Logarithmic Periodicities in the Bifurcations of Type-I Intermittent Chaos

Abstract

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...

Review of Chaotic Intermittency

“…Refs. [101,102] describe a fine structure in intermittency. These papers study numerically type-I intermittency.…”