Re: planar vector field conjugated with a constant one?

Pavel Pokorny wrote:

Maarten van Reeuwijk <maarten@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Pavel Pokorny wrote:

Dear nonlinear friends

is it true that a planar continuous vector field with no equilibrium
points is topologically conjugated with a constant nonzero field?

The absence of equilibrium points mean that streamlines do not cross
throughout all space and thus there must be a smooth transformation that
aligns these to a constant flow in one direction.

HTH, MAarten

That's the way I feel it.
But can we prove it?

Pavel

Start in the origin and create a line following the flow forward and
backward. This line defines xi(x,y) = 0. Do the same perpendicular to the
flow which defines eta(x, y) = 0. From every point of eta=0, release
imaginary particles and have them follow the flow, and the distance
traveled defines eta(x,y).
Do the same for xi(x,y), but now they travel perpendicular to the flow.
If the resulting system is invertible, i.e. x(xi, eta) and y(xi, eta) can be
defined, this is a one-to-one mapping and a proper coordinate
transformation. You will see that in the (xi, eta) coordinate system the
flow is parallel, if there are no fixed points in the system. This is
probably what you mean with 'topologically' identical.

HTH, Maarten
--
===================================================================
Maarten van Reeuwijk dept. of Multiscale Physics
Phd student Faculty of Applied Sciences
maarten.ws.tn.tudelft.nl Delft University of Technology
.