Talk:Limit-cycle
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[edit] Definition
A few days ago I replaced the erroneous definition of limit cycle with the correct one.
The gist of the erroneous definition, also appearing in MathWorld, is repeated many places on the Web. But a look at any good book on ODE's should persuade any doubter that for a closed trajectory C to be a limit cycle, there *must* be another trajectory that in forward time -- or else in backward time -- spirals in to C.
Check out a good O.D.E. book such as any of these:
1. Solomon Lefschetz (Princeton, 1985)
2. Philip Hartman (SIAM, 2002)
3. Witold Hurewicz (Dover, 2002)
4. Lawrence Perko (Springer, 2006).
The one popular math site I know of that has it right is PlanetMath, at <http://planetmath.org/encyclopedia/UnstableLimitCycle.html>.
Someone's complete re-edit (of my editing) of this article sends it back to an incorrect definition again: "As the dynamical system evolves" is a phrase that has no precise -- or even imprecise -- mathematical meaning that makes sense in this context.
If the closed trajectory C is a limit cycle, there *must* (not merely "might") be another trajectory that spirals into C as t -> oo, or else as t -> -oo.
(In lieu of an an editing war, I prefer to convince whoever else is editing this article of the correctness of what I'm saying, so please look at a book such as the above.)
The mis-definition of limit cycle here, until it is fixed, also invalidates the article Poincaré-Bendixson theorem.Daqu 16:24, 6 March 2006 (UTC)
- Some well-intentioned but unknowledgeable person who does not realize how weak is his/her comprehension of limit-cycles has been repeatedly adding
erroneous and/or meaningless phrases to the article. PLEASE check with an expert before wrecking the article for a third time. IF IN DOUBT: please discuss it here before making another mess that someone else must clean up.Daqu 05:11, 16 April 2006 (UTC)
Yet again, someone who is not familiar with the subject matter has introduced two false statements, which I have removed: that a limit cycle is an attractor (true only if it is an attracting limit cycle) or that it is the limit set of the flow.
Please discuss it here first before screwing things up. It's getting tiring for me to fix other people's mistakes.
