Talk:Limit cycle

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.

Stable / Unstable

The current explanation is self-contraddicting. In particular the sentence:

` In all other cases it is neither "stable" nor "unstable" '.

should be removed. "In all other cases" refers to possibility that there is a trajectory approaching the limit-cycle neither for t → + ∞, nor there is for t → - ∞. Then, this is not a limit-cycle, but just a periodic orbit.

As the article states correctly in the first sentence, there must be a converging(diverging) orbit to the limit-cycle for the latter to be called so.

More precisely (from http://www.scholarpedia.org/article/Limit_cycle):

“A periodic orbit $\Gamma$ on a plane (or on a two-dimensional manifold) is called a limit cycle if it is the $\alpha$-limit set or $\omega$-limit set of some point $z$ not on the periodic orbit, that is, the set of accumulation points of either the forward or backward trajectory through $z$, respectively, is exactly $\Gamma$.”

If no other issue arises, I will soon delete the sentence

` In all other cases it is neither "stable" nor "unstable" '.

77.99.150.36 (talk) 12:18, 28 March 2010 (UTC)

First picture

Hi, a limit cycle is a closed trajectory with at least one other trajectory spiraling into it. This is, what is said in the first sentence of this article. But the first picture does only show a closed trajectory without another trajectory spiraling into it. Please correct somebody this contradiction. I am no expert in nonlinear systems --188.103.105.136 (talk) 08:49, 28 March 2011 (UTC)

Visualization of four limit cycles. 16th Hilbert problem

There is a discussion of visulization of four limit cycles in two-dimensional quadratic polynomial system (in th framework of 16th Hilbert problem). Four limit cycles in two-dimensional quadratic polynomial system (16th Hilbert problem): x' = −(a1x2 + b1xy + c1y2 + α1x + β1y), y' = −(a2x2 + b2xy + c2y2 + α2x + β2y) for the coefficients a1 = b1 = β1 = −1, c1 = α1 = 0, b2 = −2.2, c2 = −0.7, a2 = 10, α2 = 72.7778, and β2 = −0.0015. Limit cycles L1,L2,L3,L4 (green color represents stable and red represents unstable). One limit cycle L4 (self-excited periodic attractor) around an unstable equilibrium (red dot) is shown in (a), while the localization of three nested limit cycles (L1,2,3; L2 is a hidden periodic attractor) around stable zero equilibrium (green dot) is presented in (b).

Four limit cycles in two-dimensional quadratic polynomial system (16th Hilbert problem)

nk (talk) 11:46, 22 March 2019 (UTC)

References

Applications

Hi, everybody. A new application of limit cycles has been recently published in a notable journal (https://doi.org/10.1007/s11071-020-05928-5) in connection with classical electrodynamics. I am the author of the paper, and therefore I am not the person allowed to upload the reference, since a COI is at stake. But perhaps somebody could simply add a modest line in the applications section say something like:

• Charged accelerated bodies can experience limit cycle behavior as a consequence of self-interactions

The notion of limit cycle has not received sufficient attention among the community of fundamental physics and it can help to introduce this trascendental concept of nonlinear dynamical systems to such community. Sincerely Alvaro12Lopez (talk) 09:44, 13 September 2020 (UTC).

-Oh my god! You derived Newton's laws from electrodynamics. Relativistic energy from the Liénard-Wiechert potential, as well. This is amazing. All theories based on kinetic energy plus potential cannot be fundamental. Kinetic energy is electrodynamic potential energy. Had never seen. And retarded potentials lead to retarded differential equations? So obvious, once seen. And your derivation of the quantum potential is splendid and totally analytic using just electromagnetic mass. Mass is not fundamental, just rest energy: like Einstein said. Did you know he believed in electromagnetic mass? Anyway, this is the first explanation to the wave-particle duality I have really seen. I congratulate you, it is a real change of paradigm. I had never heard of self-oscillation, neither, but seems like a widespread phenomenon of nonlinear dynamical systems. It even has a Wikipedia page... I have been trying to introduce the application you propose, but some user named Tercer is undoing it all the time. There is great ignorance about nonlinear dynamics and chaos theory, so be patient. I will keep trying here and there, and spread your ideas. Congrats!! Oleg.