Consider a 2D (or a higher dimension) chaotic map like Tinkerbell map:
x(n+1) = x(n)2 -y(n)2 +a x(n) +b y(n)
y(n+1) = 2x(n)y(n) +c x(n) +d y(n)
Suppose that we don’t know the above equations and we only have access to the time series of “x” (long enough, clean). Can we calculate the Largest Lyapunov Exponent?
In addition:
Can we calculate the all of the Lyapunov Exponents?
What is the effect of noise?
What is the effect of short time series?
I appreciate any references, codes, and comments.
Thank you in advance
Dear Sir,
I strongly recommend you the the papers:
1. Wolf, A., Swift, J.B., Swanney, H.L., Vastano, J.A., Determining Lyapunov exponents from a time series, Physica D, vol. 16, pg. 285, 1985.
2. Benettin, G., Galgani, L., Giorgelli, A., Strelcyn, J.M., Lyapunov characteristic exponents for smooth dynamical systems: a method for computing all of them, Part 1: theory, Part 2: numerical applications, Meccanica, Vol. 15, pag. 9 – 30, 1980.
I sent you a code in Matlab.
All the best,
prof. Deleanu
Dear Jafar,
It is very much possible to compute ALL of the Lyapunov exponents for time series of known dynamical systems. There are several algorithms such as Wolf et al,, Rosenstein, Kantz and other methods. You will need to compute the delay and embedding dimension. Rosenstein is very robust against noise and efficient for short time series. Will send some papers and codes later.
Thanks
Dear Sajad,
Some comments and caveats on Wolf and Benettin algorithms:
1)Augustova, P., Beran, Z., and Celikovsky, S. (2015). ISCS 2014: Interdisciplinary Symposium on Complex Systems, Emergence, Complexity and Computation (Eds.: A. Sanayei et al.), chapter On Some False Chaos Indicators When Analyzing Sampled Data, pages 249{258. Springer.
2) Tempkin, J. and Yorke, J. (2007). Spurious Lyapunov exponents computed from data. SIAM Journal on Applied Dynamical Systems, 6(2):457{474.
3) http://link.springer.com/article/10.1007%2Fs11071-016-2678-4
4) http://arxiv.org/pdf/1510.03835v2.pdf
WBR
NK
Cite
2 Recommendations
Juvencio Alberto Betancourt-Mar
Mexican Institute of Complex Systems
Dear Sajad Jafari:
It is posible to calculate the largest lyapunov exponents from time series. The spectrum is more difficult, but it is posible, too. You can review the effect of noise and short time series in this book and paper:
1. H. Kantz and T. Schreiber, ``Nonlinear Time Series Analysis'', 2nd edition, Cambridge University Press, Cambridge (2004).
2. R. Hegger, H. Kantz, and T. Schreiber, Practical implementation of nonlinear time series methods: The TISEAN package, CHAOS 9, 413 (1999)
The TISEAN package implements the Kantz, Rosenstein and Sano and Sawada algorithms:
http://www.mpipks-dresden.mpg.de/~tisean/Tisean_3.0.1/index.html
Sincerely,
Juvencio Alberto Betancourt Mar
Cite
Y. Azizi
@Dumitru, does your program reconstruct dynamics? I have some issue with this part in Matlab
Cite
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