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.Abstract:
Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrodinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. 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. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.read more
Citations
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Quasiperiodically forced interval maps with negative Schwarzian derivative
TL;DR: In this article, the authors study quasiperiodically forced interval maps which are monotonically increasing and have negative Schwarzian derivative and derive some basic results which only require monotonicity, and give a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems.
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Old and new results on strange nonchaotic attractors
TL;DR: Classical and new results concerning the topological structure of skew-products semiflows, coming from nonautonomous maps and differential equations, are combined in order to establish rigorous conditions giving rise to the occurrence of strange nonchaotic attractors on 𝕋d × ℝ.
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Applicability of 0-1 test for strange nonchaotic attractors
TL;DR: In this paper, the 0-1 test was used to detect the transition from quasiperiodic to chaotic motion via SNAs in terms of the golden mean number of the translation variables.
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The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations
TL;DR: In this paper, a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems is proposed.
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Applicability of 0-1 Test for Strange Nonchaotic Attractors
TL;DR: The recently introduced 0-1 test can successfully distinguish between strange nonchaotic attractors (SNAs) and periodic/quasiperiodic/chaotic attractsors, by suitably choosing the arbitrary parameter associated with the translation variables in terms of the golden mean number.
References
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Jean-Pierre Eckmann,David Ruelle +1 more
TL;DR: A review of the main mathematical ideas and their concrete implementation in analyzing experiments can be found in this paper, where the main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions).
Book
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TL;DR: In the new edition of this classic textbook, the most important change is the addition of a completely new chapter on control and synchronization of chaos as mentioned in this paper, which will be of interest to advanced undergraduates and graduate students in science, engineering and mathematics taking courses in chaotic dynamics, as well as to researchers in the subject.