Theory and Computation of Covariant Lyapunov Vectors
- 1.2k Downloads
Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.
KeywordsCovariant Lyapunov vectors Characteristic Lyapunov vectors Forward and backward Lyapunov vectors Lyapunov exponents Lyapunov analysis Tangent space High-dimensional chaos
The research leading to the results has received funding from the European Community’s Seventh Framework Programme FP7/2007–2013 under grant agreement No. HEALTH-F2-2009-241526, EUTrigTreat. P.V.K. acknowledges support from RFBR-DFG under Grant No. 08-02-91963.
- Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK users’ guide (1999) Google Scholar
- Bochkanov, S., Bystritsky, V.: ALGLIB NET. Electronic resource. http://www.alglib.net (1999)
- Kuptsov, P.V.: Vychislenie pokazateley Lyapunova dlya raspredelennyh sistem: preimuschestva i nedostatki razlichnyh chislennyh metodov. Izv. Vuz. Prik. Nelinejn. Din. 5 (2010). [Computation of Lyapunov exponents for spatially extended systems: advantages and limitations of various numerical methods, Appl. Nonlinear Dyn. 5 (2010) (in Russian)] Google Scholar
- Legras, B., Vautard, R.: A guide to Lyapunov vectors. In: Palmer, T. (ed.) Predictability Seminar Proc., ECWF Seminar, vol. 1, pp. 135–146. European Centre for Medium-Range Weather Forecasts, Reading (1996) Google Scholar
- Pazó, D., Rodríguez, M.A., López, J.M.: Spatio-temporal evolution of perturbations in ensembles initialized by bred, Lyapunov and singular vectors. Tellus 62A, 10–23 (2010) Google Scholar
- Samelson, R.M., Wolfe, C.L.: Lyapunov vectors for large systems. In: Exploring Complex Dynamics in High-Dimensional Chaotic Systems: From Weather Forecasting to Oceanic Flows. MPIPKS, Dresden (2010). http://www.pks.mpg.de/~ecodyc10/Contributions/Samelson.pdf Google Scholar