Quantifying the complexity of the delayed logistic map

2011 ◽  
Vol 369 (1935) ◽  
pp. 425-438 ◽  
Author(s):  
Cristina Masoller ◽  
Osvaldo A. Rosso
Keyword(s):  
Probability Distribution Function ◽  
Logistic Map ◽  
Nonlinear Feedback ◽  
Complementary Information ◽  
Complexity Measures ◽  
Pattern Complexity ◽  
Statistical Complexity ◽  
Ordinal Patterns ◽  
Forbidden Patterns ◽  
Ordinal Pattern

Statistical complexity measures are used to quantify the degree of complexity of the delayed logistic map, with linear and nonlinear feedback. We employ two methods for calculating the complexity measures, one with the ‘histogram-based’ probability distribution function and the other one with ordinal patterns. We show that these methods provide complementary information about the complexity of the delay-induced dynamics: there are parameter regions where the histogram-based complexity is zero while the ordinal pattern complexity is not, and vice versa. We also show that the time series generated from the nonlinear delayed logistic map can present zero missing or forbidden patterns, i.e. all possible ordinal patterns are realized into orbits.

Characterizing dynamical transitions by statistical complexity measures based on ordinal pattern transition networks

2021 ◽  
Vol 31 (3) ◽  
pp. 033127
Author(s):  
Min Huang ◽  
Zhongkui Sun ◽  
Reik V. Donner ◽  
Jie Zhang ◽  
Shuguang Guan ◽  
...  
Keyword(s):  
Complexity Measures ◽  
Statistical Complexity ◽  
Ordinal Pattern ◽  
Transition Networks

scholarly journals Pseudorandom Number Generator (PRNG) Design Using Hyper-Chaotic Modified Robust Logistic Map (HC-MRLM)

Electronics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 104 ◽  
Author(s):  
Muhammad Irfan ◽  
Asim Ali ◽  
Muhammad Asif Khan ◽  
Muhammad Ehatisham-ul-Haq ◽  
Syed Nasir Mehmood Shah ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Logistic Map ◽  
Random Numbers ◽  
Pseudorandom Number Generator ◽  
Control Of Chaos ◽  
Randomness Test ◽  
Novel Method ◽  
Chaotic Parameter

Robust chaotic systems, due to their inherent properties of mixing, ergodicity, and larger chaotic parameter space, constitute a perfect candidate for cryptography. This paper reports a novel method to generate random numbers using modified robust logistic map (MRLM). The non-smooth probability distribution function of robust logistic map (RLM) trajectories gives an un-even binary distribution in randomness test. To overcome this disadvantage in RLM, control of chaos (CoC) is proposed for smooth probability distribution function of RLM. For testing the proposed design, cryptographic random numbers generated by MRLM were vetted with National Institute of Standards and Technology statistical test suite (NIST 800-22). The results showed that proposed MRLM generates cryptographically secure random numbers (CSPRNG).

scholarly journals QUANTIFYING EMERGENCE IN TERMS OF PERSISTENT MUTUAL INFORMATION

2010 ◽  
Vol 13 (03) ◽  
pp. 327-338 ◽  
Author(s):  
ROBIN C. BALL ◽  
MARINA DIAKONOVA ◽  
ROBERT S. MACKAY
Keyword(s):  
Mutual Information ◽  
Chaotic Behavior ◽  
Past History ◽  
Logistic Map ◽  
Period Doubling ◽  
Emergent Behavior ◽  
Complexity Measures ◽  
Spatially Extended ◽  
Long Time ◽  
Definition Of

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long-term prediction, which we propose as the primary signature of (Strong) Emergent Behavior. The key feature of our definition of PMI is the omission of an interval of "present" time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behavior or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map, we compare predicted with measured long-time PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well-defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalization of the notion of ordered phases.

scholarly journals The Refined Composite Downsampling Permutation Entropy Is a Relevant Tool in the Muscle Fatigue Study Using sEMG Signals

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1655
Author(s):  
Philippe Ravier ◽  
Antonio Dávalos ◽  
Meryem Jabloun ◽  
Olivier Buttelli
Keyword(s):  
Biceps Brachii ◽  
Structural Information ◽  
Spectral Band ◽  
Electric Activity ◽  
Real Data ◽  
Voluntary Contraction ◽  
Permutation Entropy ◽  
Complexity Measures ◽  
Ordinal Patterns ◽  
Semg Signals

Surface electromyography (sEMG) is a valuable technique that helps provide functional and structural information about the electric activity of muscles. As sEMG measures output of complex living systems characterized by multiscale and nonlinear behaviors, Multiscale Permutation Entropy (MPE) is a suitable tool for capturing useful information from the ordinal patterns of sEMG time series. In a previous work, a theoretical comparison in terms of bias and variance of two MPE variants—namely, the refined composite MPE (rcMPE) and the refined composite downsampling (rcDPE), was addressed. In the current paper, we assess the superiority of rcDPE over MPE and rcMPE, when applied to real sEMG signals. Moreover, we demonstrate the capacity of rcDPE in quantifying fatigue levels by using sEMG data recorded during a fatiguing exercise. The processing of four consecutive temporal segments, during biceps brachii exercise maintained at 70% of maximal voluntary contraction until exhaustion, shows that the 10th-scale of rcDPE was capable of better differentiation of the fatigue segments. This scale actually brings the raw sEMG data, initially sampled at 10 kHz, to the specific 0–500 Hz sEMG spectral band of interest, which finally reveals the inner complexity of the data. This study promotes good practices in the use of MPE complexity measures on real data.

scholarly journals Application of Information and Complexity Measures to Neutron Stars Structure

2019 ◽  
Vol 18 ◽  
pp. 163
Author(s):  
K. Ch. Chatzisavvas ◽  
V. P. Psonis ◽  
C. P. Panos ◽  
Ch. C. Moustakidis
Keyword(s):  
Gravitational Field ◽  
Neutron Star ◽  
Neutron Stars ◽  
Short Range ◽  
Nuclear Force ◽  
Low Complexity ◽  
Complexity Measures ◽  
Astronomical Observations ◽  
Equation Of States ◽  
Statistical Complexity

We apply several information and statistical complexity measures to neutron stars structure. Neutron stars is a classical example where the gravitational field and quantum behaviour are combined and produce a macroscopic dense object. We concentrate our study on the connection between complexity and neutron star properties, like maximum mass and the corresponding radius, applying a specific set of realistic equation of states. Moreover, the effect of the strength of the gravitational field on the neutron star structure and consequently on the complexity measure is also investigated. It is seen that neutron stars, consistent with astronomical observations so far, are ordered systems (low complexity), which cannot grow in complexity as their mass increases. This is a result of the interplay of gravity, the short-range nuclear force and the very short-range weak interaction.

TIME SERIES FROM THE ORDINAL VIEWPOINT

2007 ◽  
Vol 07 (02) ◽  
pp. 247-272 ◽  
Author(s):  
KARSTEN KELLER ◽  
MATHIEU SINN ◽  
JAN EMONDS
Keyword(s):  
Time Series ◽  
Time Series Analysis ◽  
New Approach ◽  
Ordinal Information ◽  
Invariance Properties ◽  
Series Analysis ◽  
Order Relations ◽  
Ordinal Patterns ◽  
Ordinal Time Series ◽  
Ordinal Pattern

Ordinal time series analysis is a new approach to the investigation of long and complex time series, which bases on ordinal patterns describing the order relations between the values of a time series. In this paper we consider ordinal time series analysis from the conceptional viewpoint. In particular, we introduce ordinal processes as models for ordinal time series analysis and discuss the structure of ordinal pattern distributions obtained from them. Special emphasis is on the relation of ordinal time series analysis to symbolic dynamics and to a transformation extracting the whole ordinal information contained in a time series. Finally, we consider invariance properties of ordinal time series analysis.

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