^{1,a)}, Sergio Elaskar

^{2,b)}and Valeri A. Makarov

^{3,c)}

### Abstract

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 cases of intermittency.

This work has been supported by the former Spanish Ministry of Science and Innovation (Project No. FIS2010-20054), by the CONICET (Project No. PIP 11220090100809), by the Russian Ministry of Education and Science (Contract No. 14.B37.21.1237), and by grants of the National University of Córdoba and MCyT of Córdoba, Argentina.

I. INTRODUCTION

II. ASSESSMENT OF RPD FUNCTION

A. Fitting linear model to experimental data

B. How to deal with short data sets

III. LENGTH OF LAMINAR PHASE

IV. LAUGESEN TYPE-III INTERMITTENCY

A. Estimation of RPD

B. Length of laminar phase

V. PIKOVSKY INTERMITTENCY

A. Non-overlapping case

B. Fitting short data sets

C. Slightly overlapping case

D. Strongly overlapping case

VI. CONCLUSIONS

### Key Topics

- Intermittency
- 57.0
- Data sets
- 23.0
- Attractors
- 10.0
- Chaotic dynamics
- 5.0
- Probability theory
- 5.0

## Figures

Sketch of the map (3) , (29) exhibiting anomalous type-III intermittency. Thick arrow illustrates mapping of points from the chaotic region (around the maximum of F(x)) into the region with practically zero tangent of F(x). Then thin arrow indicates the following reinjection of these points into the laminar region.

Sketch of the map (3) , (29) exhibiting anomalous type-III intermittency. Thick arrow illustrates mapping of points from the chaotic region (around the maximum of F(x)) into the region with practically zero tangent of F(x). Then thin arrow indicates the following reinjection of these points into the laminar region.

Analysis of the anomalous Laugesen type-III intermittency (map (3) , (29) : , and the laminar interval ). (a) Assessment of the RPD by numerical simulation. Dots correspond to M(x) evaluated by (6) and dashed line corresponds to the least squares fit. The dashed line with slope corresponds to the uniform RPD. (b) Numerical RPD. Dashed curve corresponds to (8) with the parameters found in (a). (c) Probability density of the length of the laminar phase. Dashed line corresponds to (22) .

Analysis of the anomalous Laugesen type-III intermittency (map (3) , (29) : , and the laminar interval ). (a) Assessment of the RPD by numerical simulation. Dots correspond to M(x) evaluated by (6) and dashed line corresponds to the least squares fit. The dashed line with slope corresponds to the uniform RPD. (b) Numerical RPD. Dashed curve corresponds to (8) with the parameters found in (a). (c) Probability density of the length of the laminar phase. Dashed line corresponds to (22) .

Characteristic relations of the averaged length of the laminar phase vs ε for the map (3) , (29) . Circles and triangles show numerical data. For , the solid line has slope in agreement (within 6% of relative error) with the analytical value 0.948 given by (25) . For , the horizontal dashed line shows the asymptotic behavior of , with .

Characteristic relations of the averaged length of the laminar phase vs ε for the map (3) , (29) . Circles and triangles show numerical data. For , the solid line has slope in agreement (within 6% of relative error) with the analytical value 0.948 given by (25) . For , the horizontal dashed line shows the asymptotic behavior of , with .

Second iteration of the map (31) demonstrating the Pikovsky type-II intermittency. (a) Non-overlapping case with a gap between two reinjection intervals. Arrows show two routs of reinjection into two disjoin intervals Il and Ir for the upper laminar region. Dots mark positions of the fixed points. There are two chaotic attractors in the map. (b) Slightly overlapping case. Reinjection intervals Il and Ir overlap. There exists single chaotic attractor. (c) Time evolution of the map corresponding to the case (b). Bottom subplot shows zoomed trajectory with two laminar phases near two unstable fixed points ( ).

Second iteration of the map (31) demonstrating the Pikovsky type-II intermittency. (a) Non-overlapping case with a gap between two reinjection intervals. Arrows show two routs of reinjection into two disjoin intervals Il and Ir for the upper laminar region. Dots mark positions of the fixed points. There are two chaotic attractors in the map. (b) Slightly overlapping case. Reinjection intervals Il and Ir overlap. There exists single chaotic attractor. (c) Time evolution of the map corresponding to the case (b). Bottom subplot shows zoomed trajectory with two laminar phases near two unstable fixed points ( ).

Analysis of the Pikovsky intermittency in the non-overlapping (top row, , two chaotic attractors) and slightly overlapping (bottom row, , single chaotic attractor) cases. Results are shown for the second iteration of the map (31) . (a) and (d) Numerical data (dots) for two branches of M(x) computed using (6) for reinjections in the intervals Il and Ir . Dashed gray lines show the corresponding least mean square fits, which then used to plot and . Dashed line with slope corresponds to the uniform RPD. (b) and (e) RPDs for Il and Ir . Numerical data (dots) and pdfs evaluated by (8) (dashed curves). (c) and (f) Probability density of the length of laminar phase for the interval Ir (for Il the pdf is similar). Dashed curve corresponds to (22) .

Analysis of the Pikovsky intermittency in the non-overlapping (top row, , two chaotic attractors) and slightly overlapping (bottom row, , single chaotic attractor) cases. Results are shown for the second iteration of the map (31) . (a) and (d) Numerical data (dots) for two branches of M(x) computed using (6) for reinjections in the intervals Il and Ir . Dashed gray lines show the corresponding least mean square fits, which then used to plot and . Dashed line with slope corresponds to the uniform RPD. (b) and (e) RPDs for Il and Ir . Numerical data (dots) and pdfs evaluated by (8) (dashed curves). (c) and (f) Probability density of the length of laminar phase for the interval Ir (for Il the pdf is similar). Dashed curve corresponds to (22) .

Fitting the RPD model to short data sets. (a) Mean and standard error of least squares estimation of m averaged over 100 independent experiments using Eq. (6) . Horizontal dashed line marks the exact value of . The inset illustrates a representative example of least squares fitting of 30 data points. The data and straight line underestimate the exact value of m. (b) Mean and standard error for m estimated by using the modified scheme (17) . There exists no significant bias in the estimate even for relatively short data sets. The inset illustrates typical distributions of the error (11) obtained for ordinary least squares using Eq. (6) (dots) and modified (triangles) methods.

Fitting the RPD model to short data sets. (a) Mean and standard error of least squares estimation of m averaged over 100 independent experiments using Eq. (6) . Horizontal dashed line marks the exact value of . The inset illustrates a representative example of least squares fitting of 30 data points. The data and straight line underestimate the exact value of m. (b) Mean and standard error for m estimated by using the modified scheme (17) . There exists no significant bias in the estimate even for relatively short data sets. The inset illustrates typical distributions of the error (11) obtained for ordinary least squares using Eq. (6) (dots) and modified (triangles) methods.

Characteristic relation of the average length of the laminar phase vs ε. Dots correspond to numerical data, whereas the curve marked as slight overlap refers to numerical integration of Eq. (23) using (33) as RPD. The asymptotic behavior is given by dashed line (marked by a) with the slope −0.5 ( ). The straight line (marked by b) with the slope −0.5 ( ) matches the numerical data for the strongly overlapping case considered in Ref. 19 .

Characteristic relation of the average length of the laminar phase vs ε. Dots correspond to numerical data, whereas the curve marked as slight overlap refers to numerical integration of Eq. (23) using (33) as RPD. The asymptotic behavior is given by dashed line (marked by a) with the slope −0.5 ( ). The straight line (marked by b) with the slope −0.5 ( ) matches the numerical data for the strongly overlapping case considered in Ref. 19 .

RPD for the Pikovsky's map in the strongly overlapping case. Dots correspond to numerical simulations and the curve is obtained by Eq. (35) with the fitted values for the reinjection on Il : . The corresponding values for Ir are . In this case, .

RPD for the Pikovsky's map in the strongly overlapping case. Dots correspond to numerical simulations and the curve is obtained by Eq. (35) with the fitted values for the reinjection on Il : . The corresponding values for Ir are . In this case, .

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