scholarly journals Tracking bifurcation curves in the Hénon map from only time-series datasets

2019 ◽  
Vol 10 (2) ◽  
pp. 268-278
Author(s):  
Yoshitaka Itoh ◽  
Masaharu Adachi
Keyword(s):  
Time Series ◽  
Hénon Map ◽  
Henon Map

scholarly journals Time series classification based on triadic time series motifs

2019 ◽  
Vol 33 (21) ◽  
pp. 1950237
Author(s):  
Wen-Jie Xie ◽  
Rui-Qi Han ◽  
Wei-Xing Zhou
Keyword(s):  
Time Series ◽  
Logistic Map ◽  
Time Series Classification ◽  
Similarity Coefficients ◽  
Motif Analysis ◽  
Hénon Map ◽  
Henon Map ◽  
Motif Occurrence ◽  
Generalized Hénon Map ◽  
Dynamic Time

It is of great significance to identify the characteristics of time series to quantify their similarity and classify different classes of time series. We define six types of triadic time-series motifs and investigate the motif occurrence profiles extracted from the time series. Based on triadic time series motif profiles, we further propose to estimate the similarity coefficients between different time series and classify these time series with high accuracy. We validate the method with time series generated from nonlinear dynamic systems (logistic map, chaotic logistic map, chaotic Henon map, chaotic Ikeda map, hyperchaotic generalized Henon map and hyperchaotic folded-tower map) and retrieved from the UCR Time Series Classification Archive. Our analysis shows that the proposed triadic time series motif analysis performs better than the classic dynamic time wrapping method in classifying time series for certain datasets investigated in this work.

IDENTIFICATION OF TRUE AND SPURIOUS LYAPUNOV EXPONENTS FROM TIME SERIES

1992 ◽  
Vol 02 (01) ◽  
pp. 155-165 ◽  
Author(s):  
ULRICH PARLITZ
Keyword(s):  
Time Series ◽  
Lyapunov Exponents ◽  
Original Data ◽  
Basis Functions ◽  
Hénon Map ◽  
Lorenz Model ◽  
Henon Map ◽  
Nonlinear Approximations ◽  
Flow Map ◽  
Reversed Time

A new method for the identification of true and spurious Lyapunov exponents computed from time series is presented. It is based on the observation that the true Lyapunov exponents change their signs upon time reversal whereas the spurious exponents do not. Furthermore by comparison of the spectra of the original data and the reversed time series suitable values for the free parameters of the algorithm used for computing the Lyapunov exponents (e.g., the number of nearest neighbors) are determined. As an example for this general approach an algorithm using local nonlinear approximations of the flow map in embedding space by radial basis functions is presented. For noisy data a regularization method is applied in order to get smooth approximating functions. Numerical examples based on data from the Hénon map, a four-dimensional analog of the Hénon map, a quasiperiodic time series, the Lorenz model, and Duffing’s equation are given.

LYAPUNOV-EXPONENT SPECTRUM FROM NOISY TIME SERIES

2013 ◽  
Vol 23 (06) ◽  
pp. 1350103 ◽  
Author(s):  
TIAN-LIANG YAO ◽  
HAI-FENG LIU ◽  
JIAN-LIANG XU ◽  
WEI-FENG LI
Keyword(s):  
Time Series ◽  
Lyapunov Spectrum ◽  
Deterministic System ◽  
Hénon Map ◽  
Henon Map ◽  
Noisy Time Series ◽  
Generalized Hénon Map ◽  
Novel Method ◽  
Influence Of Noise ◽  
Lyapunov Exponent Spectrum

Since all kinds of noise exist in signals from real-world systems, it is very difficult to exactly estimate Lyapunov exponents from this time series. In this paper, a novel method for estimating the Lyapunov spectrum from a noisy chaotic time series is presented. We consider the higher-order mappings from neighbors into neighbors, rather than the mappings from small displacements into small displacements as usual. The influence of noise on the second-order mappings is researched, and an averaging method is proposed to cope with this noise. The mappings equations of the underlying deterministic system can be obtained from the noisy data via the method, and then the Lyapunov spectrum can be estimated. We demonstrate the performance of our algorithm for three familiar chaotic systems, Hénon map, the generalized Hénon map and Lorenz system. It is found that the proposed method provides a reasonable estimate of Lyapunov spectrum for these three systems when the noise level is less than 20%, 10% and 7%, respectively. Furthermore, our method is not sensitive to the distribution types of the noise, and the results of our method become more accurate with the increase of the length of time series.

RED NOISE ESTIMATION

2006 ◽  
Vol 16 (07) ◽  
pp. 2111-2117 ◽  
Author(s):  
ALEXANDROS LEONTITSIS
Keyword(s):  
Time Series ◽  
Time Delay ◽  
Chaotic Time Series ◽  
Noise Estimation ◽  
Correlated Noise ◽  
Hénon Map ◽  
Henon Map ◽  
Correct Estimation

This paper presents a method for the correct estimation of the red (linearly autocorrelated of order 1) noise from chaotic time series. The idea is to increase the time delay in order to have a reliable reconstruction. The results indicate that only on extremely correlated noise cases this increase helps, because otherwise the correlation integrals are not affected. The proposed method is successfully applied on time series of the Hénon map and is extended to weekly closes of the Nasdaq Cmp. index. The main advantage of the method presented here is that it can be used on time series with any kind of linearly correlated noise.

Autoregressive self-tuning feedback control of the Hénon map

1996 ◽  
Vol 54 (6) ◽  
pp. 6201-6206 ◽  
Author(s):  
Michael E. Brandt ◽  
Ahmet Ademoǧlu ◽  
Dejian Lai ◽  
Guanrong Chen
Keyword(s):  
Feedback Control ◽  
Hénon Map ◽  
Henon Map ◽  
Self Tuning

scholarly journals Detecting and stabilizing periodic orbits of chaotic Henon map through predictive control

2018 ◽  
Vol 27 (2018) ◽  
pp. 73-78
Author(s):  
Dumitru Deleanu
Keyword(s):  
Periodic Orbits ◽  
Predictive Control ◽  
Nonnegative Integer ◽  
Strange Attractor ◽  
Discrete System ◽  
Control Method ◽  
Nonlinear Mapping ◽  
Unstable Periodic Orbits ◽  
Hénon Map ◽  
Henon Map

The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to detect and stabilize all the UPOs up to a maximum length of the period. The role played by each involved parameter is investigated and additional results to those reported in the literature are presented.

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