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Flow Topology

Flow Topology. CS 419 Scientific Visualization John C. Hart Source: Notes on the Topology of Vector Fields and Flows by Dan Asimov. Example: Phase Space. <Position, Velocity> Trash can lid Pendulum phase Angle (periodic) Angular velocity

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Flow Topology

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  1. Flow Topology CS 419 Scientific Visualization John C. Hart Source:Notes on the Topology of Vector Fields and FlowsbyDan Asimov

  2. Example: Phase Space • <Position, Velocity> • Trash can lid • Pendulum phase • Angle (periodic) • Angular velocity • Equations of motion arevector field on phase space • Characterized by topology • Stable minimum at restwith lid down • Unstable saddle whenlid balanced upside down AngularVelocity Angle

  3. Flow Notation p • Notation • ap: trace of point p • ap(t): trace point at time t • Consistency condition: • Let q = ap(s) • Then aq(t) = ap(s+t) • More notation: f(p,t) = ap(t) • Consistency: f(f(p,s),t) = f(p,s + t) • C1: f differentiable wrt p and t • Flow: ft(p) = f(p,t) with t constant • Visual flow: correspondence between pixels from one frame to the next ap q

  4. Example: Linear Flow • If V(p) = Mp • Then f(p,t) = etMp • How do you raise e to a matrix power? • Recall: ex = Sxi/i! • So just replace x with M eM = S (1/i!) Mi • Example: M is a 90º rotation

  5. Never Crossthe Streams! • Traces of differentiable vector fields never converge • May appear to converge as t • Example: V(x,y) = (1,3y2/3) • Horizontal trace: V(x,0) = (1,0) • Vertical trace: t(t,t3) • Horizontal and vertical traces cross! • Because V not everywhere differentiable V(x,y) = (1,3y2/3)

  6. Trace Classification • Trace: fp(t): R Rn • Regular trace • fp(t) is one-to-one • Trace never visits same point twice • Closed orbit • fp(t + s) = fp(t) • Trace is a simple closed curve • Stationary point • fp(t) = p • Vector field vanishes • Aka: equilibrium, singularity, fixed point, zero (and critical point if gradient vector field)

  7. Regular Traces • Fundamental Theorem: If p is a point on a regular trace, then the neighboring traces in a neighborhood of p can be continuously deformed into a set of parallel lines. • Topology of a sufficiently small neighborhood of a regular trace is same as a straight flow

  8. Closed Orbits • Notation • x(t) = fp(t) for some p • w(x) = lim x(t) as t • a(x) = lim x(-t) as t • Limit cycle: closed orbit C s.t.x • C  w(x) • (or C  a(x)) • but x C • If, for a planar flow, w(x) is bounded, non-empty and contains no stationary point, then w(x) is a limit cycle • Limit cycles are stable x(t) If a trace is bounded, then • w(x) is not empty • w(x) is closed • w(x) is invariant • w(x) is connected

  9. Stationary Points • Isolated if only stationary point in a sufficiently small neighborhood • Classified by Jacobian (matrix of partial derivatives) v(t) = (u(t),v(t)) • Stationary point is hyperbolic if real parts of eigenvalues of Jacobian are non-zero • Locally quadratic • Stationary points are stable

  10. 2-D Stationary Point Classification • Real eigenvalues + + Source – – Sink + – Saddle • Complex congugate eigenvalues + + Spiral source – – Spiral sink • Spiral direction given by curl(v) curl(v) = u/y– v/x + counterclockwise – clockwise Eugene Zhang, 2004

  11. 3-D Stationary Point Classification • Real eigenvalues + + + Source – – – Sink + + – and + – – Saddles • Real and complex eigenvalues + + + Spiral source – – – Spiral sink + + – and + – – Spiral saddles

  12. Volume PreservingFlow Topology Gradient(curl-free) • Divergence free:   v = 0 • No set can flow onto a proper subset • No sources or sinks • Just saddles and centers • Centers • Stationary point in 2-D whose neighbors are all closed orbits • Eigenvalues purely imaginary (necessary but not sufficient) • Not hyperbolic • Stable for volume-preserving vector fields mixed VolumePreserving(div-free) Eugene Zhang, 2004

  13. Poincare Map q • Poincare map of an orbit C maps a subset of a perpendicular disk to the disk by following neighboring orbits • Assign coordinates to disk • Poincare map is a flow on the disk • Jacobian of Poincare map classifies orbits • Orbit is hyperbolic if eigenvalues not unit length • Structurally stable p C

  14. Orbit Classification • Magnitude of Poincare Jacobian eigenvalues • Real eigenvalues >1 >1 Source orbit <1 <1 Sink orbit >1 <1 Saddle orbit • Complex conjugate eigenvalues >1 spiral source orbit <1 spiral sink orbit • Product of eigenvalues always positive

  15. Topology of Orbits • Twisted orbits • Orbits might twist around each other • E.g. Ribbon visualizations • Knotted orbits • An orbit may not be continuously deformed into a circle without self-intersecting • Linked orbits • The neighborhoods of linked orbits may not be continuously deformed into separated spaces

  16. Seifert Conjecture • Consider a vector field in a solid torus • No stationary points • Vectors perpendicular to torus surface • Vectors point inward at boundary • Can such a vector field have no closed orbits?

  17. Stable Manifolds • Critical element (stationary point or closed orbit) • Stable manifold of a critical element • All points flowing into a critical element in positive time • Aka basin of attraction • Unstable manifold • All points flowing into a critical element in negative time • All points flowing out of a critical element • Stable/Unstable manifold invariant under flow Morse-Smale complex from Edelsbrunner et al., 2003

  18. Stable Manifolds of Stationary Points • 2-D • Source: just the source point • Sink: basin • Saddle: saddle point and the two traces in the negative eigenvector directions • 3-D • Other saddle: surface extending from the plane spanned by the two eigenvectors

  19. Stable Manifolds of Closed Orbits • Source closed orbit • 1-D stable manifold • The orbit itself • Saddle closed orbit • 2-D stable manifold • Surface (cylinder) extendingfrom the orbit • Sink closed orbit • 3-D stable manifold • Solid torus around the orbit V(x,y,z) = (y, z, 3.2x – 2y – z – x2) Hinke Osinga, 1999

  20. Wandering • A point is wandering if its neighborhood never flows back into itself • Non-wandering set is flow invariant • Any critical element of a vector field is necessarily contained in its non-wandering set • If, over a bounded domain, a vector field is volume preserving and has no exiting traces, then the entire domain is non-wandering

  21. Morse-Smale Flows • Structurally stable flows • Vector field is Morse-Smale if • It has finite # of hyperbolic critical elements • The non-wandering set is the union of closed orbits and stationary points • Stability manifolds intersect only transversely (no “merging”) Simplified gradient field topologyGyulassy, Vis05

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