Talk:Hyperbolic equilibrium point

The word "hyperbolic" vs. saddle point

The description in the introduction given to why a fixed point is called hyperbolic is a mischaracterization. A saddle point has orbits that look like hyperbolas, but a source or sink (both hyperbolic) does not. Cf. Strogatz, who writes:

"[Hyperbolic] is an unfortunate name -- it sounds like it should mean "saddle point" -- but is has become standard."

If no one objects in the near future, I'm going to edit the introduction to reflect this. I'll keep the image and all that, because I think it's even better to know where the name hyperbolic comes from, and why not all hyperbolic fixed points have orbits that look like hyperbolas. Bradweir (talk) 22:48, 15 October 2010 (UTC)

• I believe that if you rotate a source or sink through imaginary space, it will become a hyperbola. I am not sure, however. I am also not sure if this is the actual definition. 67.170.103.34 (talk) 22:16, 9 December 2010 (UTC)

• Please be more precise. A source/sink is a point, which you can't rotate. You rotate vectors. If you're rotating the trajectories in R², and viewing it as the complex plane, then there's no way you can rotate the trajectories into hyperbolas, because this transformation wouldn't be continuous. Btw, usually the term imaginary numbers is reserved for the purely complex numbers, i.e. a real multiplied by i, whereas the space of all numbers a + ib is the complex numbers. Bradweir (talk) 03:30, 21 February 2011 (UTC)

• it is not precised that the saddle point has eigenvalues with zero imaginary part. And the exemple does not contain any hyperpolic points (saddles). The point p=(0,0) described in the example is stable focus... — Preceding unsigned comment added by 82.228.182.165 (talk) 21:44, 16 June 2011 (UTC)

• I have no idea how to parse this. Please write in complete, coherent sentences and use English words (this is an English Wikipedia page). Bradweir (talk) 04:25, 12 July 2011 (UTC)

• I suspect the term "hyperbolic point" comes from 2D Hamiltonian systems, because in that case a hyperbolic point is necessarily a saddle point, and the trajectories (in the linearized system) are true hyperbolas due to the pairing of the eigenvalues.

• That sounds plausible. However, the sentence we're discussing is still false. Do you have any citations that suggest this is the origin of the name? If it's just a supposition it doesn't belong in the article, even though I'm inclined to believe you. Bradweir (talk) 04:25, 12 July 2011 (UTC)

Completely different definition..?

Are you certain that's the correct definition of an hyperbolic fixed point? As much as I know, the definition should be "The real part of all eigenvalues of the Jacobian is nonzero." For example, Academic paper on dynamic systems — Preceding unsigned comment added by 109.66.80.241 (talk) 20:59, 13 February 2015 (UTC)

• Both definitions describe the same thing: https://controls.engin.umich.edu/wiki/index.php/EigenvalueStability 84.156.89.63 (talk) 20:48, 28 October 2015 (UTC)

• • No, they don't. The important thing is that the pole is marginally stable. It's marginally stable when the real component is zero. If it's on the unit circle, that would imply that you have taken the Z transform or similar. Fredo699 (talk) 14:01, 12 January 2024 (UTC)

Diffy Q too hard

Equilibrium is something that is natural in dynamic systems. Giving it a name or a stranded has ZERO relevance to the subject. Also please stop quoting differential equations that most people can't make sense of (Including Wiki editors).

In real physics we have a word we use called KISS. "Keep it Simple Stupid". I suggest that this is what needs to be done if you want Wikipedia to make sense to Normal people. "ALL" Differential equations need to be removed from Wikipedia if you want a encyclopedia that makes sense. Strogatz was a fool trying to dispute what he did not understand. We loose so much real science when people do this. It happens all the time on Wikipedia too. When editors don't understand something that is contributed so instead of asking questions before they edit something. You are so fast to delete it. If you really did know the answer or had any understanding what so ever of the subject the you would be able to "EDIT", proof-read or even add to it. But, no you vandalise then accuse the contributor of being the vandal ?. HELLO ?. You need to read a dictionary before you call yourselves editors because you don't even know the meaning of the word editor like so meny other words you debate. That's why most Wikipedia so-called editors don't even know what a editor does. I call people like that "Reverted". There is a big difference between editing and contributing so you need to be more professional and show a little respect. — Preceding unsigned comment added by GCHQ-Mi-6 (talk • contribs) 01:39, 27 June 2017 (UTC)