 introduced an efficient method for periodic orbit detection from 2D/2.5D vector fields and defined a more complete vector field topology by including periodic orbits into the topology construction. In the meantime, saddle-saddle connector has been introduced by Theisel et al 15 for 3D vector field topology visualization to reduce the occlusion issue.…”

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

“…[37]), and also in an automatized method for the computation of the Conley index from index pairs constructed as cubical sets in R n [41,48], used e.g. in a method for editing vector fields and extraction of periodic orbits [8], and also very helpful in an automatic method for classification of dynamics in multi-parameter systems [4,7], which found applications e.g. in population biology [38] and in physics of plasmas [49].…”

Conley's Fundamental Theorem for a Class of Hybrid Systems

“…First, the decomposition establishes a clear, deterministic notion of steady state behavior that, no matter how complicated its temporal manifestation [44,62,65], imposes a computationally effective [51] spatial partition into attractor basins [69] whose topology persists under small perturbations. The passage from signal to symbol afforded by such partitions has great value for analyzing natural systems [6,33], and has encouraged slowly growing use in the synthesis of engineered systems as well [1,22,41,42]. Second, interpreted as a universal converse Lyapunov theorem, and a global analogue to the classical counterpart addressing a specific basin [52], the long-established value for classical [88], multistable [30], and hybrid control systems theory [36] is leveraged by a steadily advancing literature on constructive methods for their eventual feedback closed loops [10,34].…”