A fast algorithm for estimating Lyapunov exponents from time series

Computation of the Lyapunov exponents in the compass-gait model under OGY control via a hybrid Poincaré map

“…Stabilization of the one-periodic fixed point x − * of the uncontrolled hybrid Poincaré map (6) lies in the stabilization of the linearized Poincaré map (12). According to [72], we introduce the following state-feedback controller:…”

“…Generally, the Lyapunov exponents are used for chaos identification in nonlinear dynamical systems and as a powerful tool for analyzing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. Many research studies focused on the design of numerical methods to estimate the spectrum of Lyapunov exponents or the largest one from time series or data sets [3][4][5][6][7][8][9][10][11][12][13][14][15]. In addition, several papers dealt with the design of numerical/analytical methods for the computation of the spectrum of Lyapunov exponents.…”

Nonlinear analysis using Lyapunov exponents in breast thermograms to identify abnormal lesions

“…Otherwise, this computation will lead only to the largest LEs. To deal with these problems, the QR factorisation algorithm, which decomposes the matrix into an orthogonal and an upper triangular matrix, is used for approximation of LEs [13][14][15][16]. The steps involved in this method can be summarized as follows:…”

Simulation and Visualization of Chaotic Systems

“…In the case of unknown time series embedded in geometrical spaces with arbitrary values for the embedding dimension d and the time delay τ, this simplified procedure cannot be applied; in this case, the estimation of the Lyapunov exponent is performed by using a lot of different algorithms (Bremen, Udwadia, & Proskurowski, 1997;Brown, Bryant, & Abarbanel, 1991;Bryant, Brown, & Abarbanel, 1990;Chistianssen & Rugh, 1997;Diakonos, Pingel, & Schmelcher, 2000;Hegger, 1999;Oiwa & Fielder, 1998;Oiwa & Fielder, 2002;Pyragas, 1997;Rosenstein, Collins, & de Luca, 1993;Sano & Sawada, 1985;Wright, 1984) each one of them has its own advantages and disadvantages and its own speed and accuracy. In this project this task is performed by the Wolf's algorithm (Wolf, Swift, Swinney, & Vastano, 1985), that allows the estimation of the largest positive Lyapunov exponent of an unknown time series; the main idea behind this algorithm, is the selection of a trajectory point, the identification of its nearest spatial neighbor and the measurement of their distance as they evolve in time by t e (this parameter is known as evolution time).…”