Renormalization of correlations and spectra of a strange non-chaotic attractor

@article{Feudel1996RenormalizationOC,
  title={Renormalization of correlations and spectra of a strange non-chaotic attractor},
  author={Ulrike Feudel and Arkady Pikovsky and Antonio Politi},
  journal={Journal of Physics A},
  year={1996},
  volume={29},
  pages={5297-5311}
}
We study a simple nonlinear mapping with a strange nonchaotic attractor characterized by a singular continuous power spectrum. We show that the symbolic dynamics is exactly described by a language generated from a suitable inflation rule. We derive renormalization transformations for both the power spectrum and the autocorrelation function, thus obtaining a quantitative description of the scaling properties. The multifractal nature of the spectrum is also discussed. 

Figures and Tables from this paper

Periodic orbits of renormalisation for the correlations of strange nonchaotic attractors
We calculate all piecewise-constant periodic orbits (with values ±1) of the renormalisation recursion arising in the analysis of correlations of the orbit of a point on a strange nonchaotic
The non-smooth pitchfork bifurcation: a renormalization analysis
We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions,
Self-similar correlations in a barrier billiard
RENORMALIZATION IN QUASIPERIODICALLY FORCED SYSTEMS
We review our recent rigorous results on renormalization in a variety of quasiperiodically forced systems. Our results include a description of (i) self-similar fluctuations of localized states in
Renormalization analysis of correlation properties in a quasiperiodically forced two-level system
TLDR
A rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system and calculates explicitly the asymptotic height of the main peaks in the correlation function.
Renormalization of correlations in a quasiperiodically forced two-level system for a general class of modulation function
We provide a renormalization analysis of correlations in a quasi-periodically forced two-level system in a time dependent field with periodic kicks whose amplitude is given by a general class of
Renormalization of correlations in a quasiperiodically forced two-level system: quadratic irrationals
Generalizing from the case of golden mean frequency to a wider class of quadratic irrationals, we extend our renormalization analysis of the self-similarity of correlation functions in a
Renormalization of fluctuations for a generalized Harper equation for periodic continued fractions
A renormalization analysis is presented for a generalized Harper equation For values of the parameter ω having periodic continued fraction expansion, we construct the periodic orbits of the
...
...

References

SHOWING 1-10 OF 41 REFERENCES
Correlations and spectra of strange non-chaotic attractors
We consider correlations and spectra of strange nonchaotic attractors in quasiperiodically driven nonlinear systems. It is demonstrated that a self-similar autocorrelation function and a singular
Scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors
The analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems. Prominent roles in our approach are played by the generalized
Characterizing strange nonchaotic attractors.
TLDR
It is shown that phase sensitivity appears if there is a nonzero probability for positive local Lyapunov exponents to occur and a phase sensitivity exponent is calculated, that measures the sensitivity with respect to changes of the phase of the external force.
FRACTAL AND DYNAMICAL PROPERTIES OF THE KICKED HARPER MODEL
We review recent work on the so-called kicked Harper model, which can be viewed either as a model system in the framework of quantum chaos, or as a pulsed version of the Harper model, which has been
Self-similarity and localization.
TLDR
It is shown that the whole localized phase is characterized by a single strong coupling fixed point of the renormalization equations, which describes the generalized Harper model with next nearest neighbor interaction below a certain threshold.
Scaling properties of a structure intermediate between quasiperiodic and random
We consider a one-dimensional structure obtained by stringing two types of “beads” (short and long bonds) on a line according to a quasiperiodic rule. This model exhibits a new kind of order,
A structure intermediate between quasi-periodic and random
We consider an infinite chain of atoms, where the bond lengths between neighbouring sites take two values, according to a quasi-periodic rule, associated to a circle map with an irrational rotation
Structure and electronic properties of Thue-Morse lattices.
We study a one-dimensional system which is neither periodic, quasiperiodic, nor random. We find that the structure factor of this system consists of a set of peaks whose heights scale with L, the
...
...