22,907 research outputs found
Renormalized entropy for one dimensional discrete maps: periodic and quasi-periodic route to chaos and their robustness
We apply renormalized entropy as a complexity measure to the logistic and
sine-circle maps. In the case of logistic map, renormalized entropy decreases
(increases) until the accumulation point (after the accumulation point up to
the most chaotic state) as a sign of increasing (decreasing) degree of order in
all the investigated periodic windows, namely, period-2, 3, and 5, thereby
proving the robustness of this complexity measure. This observed change in the
renormalized entropy is adequate, since the bifurcations are exhibited before
the accumulation point, after which the band-merging, in opposition to the
bifurcations, is exhibited. In addition to the precise detection of the
accumulation points in all these windows, it is shown that the renormalized
entropy can detect the self-similar windows in the chaotic regime by exhibiting
abrupt changes in its values. Regarding the sine-circle map, we observe that
the renormalized entropy detects also the quasi-periodic regimes by showing
oscillatory behavior particularly in these regimes. Moreover, the oscillatory
regime of the renormalized entropy corresponds to a larger interval of the
nonlinearity parameter of the sine-circle map as the value of the frequency
ratio parameter reaches the critical value, at which the winding ratio attains
the golden mean.Comment: 14 pages, 7 figure
Mixing Bandt-Pompe and Lempel-Ziv approaches: another way to analyze the complexity of continuous-states sequences
In this paper, we propose to mix the approach underlying Bandt-Pompe
permutation entropy with Lempel-Ziv complexity, to design what we call
Lempel-Ziv permutation complexity. The principle consists of two steps: (i)
transformation of a continuous-state series that is intrinsically multivariate
or arises from embedding into a sequence of permutation vectors, where the
components are the positions of the components of the initial vector when
re-arranged; (ii) performing the Lempel-Ziv complexity for this series of
`symbols', as part of a discrete finite-size alphabet. On the one hand, the
permutation entropy of Bandt-Pompe aims at the study of the entropy of such a
sequence; i.e., the entropy of patterns in a sequence (e.g., local increases or
decreases). On the other hand, the Lempel-Ziv complexity of a discrete-state
sequence aims at the study of the temporal organization of the symbols (i.e.,
the rate of compressibility of the sequence). Thus, the Lempel-Ziv permutation
complexity aims to take advantage of both of these methods. The potential from
such a combined approach - of a permutation procedure and a complexity analysis
- is evaluated through the illustration of some simulated data and some real
data. In both cases, we compare the individual approaches and the combined
approach.Comment: 30 pages, 4 figure
Increment entropy as a measure of complexity for time series
Entropy has been a common index to quantify the complexity of time series in
a variety of fields. Here, we introduce increment entropy to measure the
complexity of time series in which each increment is mapped into a word of two
letters, one letter corresponding to direction and the other corresponding to
magnitude. The Shannon entropy of the words is termed as increment entropy
(IncrEn). Simulations on synthetic data and tests on epileptic EEG signals have
demonstrated its ability of detecting the abrupt change, regardless of
energetic (e.g. spikes or bursts) or structural changes. The computation of
IncrEn does not make any assumption on time series and it can be applicable to
arbitrary real-world data.Comment: 12pages,7figure,2 table
Recurrence Plot Based Measures of Complexity and its Application to Heart Rate Variability Data
The knowledge of transitions between regular, laminar or chaotic behavior is
essential to understand the underlying mechanisms behind complex systems. While
several linear approaches are often insufficient to describe such processes,
there are several nonlinear methods which however require rather long time
observations. To overcome these difficulties, we propose measures of complexity
based on vertical structures in recurrence plots and apply them to the logistic
map as well as to heart rate variability data. For the logistic map these
measures enable us not only to detect transitions between chaotic and periodic
states, but also to identify laminar states, i.e. chaos-chaos transitions. The
traditional recurrence quantification analysis fails to detect the latter
transitions. Applying our new measures to the heart rate variability data, we
are able to detect and quantify the laminar phases before a life-threatening
cardiac arrhythmia occurs thereby facilitating a prediction of such an event.
Our findings could be of importance for the therapy of malignant cardiac
arrhythmias
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