Reinjection probability density in type-III intermittency

Elaskar, Sergio; Río, Ezequiel del; Donoso, José

doi:10.1016/j.physa.2011.03.016

Cited by 31 publications

(43 citation statements)

References 17 publications

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“…R, where x 0 is the fixed point of (3) that defines the laminar phase of intermittency. In the previous works 15,16 to define the starting point, we used x s ¼ x 0 . Here, we generalize our approach and set…”

Section: A Fitting Linear Model To Experimental Datamentioning

confidence: 99%

“…Following Ref. 16, we introduce a continuous function cðx n Þ ¼ x 2 n and approximate the dynamics of the laminar phase by…”

Section: Length Of Laminar Phasementioning

confidence: 99%

“…As we mentioned above, in general, the RPD is determined by the behavior of trajectories within chaotic regime in a vicinity of a point of the Poincare map with infinite or zero tangent. 15,16 Let us now show how it works in this particular case. The map (3), (29) has single unstable (e > 0) fixed point at x 0 ¼ 0.…”

Section: Laugesen Type-iii Intermittencymentioning

confidence: 99%

“…16 based on expansion of trajectories around the maximum of F(x). Indeed, here the function F(x) has vanishing tangent for jxj ) 1, and points around its maximum are mapped into a small region in the laminar zone (Fig.…”

Section: Laugesen Type-iii Intermittencymentioning

confidence: 99%

“…The generalized RPD includes the uniform reinjection as a particular case m ¼ 1=2. 15,16 We showed that the shape of the generalized RPD is determined by the behavior of trajectories within chaotic regime in a vicinity of a point in the Poincare map with infinite or zero tangent. Later it has been shown that this mechanism is robust against the external noise.…”

Section: Introductionmentioning

confidence: 99%

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Theory of intermittency applied to classical pathological cases

Río

Elaskar

Makarov

2013

Chaos

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State-by-state assignment of the bending spectrum of acetylene at 15000 cm1: A case study of quantumclassical correspondence J. Chem. Phys. 111, 600 (1999); 10.1063/1.479341 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. The classical theory of intermittency developed for return maps assumes uniform density of points reinjected from the chaotic to laminar region. Though it works fine in some model systems, there exist a number of so-called pathological cases characterized by a significant deviation of main characteristics from the values predicted on the basis of the uniform distribution. Recently, we reported on how the reinjection probability density (RPD) can be generalized. Here, we extend this methodology and apply it to different dynamical systems exhibiting anomalous type-II and type-III intermittencies. Estimation of the universal RPD is based on fitting a linear function to experimental data and requires no a priori knowledge on the dynamical model behind. We provide special fitting procedure that enables robust estimation of the RPD from relatively short data sets (dozens of points). Thus, the method is applicable for a wide variety of data sets including numerical simulations and real-life experiments. Estimated RPD enables analytic evaluation of the length of the laminar phase of intermittent behaviors. We show that the method copes well with dynamical systems exhibiting significantly different statistics reported in the literature. We also derive and classify characteristic relations between the mean laminar length and main controlling parameter in perfect agreement with data provided by numerical simulations. Intermittency is a particular route to the deterministic chaos characterized by spontaneous transitions between laminar and chaotic dynamics. It is observed in a variety of different dynamical systems in Physics, Neuroscience, and Economics. Frequently, there is no feasible mathematical model for the process under study. Then reliable quantification of main characteristics of the intermittent process (e.g., the length of laminar phase) from experimental data is a challenging problem. The classical theory of intermittency has significant pitfalls. Though it works fine in some model systems, there exist a number of so-called pathological cases that deviate significantly from the classical predictions. In this work, we address the problem of unification of anomalous and standard intermittencies under single framework. The unified model can be fitted to experimental or numerical data. We note that to accomplish this step no a priori knowledge is required. We propose a procedure that can cope with reduced data sets consisting of several dozens of points. This makes our methodology useful for real-life applications. Using the experimentally obtained measures, we can classify intermittent processes into different theoretical types. We thoroughly test our method on two particular but canonical case...

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Section: A Fitting Linear Model To Experimental Datamentioning

confidence: 99%

“…Following Ref. 16, we introduce a continuous function cðx n Þ ¼ x 2 n and approximate the dynamics of the laminar phase by…”

Section: Length Of Laminar Phasementioning

confidence: 99%

Section: Laugesen Type-iii Intermittencymentioning

confidence: 99%

Section: Laugesen Type-iii Intermittencymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%