# Talk:Lyapunov exponent

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## Numerical computation

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)

# Incorrect Formula for Maximal Lyapunov Exponent

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)

Strogatz, Steven H "Nonlinear Dynamics and Chaos" (Ch. 10.5) defines the maximal Lyapunov exponent with the just the limit of t to infinity, as does many others.--Waxsin 22:36, 2 October 2007 (UTC)

The definition of Lyapunov-Exponents implicitly assumes that the limit for t -> inf exists, which might not be valid for finite disturbances in case of strange attractors: The difference of two initial conditions is then bounded by (twice) the attractor size, and hence limited; in that case, the formula will result in a Lyapunov Exponent of 0. An additional condition could be given by saying that ${\displaystyle \delta Z(t)}$ must always be small compared to the variance of the trajectory or stuff like that, but that's somewhat more complicated than adjusting the definition. Did Strogaz etc. explicitly mention Lyapunov exponents in strange attractors? Moritz, 27.Sept. 2010

Here there is a lot of confusion, the correct order of limits is the opposite of the one actually reported

in the definition, first one should ensure that the perturbation is small, than one can take


the limit t going to infinite, this is reported in any text book see M. Cencini et al, Chaos (world Scientific , 2009). I am going to correct it —Preceding unsigned comment added by 79.35.154.151 (talk) 17:54, 12 November 2010 (UTC)

The current maximal Lyapunov exponent definition

${\displaystyle \lambda =\lim _{t\to \infty }\lim _{\delta \mathbf {Z} _{0}\to 0}{\frac {1}{t}}\ln {\frac {|\delta \mathbf {Z} (t)|}{|\delta \mathbf {Z} _{0}|}}.}$

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 ${\displaystyle \max }$ or ${\displaystyle \sup }$ somewhere to get the maximal element of the Lyapunov spectrum. --Jaan Vajakas (talk) 17:41, 22 July 2012 (UTC)

# Calculation of Lyapunov exponent and strange attractors

If it is considered time-varying linearization (as it is necessary for investigation of chaotic behavior of trajectories on an attractor)

${\displaystyle J^{t}(x(t,x_{0}))=\left.{\frac {df^{t}(x)}{dx}}\right|_{x=x(t,x_{0})}}$

then positiveness of the largest Lyapunov exponent (as it is calculated above by the matrix of the first approximation ${\displaystyle J^{t}(x(t,x_{0}))}$, here ${\displaystyle x(t,x_{0})}$ --- 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. (talkcontribs) 07:39, 24 September 2011 (UTC)

## A bit too abstract

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)

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)

## Positive Lyapunov exponent

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)

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)

# Hammiltonian systems

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)

## Origin of name 'Lyapunov exponent'

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)

## Unclear use of "initial separation"

${\displaystyle \delta \mathbf {Z} _{0}}$

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)

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