Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator

Jeffries, C. D.; Pérez, José María Herrera

doi:10.1103/physreva.26.2117

Cited by 132 publications

(34 citation statements)

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“…Type-I appears with an inverse tangent bifurcation, Type-II with a Hopf bifurcation and Type-III is associated with a period doubling bifurcation. Experimentally, all three types of intermittency have been observed in a variety of hydrodynamical and electrical systems [33][34][35].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear flow of wormlike micellar gels: Regular and chaotic time-dependence of stress, normal force and nematic ordering

Ganapathy

Sood

2008

Journal of Non-Newtonian Fluid Mechanics

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Section: Introductionmentioning

confidence: 99%

Nonlinear flow of wormlike micellar gels: Regular and chaotic time-dependence of stress, normal force and nematic ordering

Ganapathy

Sood

2008

Journal of Non-Newtonian Fluid Mechanics

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“…Iterations of the map near the value of the virtual fixed points are identified with the laminar events while the reinjection iterates correspond to the turbulent bursts that occur in a erratic manner. Such bifurcation has been reported in many experimental systems [13,14] and it is abundant in the logistic and other simple maps. Renormalization group procedure to deal with type-I intermittency has already been used [15] and give analytical asymptotic expressions for statistical averages in the maps.…”

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confidence: 99%

Logarithmic Periodicities in the Bifurcations of Type-I Intermittent Chaos

Cavalcante

Leite

2004

Phys. Rev. Lett.

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The critical relations for statistical properties on saddle-node bifurcations are shown to display undulating fine structure, in addition to their known smooth dependence on the control parameter. A piecewise linear map with the type-I intermittency is studied and a log-periodic dependence is numerically obtained for the average time between laminar events, the Lyapunov exponent and attractor moments. The origin of the oscillations is built in the natural probabilistic measure of the map and can be traced back to the existence of logarithmically distributed discrete values of the control parameter giving Markov partition. Reinjection and noise effect dependences are discussed and indications are given on how the oscillations are potentially applicable to complement predictions made with the usual critical exponents, taken from data in critical phenomena. Dynamical bifurcations, when the qualitative behavior of a natural phenomena changes due to the variation of a control parameter, is a subject described by a well founded mathematical theory [1]. Their extension from equilibrium thermodinamic phase transitions to the bifurcations in non-equilibrium systems is applied within and outside physics [2,3]. Some bifurcations are abrupt, like first order equilibrium transitions, and others follow critical relations with a continuous dependence in the control parameter. Sufficiently close to the bifurcation the system properties follow characteristic functions giving rise to critical exponents which are signatures of the phenomena. Log-periodic oscillations modulating critical functional relations have been reported in studies of bifurcations [3,4]. As reviewed by Sornette [3], many ongoing research on applications are active, among them the prediction of catastrophic events. Transient chaos [5] and other escape phenomena [6] also display such log-periodicities. In simple maps, the period doubling Feigenbaum cascade is an early example where log-periodic behavior appears [7]. In non attracting sets of two dimensional maps, the topological entropy have been shown to present log-periodic oscillations [8].Among the dynamical bifurcation phenomena many can be cast on the class of intermittent chaos, ranging from the onset of turbulence [9] to synchronism of chaotic systems [10,11]. As originally proposed by Pomeau and Manneville [2, 9, 12], these instabilities can be modeled with simple one-dimensional maps and classified as type-I, II, and III, according to the Floquet multipliers of the map crossing the unity circle in the complex plane at 1, at a pair complex conjugate values or at −1, respectively. Type I, which will be discussed here, occurs when a saddle-node or tangent bifurcation is approached. Iterations of the map near the value of the virtual fixed points are identified with the laminar events while the reinjection iterates correspond to the turbulent bursts that occur in a erratic manner. Such bifurcation has been reported in many experimental systems [13,14] and it is abundant in the logistic and other simple ma...

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“…Let us note that equation (10) reduces to the condition [16] JL = TJ v in the limit 6 -+ 0 along the asymptotic path (7). In the neighborhood of the critical surface defined by equation (10), the use of both the center manifold theorem and the normal form techniques [15,19,25] (6).…”

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confidence: 99%

“…In the neighborhood of the critical surface defined by equation (10), the use of both the center manifold theorem and the normal form techniques [15,19,25] (6). In that sense, the degree of structural stability [14] of this scenario to chaos is ds = " 2 -£ ".…”

mentioning

confidence: 99%

“…Phys. France 49 (1988) The introduction by Pomeau and Manneville [1] of the notion of intermittency as a specific route to weak turbulence has stimulated many theoretical [2,3], numerical [4][5][6] and experimental [7][8][9][10][11] [12] and type-III [13] intermittencies are concerned, it has been shown that not only the local instability but also the reinjection mechanism strongly condition the statistics of the laminar regions. One of the issues of this paper, which is devoted to the study of type-11 intermittency, is to point out that the nature of the reinjection process may even affect the degree of structural stability [14] of this scenario to chaos.…”

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confidence: 99%

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