Correlations and spectra of strange nonchaotic attractors

Pikovsky, Arkady; Feudel, Ulrike

doi:10.1088/0305-4470/27/15/020

Cited by 104 publications

(109 citation statements)

References 13 publications

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“…Actually the existence of SNA was first identified by Grebogi et al [12] in their work on the transition from two-frequency torus to chaos via SNA. Later on, it was found that these attractors can arise in physically relevant situations such as a quasiperiodically forced pendulum [14,22], quantum particles in quasiperiodic potentials [14], biological oscillators [15], Duffing-type oscillators [16][17][18], velocity-dependent potential systems [11], electronic circuits [19,20], and in certain maps [21][22][23][24][25][26] in different transitions to SNA including the torus doubling bifurcation and the birth of SNAs. Also, the existence of torus doubling truncation and the appearance of SNA was confirmed by an experiment consisting of a quasiperiodically forced, buckled magnetoelastic ribbon [27].…”

Section: Introductionmentioning

confidence: 99%

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations

Venkatesan

Lakshmanan

2001

Phys. Rev. E

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A simple quasiperiodically forced one-dimensional cubic map is shown to exhibit very many types of routes to chaos via strange nonchaotic attractors (SNAs) with reference to a two-parameter (A−f ) space. The routes include transitions to chaos via SNAs from both one frequency torus and period doubled torus. In the former case, we identify the fractalization and type I intermittency routes. In the latter case, we point out that atleast four distinct routes through which the truncation of torus doubling bifurcation and the birth of SNAs take place in this model. In particular, the formation of SNAs through Heagy-Hammel, fractalization and type-III intermittent mechanisms are described. In addition, it has been found that in this system there are some regions in the parameter space where a novel dynamics involving a sudden expansion of the attractor which tames the growth of period-doubling bifurcation takes place, giving birth to SNA. The SNAs created through different mechanisms are characterized by the behaviour of the Lyapunov exponents and their variance, by the estimation of phase sensitivity exponent as well as through the distribution of finite-time Lyapunov exponents. PACS number(s): 05.45.+b

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Section: Introductionmentioning

confidence: 99%

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations

Venkatesan

Lakshmanan

2001

Phys. Rev. E

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“…For the parallel problem of the autocorrelation function in strange nonchaotic attractors, there is evidence in [14] that, at least for certain quadratic irrational frequencies, the autocorrelation function displays self-similarity of the type studied here. Periodic orbits of a generalisation of the functional recurrence (1.1) provide an explanation of this phenomenon in both settings, and will be the subject of a forthcoming paper.…”

Section: Resultsmentioning

confidence: 68%

Renormalization analysis of correlation properties in a quasiperiodically forced two-level system

Mestel

Osbaldestin

2002

Journal of Mathematical Physics

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We give a rigorous renormalization analysis of the self-similarity of correlation functions in a quasiperiodically forced two-level system. More precisely, the system considered is a quantum two-level system in a time-dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency. Mathematically, our analysis consists of a description of all piecewise-constant periodic orbits of an additive functional recurrence. We further establish a criterion for such orbits to be globally bounded functions. In a particular example, previously only treated numerically, we further calculate explicitly the asymptotic height of the main peaks in the correlation function.

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“…We conjecture that the spectrum of the process has a singular continuous component [Queffélec, 1987]. Such spectra have been studied recently for complex nonchaotic systems like strange nonchaotic attractors [Pikovsky & Feudel, 1994], substitution sequences [Zaks et al, 1998], fluid flows [Zaks, 2001], etc. Unfortunately, in the present case it is hardly possible to extract the singular continuous component from the spectrum, because it has also very strong discrete components.…”

Section: Correlationsmentioning

confidence: 95%

Dynamics Between Order and Chaos in a Simple Reentrant Model of Production Dynamics

Katzorke

Pikovsky

2003

Int. J. Bifurcation Chaos

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We consider a simple reentrant model of a manufacturing process, consisting of one machine at which two different types of items have to be processed. The model is completely deterministic: all delivery and processing times are fixed, and are generally incommensurate. Dependent on the arrival and processing times, a queue of waiting items grows, remains constant or disappears. We demonstrate that the dynamics of the system crucially depends on the queue type. Complexity is most observed for the case of growing queue. We characterize this dynamics between order and chaos with the T-entropy and the autocorrelation function.

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Correlations and spectra of strange nonchaotic attractors

Cited by 104 publications

References 13 publications

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations

Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations

Renormalization analysis of correlation properties in a quasiperiodically forced two-level system

Dynamics Between Order and Chaos in a Simple Reentrant Model of Production Dynamics

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