Abstract
What physical quantity can we use to capture and describe the multitude of invariant sets and the local structure of unstable manifolds W u that determine the form and structure of chaos? By introducing the expansion rate of neighboring orbits, which expresses the stretching and folding of segments of W u, and the local dimension, which describes the self-similarity of the nested structure of strange attractors, the geometric and statistical descriptions given in terms of chaotic orbits through the fluctuations of these quantities can be unified. We show that the chaotic bifurcations and tangency structure of unstable manifolds can be directly understood in terms of the spectrum ψ(Λ), and also that a fixed relationship exists between the spectra of this expansion rate and the local dimension.
This is a preview of subscription content, log in via an institution to check access.
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
Eckmann, J.-P., Procaccia, I. (1986) Fluctuations of dynamical scaling indices in nonlinear systems. Phys. Rev. A 34, 659–661
Ellis, R.S. (1985) Entropy, Large Deviations, and Statistical Mechanics. Springer, Berlin, Heidelberg
Fujisaka, H., Inoue, M. (1987) Statistical-thermodynamics formalism of self- similarity. Prog. Theor. Phys. 77, 1334–1343
Grassberger, P., Badii, R., Politi, A. (1988) Scaling laws for invariant measures on hyperbolic and non-hyperbolic attractors. J. Stat. Phys. 51, 135–178
Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.I. (1986) Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33, 1141–1151
Hata, H., Horita, T., Mori, BL, Morita, T., Tomita, K. (1988) Characterization of local structures of chaotic attractors in terms of coarse-grained local expansion rate. Prog. Theor. Phys. 80, 809–826
Hata, H., Horita, T., Mori, H., Morita, T., Tomita, K. (1989) Singular local structures of chaotic attractors and q-phase transitions of spatial scaling structures. Prog. Theor. Phys. 81, 11–16
Horita, T., Hata, H., Mori H., Morita, T., Tomita, K., Kuroki, S., Okamoto, H. (1988a) Local structures of chaotic attractors and q-phase transitions at attractor-merging crises in the sine-circle maps. Prog. Theor. Phys. 80, 793–808
Horita, T., Hata, H., Mori, H., Morita, T., Tomita, K., (1988b) Singular local structures of chaotic attractors due to collisions with unstable periodic orbits in two-dimensional maps. Prog. Theor. Phys. 80, 923–928
Jensen, M.H., Kadanoff, L.P., Libchaber, A., Procaccia, I., Stavans, J. (1985) Global universality at the onset of chaos: results of a forced Rayleigh-Bénard experiment. Phys. Rev. Lett. 55, 2798–2801
Mori, H., Hata, H., Horita, T., Kobayashi, T. (1989a) Statistical mechanics of dynamical sytsems. Prog. Theor. Phys. Suppl. 99, 1–63
Mori, N., Kobayashi, T., Hata, H. Morita, T., Horita T., Mori, H. (1989b) Scaling structures and statistical mechanics of type-I intermittent chaos. Prog. Theor. Phys. 81, 60 - 77
Morita, T., Hata, H., Mori, H., Horita, T., Tomita, K. (1987) On partial dimensions and spectra of singularities of strange attractors. Prog. Theor. Phys. 78, 511–515
Morita, T., Hata, H., Mori, H., Horita, T., Tomita, K. (1988) Spatial and temporal scaling properties of strange attractors and their representations by unstable periodic orbits. Prog. Theor. Phys. 79, 296–312
Ott, E., Grebogi, C., Yorke, J. A. (1989) Theory of first order phase transitions for chaotic attractors of nonlinear dynamical systems. Phys. Lett. A 135, 343–348
Ruelle, D. (1978) Thermodynamic Formalism. Addison-Wesely, Reading, M. Sano, M., Sato S., Sawada, Y. (1986) Global Spectral Characterisitics of Chaotic Dynamics. Prog. Theor. Phys. 76, 945–948
Sano, M., Sato S., Sawada, y. (1986) Global Spectral Characteristics of Chaotic Dynamics. Prog. Theor. Phys. 76, 945–948
Shenker, S.J. (1982) Scaling behavior in a map of a circle onto itself; empirical results. Physica D 5, 405–411
Takahashi, Y. (1980) Chaos, periodic points and entropy. Butsuri 35, 149–161 (Japanese)
Takahashi, Y., Oono, Y. (1984) Towards the statistical mechanics of chaos. Prog. Theor. Phys. 71, 851–854
Tomita, K., Hata, H., Mori, H., Morita, T. (1988) Scaling structures of chaotic attractors and g-phase transitions at crises in the Hénon and the annulus maps. Prog. Theor. Phys. 80, 953–972
The statistical thermodynamics formalism of dynamical systems discussed in this chapter was first introduced from a mathematical point of view. To understand the motivation behind the development of this formal- ism, see, for example, Takahashi (1980) and Ellis (1985). The concrete treatment given in this chapter of the multifractal dimension D(q), the two spectra f(α) and ψ(Λ), and related quantities for physical dynamical systems follows the theoretical work contained in Halsey. (1986), Morita. (1988), and Grassberger. (1988), and the line of reasoning presented in Mori. (1989a), in which a number of subjects are treated in a unified manner. The attempt to unify the geometrical and statistical descriptions of chaos presented in this chapter has its origin in the papers by Horita. (1988a), Hata. (1988), Tomita. (1988), and Mori. (1989b), which seek a statistical characterization of the bifurcations arising from fluctuations in the coarse-grained expansion rate of orbits, and the paper Horita. (1988b), which deals with the theory regarding the slope sß resulting from collision with a saddle.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mori, H., Kuramoto, Y. (1998). The Statistical Physics of Aperiodic Motion. In: Dissipative Structures and Chaos. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-80376-5_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-80376-5_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-80378-9
Online ISBN: 978-3-642-80376-5
eBook Packages: Springer Book Archive