Talk:Lyapunov exponent

message: The section on mathematical methods mentions the L-matrix, yet the theory section before uses different notation. 
Can someone clarify? 
Benjamin.friedrich (talk)  —Preceding undated comment added 12:51, 22 January 2019 (UTC)
thread_title: L-matrix

message: I removed the citation:
: Sprott J. C. Chaos and Time-Series Analysis Oxford University Press, 2003—see also online supplement Numerical Calculation of Largest Lyapunov Exponent
because the method described in that online supplement should be limited to the case when the equation of motion is not known analytically.   When the equations of motion are known, the method described in the Wikipedia article can be implemented directly. It provides the entire spectrum and better analytic control.
The computation of the Lyapunov exponents is not without its perils.  As an experiment, try computing the Lyapunov exponents for the canonical Hénon map and for the Hénon map with a = 1.39945219 and b = 0.3.   XaosBits 04:43, 9 January 2006 (UTC)
thread_title: Numerical computation

message: It should only be the limit of t to infinity, not also the limit of delta zero to infinity as the limit of delta zero to infinity alone is the definition for the short-time Lyapunov exponent according to Siopis, Christos, Barbara L. Eckstein, and Henry E. Kandrup.  “Orbital Complexity, 
Short-Time Lyapunov Exponents, and Phase Space Transport in Time-Independent Hamiltonian Systems.”  Annals of the New York Academy of Sciences, Vol. 867. (Dec. 30, 1998), pp. 41-60.--Waxsin 22:36, 2 October 2007 (UTC)
replies: text: \lambda = \lim_{t \to \infty} \lim_{\delta \mathbf{Z}_0 \to 0} 
\frac{1}{t} \ln\frac{| \delta\mathbf{Z}(t)|}{|\delta \mathbf{Z}_0|}.
only works in the case when there is only one element in the Lyapunov spectrum; e.g. in case of a one-dimensional system. Actually the word "maximal" in "maximal Lyapunov exponent" is then superfluous. In multiple dimensions, there should generally be \max or \sup somewhere to get the maximal element of the Lyapunov spectrum. --Jaan Vajakas (talk) 17:41, 22 July 2012 (UTC)
thread_title: Incorrect Formula for Maximal Lyapunov Exponent

message: If it is considered time-varying linearization (as it is necessary for investigation of chaotic behavior of trajectories on an attractor) 
:  J^t(x(t,x_0)) = \left. \frac{ d f^t(x) }{dx} \right|_{x = x(t,x_0)} 
then positiveness of the largest Lyapunov exponent (as it is calculated above by the matrix of the first approximation  J^t(x(t,x_0)) , here  x(t,x_0)  --- a trajectory on the attractor) doesn’t, in general, indicate chaos (see  -- Perron effects of Lyapunov exponent sign inversions).
In general case, additional justification of time-varying linearization and calculations of Lyapunov exponents is required (G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron effects, International Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007, pp. 1079-1107).  — Preceding unsigned comment added by Kuznetsov N.V. (talk • contribs) 07:39, 24 September 2011 (UTC)
thread_title: Calculation of Lyapunov exponent and strange attractors

message: The article is helpful for those who already have a grasp on multidimensional dynamical systems. As a layman and non-mathematician however, I'd prefer if a special, less abstract explanation was provided. I found several explanations for the Lyapunov exponent in simple one dimensional systems on the web. It should be easy for someone with a background in mathematics to add such an alternative perspective for the uninitiated to the article. Thank you! --89.48.244.241 12:26, 5 September 2006 (UTC)
replies: text: The article Lissajous orbit says "Lyapunov orbits around a libration point are curved paths that lie entirely in the plane of the two primary bodies.". The article should explain and illustrate a Lyapunov orbit. Cuddlyable3 (talk) 22:24, 9 October 2009 (UTC)
replies: text: 12 years later, but I'm seconding this, as there are links to this e.g. from articles on potential Earth-impactor asteroids that mention a "Lyapunov Time Horizon" as a feature of their somewhat chaotic motion. I presume this means the period after which the relationship between a prediction made at time zero and the object's actual motion at time T is no more accurate than the output of a random number generator (or, say, it's headed in at least 90 degrees / maybe a full 180 degrees vs the predicted vector), but having scanned through this article for sections I have any chance of understanding (my hard-maths education effectively ended around 17-18 years old), I haven't been able to confirm that... because the language is simply too jargon heavy with insufficient layperson explanation (remember, this is a public encyclopaedia) for any chance of comprehension. 146.199.0.203 (talk) 15:31, 3 March 2018 (UTC)
thread_title: A bit too abstract

message: It's easy to have non-chaotic system with positive lyapunov exponent, for instance x_{i+1}=2x_i
Actually the above case has no attractor.  The case x=0 is a fixed point (mathematics) but it is unstable, i.e. any small deviation from that value will grow over time and head towards either plus or minus infinity. Fracton (talk) 00:37, 25 September 2009 (UTC)
replies: text: Chaotic systems do not need attractors. Perturbations in the above example grow, but not exponentially, so by the formal definition the Lyapunov exponent is zero.  — Preceding unsigned comment added by 67.84.156.202 (talk) 02:02, 3 November 2017 (UTC)
thread_title: Positive Lyapunov exponent

message: The chaotic pendulum has a chaotic attractor in phase space. Therefore the statement that Hamiltonian systems do not have attractors is not right, is it? Or did I understand something wrong here? Herrkami (talk) 20:45, 28 February 2014 (UTC)
thread_title: Hamiltonian systems

message: Does anyone know who invented this term? Or better, where Lyapunov first used the idea? I doubt whether he did because the definition would mean discussing eigenvalues of matrices which was not current in his time. Certainly matrices were not used in his well-known book on stability JFB80 (talk) 18:31, 25 September 2014 (UTC)
thread_title:  Origin of name 'Lyapunov exponent'  

message: :\delta\mathbf{Z}_0
Is introduced to the reader in the first paragraph as "initial separation," and then it continues to be used in the definition of maximal Lyapunov exponent section with no clear definition.
If the definition cannot be written succinctly, maybe this object warrants either its own section or its own page.
If it has its own page, the fix is to link to it in the same sentence as it is introduced.  — Preceding unsigned comment added by 2601:19B:500:4BBA:ED36:601F:F71D:65D7 (talk) 17:46, 29 April 2016 (UTC)
thread_title:  Unclear use of "initial separation" 

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thread_title:  External links modified 

message: It's an extremely bad idea to use a math symbol before defining it, especially in the introductory paragraph to a math article in Wikipedia.
Yet this is currently the case with this article.2600:1700:E1C0:F340:405:A577:58A7:9956 (talk) 23:20, 6 March 2019 (UTC)
thread_title:  It's an extremely bad idea to use a math symbol before defining it 

message: I have now read much farther, and I am very sorry to say: This is one of the very worst mathematics articles I have ever seen in Wikipedia.
The main reason for this is that in numerous places there are bother notation and terminology that are used without being clearly defined, or without being defined at all.
I strongly recommend that this article be deleted in its entirety, to be later rewritten by someone who is both familiar with the subject matter and with how to write mathematics clearly.2600:1700:E1C0:F340:405:A577:58A7:9956 (talk) 23:37, 6 March 2019 (UTC)
thread_title:  One of the worst math articles I have seen in Wikipedia