A Two-dimensional Mapping with a Strange Attractor

Hénon, M.

doi:10.1007/978-0-387-21830-4_8

Cited by 325 publications

(351 citation statements)

References 4 publications

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“…The creation of horseshoes in families of two dimensional maps has been extensively studied, with the Henon map serving as a prototype [19]. The principal results [3,32] extend Jakobson's Theorem for one dimensional maps [8] into the setting of two dimensional diffeomorphisms with small Jacobian.…”

Section: Maximal Canardsmentioning

confidence: 94%

Bifurcations of relaxation oscillations

Guckenheimer

2004

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

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We are still far from a comprehensive theory of bifurcation in dynamical systems with multiple time scales. However, systematic application of geometric methods and study of examples have produced descriptions of varied phenomena. These lectures present a selective summary of some of what has been discovered, concentrating on periodic orbits called relaxation oscillations. We focus upon key aspects of the phenomena, avoiding mathematical details and making the analysis as simple as possible. The dynamics of reduced systems plays a central role in our discussion.

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Section: Maximal Canardsmentioning

confidence: 94%

Bifurcations of relaxation oscillations

Guckenheimer

2004

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

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“…By using a stability analysis in a two equation system, these authors find instability for ~2<N< 2.27, just confirming the results from our bifurcation diagram. Moreover they show for N=2.27 the existence of a strange attractor of the Hénon type (see Hénon, 1976) .…”

Section: Delay Effects In Dynamic Logit Modelsmentioning

confidence: 97%

Impacts of Multiple‐Period Lags in Dynamic Logit Models

Nijkamp

Reggiani

1992

Geographical Analysis

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This paper will provide an introduction to a new field of research, viz the sensitivity of the solution trajectory of a dynamic logit model (belonging to the class of discrete choice models) in the light of a multi-period lag structure. It is well known from recent advances in the area of chaos and turbulence theory that the stability of a dynamic system is critically dependent on various factors, such as threshold values of parameters, initial conditions and also the lag structure. This paper aims to identify the consequences of different lag structures in dynamic logit models (including also dynamic spatial interaction models). Various simulation experiments will be used to show that the onset of instability of the solution trajectory tends to decrease as the number of time lags increases (depending also on the growt.h rate of the system) . Acknowledgement

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“…2 form a chaotic time series. The third one is the Henon equation [31] which has a simple format described by…”

Section: Benchmark Chaotic Time Seriesmentioning

confidence: 99%

Energy Associated Tuning Method for Short-Term Series Forecasting by Complete and Incomplete Datasets

Rivero

Pucheta

Laboret

et al. 2016

Journal of Artificial Intelligence and Soft Computing Research

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This article presents short-term predictions using neural networks tuned by energy associated to series based-predictor filter for complete and incomplete datasets. A benchmark of high roughness time series from Mackay Glass (MG), Logistic (LOG), Henon (HEN) and some univariate series chosen from NN3 Forecasting Competition are used. An average smoothing technique is assumed to complete the data missing in the dataset. The Hurst parameter estimated through wavelets is used to estimate the roughness of the real and forecasted series. The validation and horizon of the time series is presented by the 15 values ahead. The performance of the proposed filter shows that even a short dataset is incomplete, besides a linear smoothing technique employed; the prediction is almost fair by means of SMAPE index. Although the major result shows that the predictor system based on energy associated to series has an optimal performance from several chaotic time series, in particular, this method among other provides a good estimation when the short-term series are taken from one point observations.

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A Two-dimensional Mapping with a Strange Attractor

Cited by 325 publications

References 4 publications

Bifurcations of relaxation oscillations

Bifurcations of relaxation oscillations

Impacts of Multiple‐Period Lags in Dynamic Logit Models

Energy Associated Tuning Method for Short-Term Series Forecasting by Complete and Incomplete Datasets

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