I'll agree that the tangent vectors to any such curve can (probably) be extended to a nonzero vector field on some open set in R^3 containing it. But my understanding of the Lorenz attractor (which resembles two LP records that are mating), is sufficiently limited, and the Lorenz attractor sufficiently complicated, that I cannot say whether this provides an example or not. —Dan
On Feb 10, 2016, at 12:07 PM, Allan Wechsler <acwacw@gmail.com> wrote:
How about an orbit under the Lorenz system of differential equations? https://en.wikipedia.org/wiki/Lorenz_system
On Wed, Feb 10, 2016 at 2:57 PM, Dan Asimov <asimov@msri.org> wrote:
Let
f: R —> R^3
be a C^oo one-to-one mapping of the reals into 3-space.
(For convenience, assume WLOG that ||f'(t)|| is never 0.)
Suppose further that
a) The image f(R) is closed and bounded in R^3;
and
b) If for some sequence t_j in R we have
lim f(t_j) = f(t) j—>oo
for some t in R, then we also have convergence of the tangent vectors:
lim f'(t_j) = f'(t) j—>oo
-------------------------------------------------
Question: Does there exist such a curve as f ???
(Note: If not for b), an easy example would be for the image f(R) to be a figure-8.)
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun