i wanna calculate Lyapunov exponent and i dont know any thing about it can you introduce for me some good source about it??
thanks
Before hopping on technical scientific articles, I would advise you to read standard books on nonlinear dynamics and chaos to understand the physical mechanisms.
1) Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Strogatz
2) Chaotic Dynamics: An Introduction Based on Classical Mechanics by Tamás Tél and Márton Gruiz
3) Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers by Robert Hilborn
4) Chaos: Making a New Science by James Gleick
The first book is a true classic for studying nonlinear dynamics and the fourth one gives you the insight without introducing a single mathematical equation!
Another interesting article is available for free on RG. https://www.researchgate.net/publication/10731142_Fundamentals_of_synchronization_in_chaotic_systems_concepts_and_applications
Article Fundamentals of synchronization in chaotic systems, concepts...
In graduate courses on analysis, you are taught that, if you integrate a differential equation of a mechanical system with two initial conditions that are close enough, the solution to the differential equation remains the same: This is the Cauchy criterion.
In chaotic systems, instead, it happens the exact opposite: if you integrate the system with two arbitrarily close initial conditions, the solutions diverge with time. If you observe this divergence in a very short time window, the divergence is exponential. The Lyapunov exponent is a measure of this divergence, in the sense that the two solutions diverge like exp(lambda*t), where lambda is the Lyapunov exponent, and t the time.
I don't know what system you are dealing with, but the basic idea behind is this.
Dear Mohammadreza
The usual test for chaos is calculation of the largest Lyapunov exponent. A positive largest Lyapunov exponent indicates chaos. When one has access to the equations generating the chaos, this is relatively easy to do. When one only has access to an experimental data record, such a calculation is difficult to impossible, and that case will not be considered here. The general idea is to follow two nearby orbits and to calculate their average logarithmic rate of separation. Whenever they get too far apart, one of the orbits has to be moved back to the vicinity of the other along the line of separation. A conservative procedure is to do this at each iteration. The complete procedure is as follows:
http://sprott.physics.wisc.edu/chaos/lyapexp.htm
http://www.msandri.it/docs/Numerical_Calculation_of_Lyapunov_Exponents.pdf
https://www.physionet.org/physiotools/lyapunov/RosensteinM93.pdf
From: Journal of Theoretical Biology 289(2011)181–196:
The role of seasonality and import in a minimalistic multi-strain dengue
model capturing differences between primary and secondary infections:
Complex dynamics and its implications for data analysis
M. Aguiar, S.Ballesteros , BW.Kooi, N.Stollenwerk
The Lyapunovexponents can be calculated using an iterated technique using the QR decomposition algorithm via Householder matrices (see Aguiar etal.,2008; Eckmann etal.,1986; Holzfuss and Lauterborn,1989; Holzfuss and Parlitz,1991).
Aguiar, M.,Kooi,B.W.,Stollenwerk,N.,2008.Epidemiology of dengue fever:
a model with temporary cross-immunity and possible secondary infection
shows bifurcations and chaotic behaviour in wide parameter regions.
Math.Model. Nat. Phenom.4,48–70.
Holzfuss, J.,Lauterborn,W.,1989.Liapunov exponents from a time series of
acoustic chaos. Phys.Rev.A39,2146–2152.
Holzfuss, J.,Parlitz,U.,1991.Lyapunov exponents from time series.Lecture Notes in Mathematics,vol.1486;1991,pp.263–270.
Eckmann, J.-P.,Kamphorst, O.S., Ruelle,D., Cilliberto,S.,1986 .Liapunov exponents from time series. Phys.Rev.A34,4971–4979.
For calculating the lyapunov exponents of a chaotic system you can try with the following paper
A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov Exponents from a Time Series," Physica D, Vol. 16, pp. 285-317, 1985.
I think before any book you must be seen the
http://chaosbook.org/
It has open book and video together!
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