This paper is devoted to the development of some dynamical indicators that allow the determination of regions and structures that separate different dynamic regimes in autonomous and non-autonomous dynamical systems. The underlying idea is closely related to the Lagrangian Coherent Structures concept introduced by Haller. In the present paper, instead of using the Cauchy-Green tensor, that determines the domains where the flow associated to a differential equation is expanding in the normal direction, the Jet Transport methodology is used. This is a semi-numerical tool, that has as basic ingredients a polynomial algebra package and a numerical integration method, allowing, at each integration step, the propagation under a flow of a neighbourhood U instead of a single initial condition. The output of the procedure is a polynomial in several variables that represents the image of U up to a selected order, containing high order terms of the variational equations. Using these high order representation, the places where the normal direction expands can be easily detected, in a similar manner as the procedures for calculating the Lagrangian Coherent Structures do. In order to illustrate the methodology, first the results obtained in the determination of the separatrices of the simple and the periodically perturbed pendulum are given. Later, the applications to the circular restricted three body problem are considered, where the aim is the detection of invariant manifolds of libration point orbits, as well as in the non-autonomous vector field defined by the elliptic restricted three body problem.
Evaluating web traffic on a web server is highly critical for web service providers since, without a proper demand forecast, customers could have lengthy waiting times and abandon that website. However, this is a challenging task since it requires making reliable predictions based on the arbitrary nature of human behavior. We introduce an architecture that collects source data and in a supervised way performs the forecasting of the time series of the page views. Based on the Wikipedia page views dataset proposed in a competition by Kaggle in 2017, we created an updated version of it for the years 2018–2020. This dataset is processed and the features and hidden patterns in data are obtained for later designing an advanced version of a recurrent neural network called Long Short-Term Memory. This AI model is distributed training, according to the paradigm called data parallelism and using the Downpour training strategy. Predictions made for the seven dominant languages in the dataset are accurate with loss function and measurement error in reasonable ranges. Despite the fact that the analyzed time series have fairly bad patterns of seasonality and trend, the predictions have been quite good, evidencing that an analysis of the hidden patterns and the features extraction before the design of the AI model enhances the model accuracy. In addition, the improvement of the accuracy of the model with the distributed training is remarkable. Since the task of predicting web traffic in as precise quantities as possible requires large datasets, we designed a forecasting system to be accurate despite having limited data in the dataset. We tested the proposed model on the new Wikipedia page views dataset we created and obtained a highly accurate prediction; actually, the mean absolute error of predictions regarding the original one on average is below 30. This represents a significant step forward in the field of time series prediction for web traffic forecasting.
The Jet Transport method has emerged as a powerful tool for the numerical integration of ordinary differential equations; it uses polynomial expansions to approximate the flow map associated to the differential equation in the neighborhood of a reference solution. One of the main drawbacks of the method is that the region of accuracy becomes smaller along the integration. In this paper we introduce a procedure to determine a ball covering the set of given initial conditions that keeps the accuracy of the integration within a selected threshold. The paper gives detailed explanations of the algorithm illustrated with some examples of applicability, as well as a comparison with a previous existing method for the same purpose.
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