Harper equation, the dissipative standard map and strange nonchaotic attractors: relationship between an eigenvalue problem and iterated maps

@article{Ketoja1997HarperET,
  title={Harper equation, the dissipative standard map and strange nonchaotic attractors: relationship between an eigenvalue problem and iterated maps},
  author={Jukka A. Ketoja and Indubala I. Satija},
  journal={Physica D: Nonlinear Phenomena},
  year={1997},
  volume={109},
  pages={70-80}
}
  • J. Ketoja, I. Satija
  • Published 1 November 1997
  • Physics, Mathematics
  • Physica D: Nonlinear Phenomena

Figures from this paper

Strange Nonchaotic Attractors From a Family of Quasiperiodically Forced Piecewise Linear Maps
TLDR
It is proved that there exists a unique strange nonchaotic attractor for some set of parameter values which is the graph of an upper semi-continuous function, which is invariant, discontinuous almost everywhere and attracts almost all orbits.
Universal mechanism for the intermittent route to strange nonchaotic attractors in quasiperiodically forced systems
To examine the universality for the intermittent route to strange nonchaotic attractors (SNAs), we investigate the quasiperiodically forced Henon map, ring map and Toda oscillator which are
Strange Nonchaotic attractors
TLDR
The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here.
Thermodynamics of critical strange nonchaotic attractors.
TLDR
The Tsallis nonextensive entropy which is known to characterize the thermodynamics of systems with leading Lyapunov exponent zero is found to be subadditive for the critical states.
Intermittency route to strange nonchaotic attractors in a non-skew-product map.
  • T. Mitsui, Y. Aizawa
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2010
TLDR
This paper derives Badard's non-skew-product map from a periodically driven continuous dynamical system with spatially quasiperiodic potential and investigates onset mechanisms of SNAs in the map, focusing on a transition route to intermittent SNAs.
Reliable Computation of Robust Response Tori on the Verge of Breakdown
TLDR
These proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with the novel computational techniques.
Critical Strange Nonchaotic Dynamics in the Fibonacci Map
TLDR
A driven quasiperiodic skew-product dynamical mapping in which orbits with all Lyapunov exponents equal to zero lie on fractal attractors form a special category of strange nonchaotic attractors (SNAs), and the scenario for their formation is described.
Strange nonchaotic attractors in Harper maps.
TLDR
It is proved that for a set of parameters of positive measure, the map possesses a SNA, but the set is nowhere dense, because by changing the parameter arbitrarily small amounts, the attractor is a smooth curve and not a S NA.
Quasiperiodically forced interval maps with negative Schwarzian derivative
In the study of quasiperiodically forced systems invariant graphs have a special significance. In some cases, it was already possible to deduce statements about the invariant graphs of certain
Bubbling route to strange nonchaotic attractor in a nonlinear series LCR circuit with a nonsinusoidal force.
TLDR
It is found that bubbles appear in the strands of the quasiperiodic attractor due to the instability induced by the additional square-wave-type force, and bubbles then enlarge and get increasingly wrinkled as a function of the control parameter.
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 56 REFERENCES
Universal criterion for the breakup of invariant tori in dissipative systems.
  • Ketoja
  • Physics
    Physical review letters
  • 1992
TLDR
Numerical investigation on the dissipative standard map suggests that this universal number could become observable in experiments, and is argued that the (minimal) slope of the critical iterated circle map is proportional to the effective Jacobian determinant.
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.
Recurrence of invariant circles in a dissipative standardlike map.
  • Kim, Hu
  • Physics
    Physical review. A, Atomic, molecular, and optical physics
  • 1991
TLDR
A dissipative standardlike map that contains a parameter z that can be used to tune the map from a piecewise-linear map to the dissipativeStandard map is studied, and the reappearance of an invariant circle after its breakup is observed.
Existence, nonexistence and universal breakdown of dissipative golden invariant tori. I. Golden critical circle maps
In this series of three papers the author rigorously formulates and proves a number of the main conjectures associated with renormalization of golden-mean quasi-periodic dynamical systems. In
Fractalization of a torus as a strange nonchaotic attractor.
  • Nishikawa, Kaneko
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1996
Fractalization of a torus and its transition to chaos in a quasiperiodically forced logistic map is reinvestigated in relation to a strange nonchaotic attractor, with the aid of a functional equation
Blowout Bifurcation Route to Strange Nonchaotic Attractors.
TLDR
It is shown that for dynamical systems with an invariant subspace in which there is a quasiperiodic torus, the loss of the transverse stability of the torus can lead to the birth of a strange nonchaotic attractor.
...
1
2
3
4
5
...