A plethora of strange nonchaotic attractors

Negi, Surendra Singh; Ramaswamy, Ramakrishna

doi:10.1007/s12043-001-0140-7

Cited by 17 publications

(7 citation statements)

References 34 publications

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“…Furthermore, explicit quasiperiodic driving is also a feature of hitherto studied systems. However, both these features are probably not necessary in order that SNAs be formed [Negi and Ramaswamy, 2000a]. In particular, driving a system with signals based on fractal sequences also appears to yield SNAs [Kuptsov, 1998].…”

Section: Discussionmentioning

confidence: 99%

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Strange Nonchaotic Attractors

Prasad

Negi

Ramaswamy

2001

Int. J. Bifurcation Chaos

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152

114

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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 Schrödinger equation for a particle in a related quasiperiodic potential, giving 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, has 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, suggest novel applications.

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Section: Discussionmentioning

confidence: 99%

“…Mathematically rigorous results are more difficult in this case, but there is some indication that it will be possible to have fractal nonchaotic attractors in a large class of systems, even in the absence of explicit quasiperiodic driving [Negi and Ramaswamy, 2000a].…”

Section: Strange Nonchaotic Dynamics: Occurrence and Characterizamentioning

confidence: 99%

Strange Nonchaotic Attractors

Prasad

Negi

Ramaswamy

2001

Int. J. Bifurcation Chaos

Self Cite

152

114

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“…In subsequent years, there have been several experimental observations of SNAs in practical systems [11][12][13][14][15]; however, all these studies presented the necessity of having quasiperiodic forcing to generate SNAs. While these previous investigations on SNA involved external quasiperiodic forcing, Negi et al [16] theoretically showed that quasiperiodic forcing is not essential for the existence of SNA, and it could happen in naturally driven systems as well without the need of external forcing. Recently, Lindner et al [17] showed the observation of SNAs in the natural system of a pulsating star KIC 5520878 network.…”

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confidence: 99%

Strange nonchaos in self-excited singing flames

Premraj¹,

Pawar²,

Kabiraj³

et al. 2020

EPL

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We report the first experimental evidence of strange nonchaotic attractor (SNA) in the natural dynamics of a self-excited laboratory-scale system. In the previous experimental studies, the birth of SNA was observed in quasiperiodically forced systems; however, such an evidence of SNA in an autonomous laboratory system is yet to be reported. We discover the presence of SNA in between the attractors of quasiperiodicity and chaos through a fractalization route in a laboratory thermoacoustic system. The observed dynamical transitions from order to chaos via SNA is confirmed through various nonlinear characterization methods prescribed for the detection of SNA.Coupled nonlinear systems exhibit various kinds of dynamical behaviours including periodic, quasiperiodic, and chaotic oscillations [1,2]. Among these dynamics, one of the commonly observed state in quasiperiodically driven nonlinear systems is a strange nonchaos. Although strange nonchaotic attractors (SNAs) show similarity to chaotic attractors by having a fractal geometrical structure, SNAs are insensitive to initial conditions unlike the chaotic attractors [3]. Grebogi et al. [4] was the first to report the possibility of SNAs in the system of quasiperiodically forced map. Afterwards, several numerical studies have demonstrated the existence of SNAs in systems such as pendulum [5], Duffing oscillator [6], logistic map [7], Henon map [8], and circular map [9].The experimental discovery of SNA was reported by Ditto et al.[10] in a quasiperiodically forced system with a buckled magnetoelastic ribbon. In subsequent years, there have been several experimental observations of SNAs in practical systems [11][12][13][14][15]; however, all these studies presented the necessity of having quasiperiodic forcing to generate SNAs. Contrary to these studies, Negi et al. [16] showed theoretically that the need of quasiperiodic forcing is not mandatory for the creation of SNAs, and it could happen in naturally driven systems as well without the need of external forcing. Recently, Lindner et al. [17] showed the observation of SNAs in the natural system of a pulsating star KIC 5520878 network. However, to the best of our knowledge, there has not been a single experimental evidence of SNA reported in self-driven laboratory systems until now.Most of the recent studies are focused on identifying the routes to generate SNAs [3,12,18]. The mechanisms for the onset of SNAs are usually classified into three types as: (i) Heagy-Hammel route -the SNAs emerges during the collision of a period doubled torus with its own unstable parent [19], (ii) fractalization route -the truncated torus gets wrinkled and forms SNAs without any interaction with the parent torus [20], and (iii) type-III intermittency route -SNAs occur when the torus doubling bifurcation is controlled by sub-harmonic bifurcations [6]. Another possibility for the occurrence of SNAs is through crisis-induced intermittency, wherein the collision of the wrinkled torus with the boundary results in sudden widening of the attra...

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“…We can however do even better than this by ensuring that all components of the system are piecewise linear. In [18] a similar piecewise linear system was studied of the form…”

Section: Making the Argument Exact: Piecewise Linear Snasmentioning

confidence: 99%

Dimensions of structurally stable and critical strange non-chaotic attractors

Adamson¹,

Osbaldestin²

2014

J. Phys. A: Math. Theor.

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We provide evidence that the box-counting dimension of a structurally stable strange non-chaotic attractor (SNA) of pinched skew product type is equal to 2 by showing that it has non-negligible area. The argument presented is made more accurate in the study of a piecewise linear SNA. Furthermore we provide evidence that the fractal dimension of a critical SNA is not equal to 2, but in fact lies between 1 and 2. We numerically calculate the box-counting dimension for several critical SNAs, providing further evidence to support this conjecture.

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A plethora of strange nonchaotic attractors

Cited by 17 publications

References 34 publications

Strange Nonchaotic Attractors

Strange Nonchaotic Attractors

Strange nonchaos in self-excited singing flames

Dimensions of structurally stable and critical strange non-chaotic attractors

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