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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.
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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
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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.
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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
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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
1053:). 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).
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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
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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
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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
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This is still an issue. The article has inconsistent notation and needs a good polishing. I'm going to tag some math & physics contributors who might be able to help in hope that someone can take on this (small but important) task.
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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
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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.
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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?
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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.
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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.
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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.--
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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.
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734:{\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}|}}.}
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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! --
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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
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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
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The computation of the Lyapunov exponents is not without its perils. As an experiment, try computing the Lyapunov exponents for the canonical
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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 Knowledge.
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If it is considered time-varying linearization (as it is necessary for investigation of chaotic behavior of trajectories on an attractor)
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The section on mathematical methods mentions the L-matrix, yet the theory section before uses different notation. Can someone clarify?
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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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
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but it is unstable, i.e. any small deviation from that value will grow over time and head towards either plus or minus infinity.
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then positiveness of the largest Lyapunov exponent (as it is calculated above by the matrix of the first approximation
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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https://web.archive.org/web/20060425194442/http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
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Here there is a lot of confusion, the correct order of limits is the opposite of the one actually reported
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It's easy to have non-chaotic system with positive lyapunov exponent, for instance x_{i+1}=2x_i
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in the definition, first one should ensure that the perturbation is small, than one can take
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can be implemented directly. It provides the entire spectrum and better analytic control.
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If you found an error with any archives or the URLs themselves, you can fix them with
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939:{\displaystyle J^{t}(x(t,x_{0}))=\left.{\frac {df^{t}(x)}{dx}}\right|_{x=x(t,x_{0})}}
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The definition of Lyapunov-Exponents implicitly assumes that the limit for t -: -->
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1049:--- a trajectory on the attractor) doesn’t, in general, indicate chaos (see --
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that are used without being clearly defined, or without being defined
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It's an extremely bad idea to use a math symbol before defining it,
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It's an extremely bad idea to use a math symbol before defining it
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http://www.mpipks-dresden.mpg.de/eprint/freddy/9702017/9702017.ps
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somewhere to get the maximal element of the Lyapunov spectrum. --
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in the introductory paragraph to a math article in Knowledge.
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for additional information. I made the following changes:
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and for the Hénon map with a = 1.39945219 and b = 0.3.
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Oxford University Press, 2003—see also online supplement
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One of the worst math articles I have seen in Knowledge
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Calculation of Lyapunov exponent and strange attractors
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Perron effects of Lyapunov exponent sign inversions
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509:867. (Dec. 30, 1998), pp. 41-60.--
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