Abstract
A spectral decomposition of the Frobenius–Perron operator is constructed for one-dimensional maps with intermittent chaos, using the method of coherent states. A technique using the spectral density function is applied to the the well-known cusp map, which generates weak type-II intermittency. Higher-order corrections are obtained to the leading 1/t long-time behavior of the x−x autocorrelation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference
M. Schroeder, Fractals, Chaos, Power Laws (W. H. Freeman, New York, 1991).
D. Sornette, Critical Phenomena in Natural Sciences (Springer-Verlag, Berlin, 2000).
P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: An explanation of 1/f noise, Phys. Rev. Lett. 59:381-384 (1987).
E. Weeks, T. Solomon, J. Urbach, and H. Swinney, Observations of anomalous diffusion and Lévy flights, in Lévy Flights and Related Topics in Physics, M. Shlesinger, G. Zaslavsky, and U. Frisch, eds. (Springer-Verlag, Berlin, 1995).
P. Manneville, Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems, J. Physique 41:1235-1243 (1980).
Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys. 74:189-197 (1980).
H. G. Schuster, Deterministic Chaos (VCH Publishers, Weinheim, 1988).
A. Lasota and M. C. Mackey, Chaos, Fractals and Noise (Springer-Verlag, New York, 1994).
D. J. Driebe, Fully Chaotic Maps and Broken Time Symmetry (Kluwer Academic Publishers, Dordrecht, 1999).
P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, 1998).
I. Antoniou and G. Lumer, eds., Generalized Functions, Operator Theory, and Dynamical Systems (Chapman & Hall/CRC Press, Boca Raton, 1999).
D. Ruelle, Resonances of chaotic dynamical systems, Phys. Rev. Lett. 56:405-409 (1986).
A. Van Der Ziel, On the noise spectra of semi-conductor noise and of flicker effect, Phys. 16:359-372 (1950).
W. Saphir and H. Hasegawa, Spectral representations of the Bernoulli map, Phys. Lett. A 171:317-322 (1992).
I. Antoniou and S. Tasaki, Spectral decomposition of the Renyi map, J. Phys. A 26:73-94 (1993).
E. C. G. Sudarshan, Coherence and chaos, in Coherent States: Past Present and Future (World Scientific Publishing, 1994).
H. H. Hasegawa and E. Luschei, Exact power spectrum for a system of intermittent chaos, Phys. Lett A 186:193-202 (1994).
Z. Kaufman, H. Lustfeld, and J. Bene, Eigenvalue spectrum of the Frobenius-Perron operator near intermittency, Phys. Rev. E 53:1416-1421 (1996).
S. Grossman and H. Horner, Long time tail correlations in discrete chaotic dynamics,Z. Phys. B 60:79-85 (1985).
P. Collet and S. Isola, On the Essential Spectrum of the Transfer Operator for Expanding Markov Maps, Comm. Math. Phys. 139:551-557 (1991).
M. J. Feigenbaum, Presentation functions, fixed points, and a theory of scaling function dynamics, J. Statist. Phys. 52:527-569 (1988).
P. C. Hemmer, The exact invariant density for a cusp-shaped return map, J. Phys. A 17:L247-L249 (1984).
G. Györgyi and P. Szépfalusy, Fully developed chaotic 1-d maps, Z. Phys B 55:179-186 (1984).
H. H. Rugh, Intermittency and regularized Fredholm determinants, Invent. Math. 135:1-24 (1999).
J. Spanier and K. B. Oldham, An Atlas of Functions (Hemisphere Publishing Corporation, New York, 1987).
A. Ben-Mizrachi, I. Procaccia, N. Rosenberg, A. Schmidt, and H. G. Schuster, Real and apparent divergencies in low-frequency spectra of nonlinear dynamical systems, Phys. Rev. A 31:1830-1840 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nelson, K., Driebe, D.J. Ensemble Dynamics of Intermittency and Power-Law Decay. Journal of Statistical Physics 111, 1183–1207 (2003). https://doi.org/10.1023/A:1023000232088
Issue Date:
DOI: https://doi.org/10.1023/A:1023000232088