Dressler, Ute; Farmer, J. Doyne Generalized Lyapunov exponents corresponding to higher derivatives. (English) Zbl 0778.58036 Physica D 59, No. 4, 365-377 (1992). Let \(f:I\to I\) be a differentiable map on some \(I\subset\mathbb{R}\). The paper introduces the expression \(\lim_{n\to\infty}{1\over n}\log|{d^ p\over dx^ p}f^ n(x)|\) to be the \(p\)th order Lyapunov exponent. The notion is extended to \(f:M\to M\), \(M\) an \(n\)-dimensional manifold, expressing the \(p\)th derivative of \(f\) by the \(p\)th derivative of the induced map in a chart. The cases of fixed points and of periodic orbits of \(f\) are discussed in some detail for both the one- and the \(n\)- dimensional case, providing expressions for the \(p\)th order exponents in terms of the first order exponents. Reviewer: H.Crauel (Saarbrücken) Cited in 3 Documents MSC: 37A99 Ergodic theory 37E99 Low-dimensional dynamical systems Keywords:differentiable map; Lyapunov exponent; fixed points; periodic orbits PDFBibTeX XMLCite \textit{U. Dressler} and \textit{J. D. Farmer}, Physica D 59, No. 4, 365--377 (1992; Zbl 0778.58036) Full Text: DOI References: [1] Auerbach, D.; Cvitanovic, P.; Eckmann, J-P.; Gunaratne, G., Exploring chaotic motion through periodic orbits, Phys. Rev. Lett., 58, 2387-2389 (1987) [2] Ute Dressler, Gottfried Mayer-Kress and W. Lauterborn, Local stability analysis of dynamical systems, in preparation.; Ute Dressler, Gottfried Mayer-Kress and W. Lauterborn, Local stability analysis of dynamical systems, in preparation. [3] Farmer, J. D.; Sidorowich, J. J., Predicting chaotic time series, Phys. Rev. Lett., 59, 845-848 (1987) [4] Farmer, J. D.; Sidorowich, J. J., Exploiting chaos to predict the future and reduce noise, Los Alamos Preprint (1988) [5] Hirsch, M. W.; Smale, S., Differential Equations, Dynamical Systems, and Linear Algebra (1979), Academic Press: Academic Press Orlando [6] Oseledec, V. I., A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Moscow Math. Soc., 19, 197 (1968) · Zbl 0236.93034 [7] T. Taylor, private communication.; T. Taylor, private communication. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.