Vector Field Topology Josh Levine

Overview Vector fields (VFs) typically used to encode many different data sets: –e.g. Velocity/Flow, E&M, Temp., Stress/Strain Area of interest: Visualization of VFs Problem: Data overload! –One solution: Visualize a “skeleton” of the VF by viewing its topology

Vector Fields A steady vector field (VF) is defined as a mapping: –v: N → TN, N a manfold, TN the tang. bundle of N In general, N = TN ≈ R n An integral curve is defined by a diff. eqn: –d  /dt = v(  (t)), with  o, t o as initial conditions –Also called streamlines

Vector Fields A phase portrait is a depiction of these integral curves: Image: A Combinatorial Introduction to Topology, Michael Henle

Critical Points A critical point is a singularity in the field such that v(x) = 0. Critical points are classified by eigenvalues of the Jacobian matrix, J, of the VF at their position: –e.g. in 2d, If J has full rank, the critical point is called linear or first-order Hyperbolic critical points have nonzero real parts

Critical Points Image: Surface representations of 2- and 3-dimensional fluid flow topology, Helman & Hesselink

Critical Points Generally: –R > 0 refers to repulsion –R < 0 refers to attraction e.g. a saddle both repels and attracts –I ≠ 0 refers to rotation e.g. a focus and a center –Note in 2d case I 1 = -I 2

Sectors & Separatrices In the vicinity of a critical point, there are various sectors or regions of different flow type: –hyperbolic: paths do not ever reach c.p. –parabolic: one end of all paths is at c.p. –elliptic: all paths begin & end at c.p. A separatrix is the bounding curve (or surface) which separates these regions

Sectors & Separatrices Images: A topology simplification method for 2D vector fields. Xavier Tricoche, Gerik Scheuermann, & Hans Hagen

Sectors & Separatrices Images: A topology simplification method for 2D vector fields. Xavier Tricoche, Gerik Scheuermann, & Hans Hagen

Planar Topology Planar topology of a VF is simply a graph with the critical points as nodes and the separatrices as edges. e.g.:

Poincaré Index Another topological invariant The index (a.k.a. winding number) of a critical point is number of VF revolutions along a closed curve around that critical point By continuity, always an integer The index of a closed curve around multiple critical points will be the sum of the indices of the critical points

Poincaré Index The index around no critical point will always be zero For first order critical points, saddle will be -1 and all others will be +1 There is a combinatorial theory that shows:

Three Dimensions In 3D, we classify critical points in a similar manner using the 3 eigenvalues of the Jacobian Broadly, there are 2 cases: –Three real eigenvalues –Two complex conjugates & one real

Three Dimensions Images: Saddle Connectors – An approach to visualizing the topological skeleton of complex 3D vector fields, Theisel, Weinkauf, Hege, and Seidel Left-to-right: Nodes, Node-Saddles, Focus, Focus-Saddles Top: Repelling variants; Bottom: Attracting variables Left-half: 3 real eigenvalues; Right-half: 2 complex eigenvalues

Three Dimensions Separatrices now become 2d surfaces and 1d curves. Thus topology of first-order critical points will be composed of the critical points themselves + curves + surfaces Images: Saddle Connectors – An approach to visualizing the topological skeleton of complex 3D vector fields, Theisel, Weinkauf, Hege, and Seidel

Vector Field Equivalence We can call two VFs equivalent by showing a diffeomorphism which maps integral curves from the first to the second and preserves orientation A VF is structural stable if any perturbation to that VF results in one which is structurally equivalent In particular, nonhyperbolic critical points (such as centers) mean a VF is unstable because an arbitrarily small perturbation can change the critical point to a hyperbolic one.

Bifurcations Consider an unsteady (time-varying) VF: –v: N  I → TN, I  R As time progresses, topological transitions, or bifurcations, will occur as critical points are created, merged, or destroyed Two main classifications, local (affecting the nature of a singular point) and global (not restricted to a particular neighborhood)

Local Bifurcations Hopf Bifurcation –A sink is transformed into a source –Creates a closed orbit around the sink: Image: Topology tracking for the visualization of time-dependent two-dimensional flows, Xavier Tricoche, Thomas Wischgol, Gerik Scheuermann, & Hans Hagen

Local Bifurcations Also, Fold Bifurcations: –Pairwise annihilation of saddle & source/sink: Image: Topology tracking for the visualization of time-dependent two-dimensional flows, Xavier Tricoche, Thomas Wischgol, Gerik Scheuermann, & Hans Hagen

Global Bifurcations Basin Bifurcation –Separatrices between two saddles “swap” –Creates a heteroclinic connection Image: Topology tracking for the visualization of time-dependent two-dimensional flows, Xavier Tricoche, Thomas Wischgol, Gerik Scheuermann, & Hans Hagen

Global Bifurcations Periodic Blue Sky Bifurcation –Between a saddle and a focus –Creates a closed orbit and a source –Passes through a homoclinic connection Image: Topology tracking for the visualization of time-dependent two-dimensional flows, Xavier Tricoche, Thomas Wischgol, Gerik Scheuermann, & Hans Hagen

Visualization Images: Stream line and path line oriented topology for 2D time-dependent vector fields, Theisel, Weinkauf, Hege, and Seidel

Conclusions By observing the topology of a VF, we present a “skeleton” of the information, i.e. the defining structure of the VF In doing so, we can consider only areas of interest such as critical points or in the unsteady case bifurcations