Abstract
This series of lectures reviews some aspects of the theory of Lyapunov characteristic exponents (LCE) and its application to spatially extended systems, namely chains of coupled oscillators and coupled map lattices (CML). After introducing the main definitions and theorems and presenting the most widely used computational algorithm (due to Benettin et al.) in Section 1, I give an account, in Section 2, of the existence of a Lyapunov spectral density in the thermodynamic limit (first conjectured by Ruelle for the Navier-Stokes equation and then numerically evidentiated for the Fermi-Pasta-Ulam oscillator chain by Politi et al.). Although not intrinsically defined, Lyapunov eigenvectors are an important tool for studying spatial development of chaos: their localization property (in the sense of Anderson theory) is presented in Section 3, together with a recent generalization known as chronotopic analysis, due to Lepri et al.. The random matrix approximation allows to obtain reasonable estimates of the LCE when chaos is well developed and correlations are weak: some examples of analytical calculation of scaling laws in the perturbation parameter for random Verlet matrices are discussed in Section 4, along the path opened by Parisi et al.. Finally, in Section 5 I discuss a recently discovered phenomenon known as coupling sensitivity (Daido), which is the sharp increase (as 1/ln(ε)) of the maximal LCE when the coupling diffusive parameter ε is switched on in CML models; the analytical treatment is quite appealing, making reference to an interesting probabilistic model, the random energy model of Derrida.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benettin, G., Physica D 13, 211 (1984).
Benettin, G., L. Galgani, A. Giorgilli & J.M. Strelcyn, Meccanica 9, 21 (1980).
Benzi, R., G. Paladin, G. Parisi & A. Vulpiani, J. Phys. A 18, 2157 (1985).
Bunimovich, L. & Y. Sinai, Nonlinearity 1, 491 (1988).
Casetti, L., R. Livi & M. Pettini, Phys. Rev. Lett 74, 375 (1995); T. Dauxois, S. Ruffo, A. Torcini, unpublished (1997).
Channel, P.J. & C. Scovel, Nonlinearity 3, 231 (1990; E. Forest, R.D. Ruth, Physica D 43, 105 (1990); H. Yoshida, Phys. Lett A, 150, 262 (1990).
Crisanti, A., G. Paladin & A. Vulpiani, Products of Random Matrices in Statistical Physics, Springer, Berlin (1993).
Daido, H., Prog. Theor. Phys. 72, 853 (1984).
De Luca, J., A.J. Lichtenberg, S. Ruffo, Phys. Rev. E 51, 2877 (1995).
Derrida, B. Phys. Rev. B 24 2613 (1981).
Eckmann, J.P. & E. Wayne, Comm. Math. Phys. 121, 147 (1989).
Eckmann, J.P. & E. Wayne, J. Stat. Phys. 50, 853 (1988).
Evans, D.J., E.G.D. Cohen & G.P. Morris, Phys. Rev. A42, 5990 (1990); E.G.D. Cohen and G. Gallavotti, Phys. Rev. Lett. 74, 2694 (1995).
II, p. 978); also reprinted in Nonlinear Wave Motion, A. C. Newell ed., Lect. Appl. Math. 15, AMS, Providence, Rhode Island (1974); also in The Many-Body Problem, D. C. Mattis ed., World Scientific, Singapore (1993).
Ford, J., Phys. Rep. 213, 271 (1992).
Giacomelli, G. & A. Politi, Europh. Lett. 15, 387 (1991).
Grassberger, P. & H. Kantz, Phys. Lett. Al23, 437 (1987).
Hoover, W.G., C.G. Tull & H.A. Posch, Phys. Lett. A131, 211 (1988).
Isola, S., A. Politi, S. Ruffo & A. Torcini, Phys. Lett. A143, 365 (1990).
Kaneko, K., Prog. Theor. Phys. 72, 480 (1984).
Kaneko, K. & T. Konishi, J. Phys. Soc. Jpn. 56, 2993 (1987).
Kolmogorov, A.N., Dokl. A.ad. Nauk SSSR, 98, 527 (1954); J. Moser, Nachr. Akad. Wiss. Göttingen Math. Phys. Kl., 2, 1 (1962); V. I. Arnol’d, Russ. Math. Surv., 18, 9 and 85 (1963).
Lepri, S., A. Politi & A. Torcini, J. Stat. Phys. 82, 1429 (1996).
Lima R. & S. Ruffo, J. Stat. Phys. 52, 259 (1988).
Livi, R., M. Pettini, S. Ruffo, M. Sparpaglione & A. Vulpiani, Phys. Rev. A31, 1039 (1985); R. Livi, M. Pettini, S. Ruffo, A. Vulpiani, Phys. Rev. A31, 2740 (1985).
Livi, R., M. Pettini, S. Ruffo & A. Vulpiani, J. Stat. Phys. 48, 539 (1987).
Livi, R., A. Politi & S. Ruffo, J. Phys. A 25, 4813 (1992).
Livi, R., A. Politi & S. Ruffo, J. Phys. A 19, 2033 (1986).
Livi, R., A. Politi, S. Ruffo & A. Vulpiani, J. Stat. Phys. 46, 147 (1987).
Mac Lachlan, R.I. & P. Atela, Nonlinearity 5, 541 (1992).
Manneville, P., in Macroscopic Modeling of Turbulent Flows, O. Pironneau ed., Lecture Notes in Physics 230, 319, Springer-Verlag, Berlin (1985).
Newman, C.M., Comm. Math. Phys. 103, 121 (1986).
Oseledec, V.I., Trans. Moscow Math. Soc. 19, 197 (1968).
Paladin, G. & A. Vulpiani, Phys. Rep. 156, 147 (1987).
Paladin, G. & A. Vulpiani, J. Phys. A 19, 1881 (1986).
Parisi, G. & A. Vulpiani, J. Phys. A 19, L425 (1986).
Pettini M. & M. Landolfi, Phys. Rev. A 41, 768 (1990).
Pichard, J.L. & G. André, Europh. Lett. 2, 477 (1986).
Pikovsky, A.S., J. Techn. Phys. Lett. 11, 672 (1985).
Ruelle, D., Ann. Inst. Poincaré 42, 109 (1985).
Ruelle, D., Commun. Math. Phys. 87, 287 (1982).
Shimada, I. & T. Nagashima, Frog. Theor. Phys. 61, 1605 (1979).
Sinai, Ya. G., A Remark Concerning the Thermodynamical Limit of Lyapunov Spectrum, preprint (1995).
Verlet, L., Phys. Rev. 159, 89 (1967).
Waller, I. & R. Kapral, Phys. Rev. A30, 2047 (1984).
Yamada, M. & K. Ohkitani, Phys. Rev. Lett. 60, 983 (1988).
Yang, W., E.J. Ding & M. Ding, Phys. Rev. Lett. 76, 1808 (1996).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Ruffo, S. (1999). Lyapunov Spectra in Spatially Extended Systems. In: Goles, E., Martínez, S. (eds) Cellular Automata and Complex Systems. Nonlinear Phenomena and Complex Systems, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9223-9_6
Download citation
DOI: https://doi.org/10.1007/978-94-015-9223-9_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5154-7
Online ISBN: 978-94-015-9223-9
eBook Packages: Springer Book Archive