While analyzing chaos, Lyapunov exponent can be obtained if we know the differential equations. However, if we only have the time series data (numerically or experimentally obtained), is it possible to determine the Lyapunov exponent? If yes, what is the procedure?
Dear Pranav,
There are different algorithms to numerically estimate the Lyapunov spectrum or the Largest Lyapunov Exponent (LLE) from the scaler time history (Let's say experimentally obtained temporal response). Two of the most popular are :
1) Wolf's algorithm and 2) Rosenstein algorithm. However, it is advisable to have a sufficiently long time history to get a converged value of the numerically estimated LLE.
The references are as follows:
1) Rosenstein, M. T., Collins, J. J., & De Luca, C. J. (1993). A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena, 65(1-2), 117-134.
2) Wolf, A., Swift, J. B., Swinney, H. L., & Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena, 16(3), 285-317.
You can also follow two of my publications for details:
1) Chandan Bose, Vikas Reddy, Sayan Gupta, Sunetra Sarkar (2017): Transient and stable chaos in dipteran flight inspired flapping flight, Journal of Computational and Nonlinear Dynamics, Volume 13, Issue 2, 021014, DOI: 10.1115/1.4038447.
2) Transition to chaos in the flow-induced vibration of a pitching-plunging airfoil at low Reynolds numbers : Ruelle-Takens-Newhouse scenario (under review).
Best regards,
Chandan
Dear Pranav
Here you have an useful paper for determining the Lyapunov exponents from time series. Also, if you work with Matlab, you'll find a Matlab script for Lyapunov exponents for a discrete system.
Best regards,
Dumitru Deleanu
the chaos can be analyzed by the strong methods such as the Melnikov method, the Global approach or Hamiltonian procedure and etc. obtaining the Lyapunov function for the non-autonomous systems (time dependent equations) usually is difficult
The chaos and hyper chaos can be analyzed by the various methods such:
1- Wolf's algorithm: Algorithms to numerically estimate the Lyapunov spectrum or the Largest Lyapunov Exponent.
2- The 0-1algorithm for chao: It works directly with the time series and daos not require any phase space reconstruction. for more detail see forexemple:
Ref: Gottwald, G. A and Melbourne, I (2009), on the implementation of the 0-1 test for chaos, SiaM. J. App. Dyn. 8, 129-145.
3. Methode for finding topological Horseshoes in dynamical system, for exemple see: Ref: Topological Horseshoe Analysis, Ultimate Boundary Estimations of a New 4D Hyperchaotic System and Its FPGA Implementation
Enzeng Dong et al., International Journal of Bifurcation and Chaos
Calculating the Lyapunov exponent for time series is not acceptable in principle in terms of its definition. The scientific community has accepted a procedure for its calculation (e.g., highly cited paper by Wolf, 1985), with some later modifications, as a “right” one. Accepting some procedures by either the part or whole scientific community does not necessarily mean that it is always justified in principle.
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