A hybrid method for computing Lyapunov exponents
The authors propose a numerical method based on the discrete QR-method combined with spatial integration to compute all the Lyapunov exponents of a dynamical system. This is a combination of the usual time averages in the computation of Lyapunov exponents (in this case a common QR discrete method) and the method of spatial averages introduced by \textit{P. J. Alston} and \textit{M. Dellnitz} [Comput. Methods Appl. Mech. Eng. 170, No.~3--4, 223--237 (1999; Zbl 0945.65142)] using the software GAIO. As explained in the work, it is based on the computation of the sequence of integrals converging to the Lyapunov exponents from the following theorem: Denoting by \(\lambda_1 \geq \ldots \geq \lambda_d\) the Lyapunov exponents (including multiplicities) of a discrete dynamical system \(g: M \rightarrow M\) (\(C^1\)-diffeomorphism on a \(d\)-dimensional submanifold \(M\) of \(\mathbb{R}^k\)), and further assuming that, \[ \lambda_j=\lim_{n \to \infty} \frac{1}{n} \ln R_{jj}(Dg^n(x)) \quad \mu \text{ a.e.}, \quad j=1,\ldots,d. \] Then one has, \[ \lambda_j=\lim_{n \to \infty} \frac{1}{n} \int \ln R_{jj}(Dg^n(x)) d\mu(x), \quad j=1,\ldots,d, \] where \(R_{jj}(A)\) stands for the \(j\)th diagonal element of the \(R\)-factor in the QR-decomposition of a nonsingular matrix \(A\) and \(Dg^n\) denotes the derivative of the \(n\)th iterate \(g^n\). In the paper, the methodology is applied to the Hénon system and to the time-1-map Lorenz system and the atmospheric model of \textit{R. V. Bekryaev} [Atmospheric and Oceanic Physics 31, No.~4, 455--464 (1996)]. An analysis of the errors under the conditions of the Oseledets theorem [\textit{V. I. Oseledets}, Trans. Mosc. Math. Soc. 19, 197--231 (1968); translation from Tr. Mosk. Mat. Obshch. 19, 179--210 (1968; Zbl 0236.93034); ] is presented (but will be proved in a forthcoming paper) and the extrapolation results to accelerate the convergence are implemented in these examples.
scientific article; zbMATH DE number 5603874
scientific article