scholarly journals Issues on Applying One- and Multi-Step Numerical Methods to Chaotic Oscillators for FPGA Implementation

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 151
Author(s):  
Omar Guillén-Fernández ◽  
María Fernanda Moreno-López ◽  
Esteban Tlelo-Cuautle
Keyword(s):  
Time Series ◽  
Numerical Methods ◽  
Numerical Method ◽  
Euler Method ◽  
Fpga Implementation ◽  
Chaotic Time Series ◽  
Time Step ◽  
Chaotic Oscillators ◽  
One Step ◽  
Selection Of

Chaotic oscillators have been designed with embedded systems like field-programmable gate arrays (FPGAs), and applied in different engineering areas. However, the majority of works do not detail the issues when choosing a numerical method and the associated electronic implementation. In this manner, we show the FPGA implementation of chaotic and hyper-chaotic oscillators from the selection of a one-step or multi-step numerical method. We highlight that one challenge is the selection of the time-step h to increase the frequency of operation. The case studies include the application of three one-step and three multi-step numerical methods to simulate three chaotic and two hyper-chaotic oscillators. The numerical methods provide similar chaotic time-series, which are used within a time-series analyzer (TISEAN) to evaluate the Lyapunov exponents and Kaplan–Yorke dimension (DKY) of the (hyper-)chaotic oscillators. The oscillators providing higher exponents and DKY are chosen because higher values mean that the chaotic time series may be more random to find applications in chaotic secure communications. In addition, we choose representative numerical methods to perform their FPGA implementation, which hardware resources are described and counted. It is highlighted that the Forward Euler method requires the lowest hardware resources, but it has lower stability and exactness compared to other one-step and multi-step methods.

scholarly journals Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1938
Author(s):  
Martín Alejandro Valencia-Ponce  ◽  
Esteban Tlelo-Cuautle ◽  
Luis Gerardo de la Fraga
Keyword(s):  
Numerical Simulation ◽  
Numerical Methods ◽  
Execution Time ◽  
Chaotic Oscillator ◽  
Time Step ◽  
Chaotic Oscillators ◽  
Swarm Optimization ◽  
One Step ◽  
Multi Step Methods

The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step h. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest h in either one-step or multi-step methods. Four chaotic oscillators are used as a case study, and the optimization of their Kaplan-Yorke dimension (DKY) is performed by applying three metaheuristics, namely: particle swarm optimization (PSO), many optimizing liaison (MOL), and differential evolution (DE) algorithms. Three representative one-step and three multi-step methods are used to solve the four chaotic oscillators, for which the estimation of the highest h is obtained from their stability analysis. The optimization results show the effectiveness of using a high h value for the six numerical methods in reducing execution time while maximizing the positive Lyapunov exponent (LE+) and DKY of the chaotic oscillators by applying PSO, MOL, and DE algorithms.

scholarly journals Sub-daily runoff simulations with parameters inferred at the daily time scale

2015 ◽  
Vol 12 (8) ◽  
pp. 7437-7467 ◽  
Author(s):  
J. E. Reynolds ◽  
S. Halldin ◽  
C. Y. Xu ◽  
J. Seibert ◽  
A. Kauffeldt
Keyword(s):  
Time Series ◽  
Numerical Method ◽  
Time Scales ◽  
Time Scale ◽  
Input Data ◽  
Flood Forecasting ◽  
Euler Method ◽  
Time Stepping ◽  
Parameter Values ◽  
Daily Time

Abstract. Concentration times in small and medium-sized watersheds (~ 100–1000 km2) are commonly less than 24 h. Flood-forecasting models then require data at sub-daily time scales, but time-series of input and runoff data with sufficient lengths are often only available at the daily time scale, especially in developing countries. This has led to a search for time-scale relationships to infer parameter values at the time scales where they are needed from the time scales where they are available. In this study, time-scale dependencies in the HBV-light conceptual hydrological model were assessed within the generalized likelihood uncertainty estimation (GLUE) approach. It was hypothesised that the existence of such dependencies is a result of the numerical method or time-stepping scheme used in the models rather than a real time-scale-data dependence. Parameter values inferred showed a clear dependence on time scale when the explicit Euler method was used for modelling at the same time steps as the time scale of the input data (1–24 h). However, the dependence almost fully disappeared when the explicit Euler method was used for modelling in 1 h time steps internally irrespectively of the time scale of the input data. In other words, it was found that when an adequate time-stepping scheme was implemented, parameter sets inferred at one time scale (e.g., daily) could be used directly for runoff simulations at other time scales (e.g., 3 or 6 h) without any time scaling and this approach only resulted in a small (if any) model performance decrease, in terms of Nash–Sutcliffe and volume-error efficiencies. The overall results of this study indicated that as soon as sub-daily driving data can be secured, flood forecasting in watersheds with sub-daily concentration times is possible with model-parameter values inferred from long time series of daily data, as long as an appropriate numerical method is used.

Evaluating Forecasting Procedures for Predicting Pacific Herring (Clupea harengus pallasi) Recruitment in British Columbia

1988 ◽  
Vol 45 (6) ◽  
pp. 928-935 ◽  
Author(s):  
M. Stocker ◽  
D. J. Noakes
Keyword(s):  
Time Series ◽  
Clupea Harengus ◽  
Environmental Data ◽  
Forecast Errors ◽  
Pacific Herring ◽  
Time Step ◽  
Absolute Deviation ◽  
Percent Error ◽  
One Step ◽  
Absolute Percent Error

The ability of four forecasting methods to generate one-step-ahead forecasts of Pacific herring (Clupea harengus pillasi) recruitment is considered in this paper. Recruitment time series for five coastal stocks and various environmental time series are employed in the analyses. Information up to and including time t is employed to estimate the parameters of each model used to forecast recruitment in year t + 1. Parameter estimates are then updated after each time step with a total of seven one-step-ahead forecasts being generated by each model for each stock. The forecast errors are compared using the five criteria: (1) root mean squared error, (2) mean absolute deviation, (3) mean absolute percent error, (4) median absolute deviation, and (5) median absolute percent error. The results of the study indicate that time series models may provide better forecasts of recruitment for the Strait of Georgia/Johnstone Strait stocks than the other competing procedures. A Ricker stock–recruitment model that takes into account environmental data appears to produce marginally better forecasts for the Central Coast and Queen Charlotte Island stocks, while all models produced equally good/bad forecasts for the Prince Rupert district stocks.

Optimal Selection of Parameters for Nonuniform Embedding of Chaotic Time Series Using Ant Colony Optimization

2013 ◽  
Vol 43 (2) ◽  
pp. 790-802 ◽  
Author(s):  
Meie Shen ◽  
Wei-Neng Chen ◽  
Jun Zhang ◽  
Henry Shu-Hung Chung ◽  
O. Kaynak
Keyword(s):  
Time Series ◽  
Ant Colony Optimization ◽  
Chaotic Time Series ◽  
Ant Colony ◽  
Optimal Selection ◽  
Selection Of

scholarly journals Local Prediction of Chaotic Time Series Based on Polynomial Coefficient Autoregressive Model

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Liyun Su ◽  
Chenlong Li
Keyword(s):  
Time Series ◽  
Autoregressive Model ◽  
Time Series Prediction ◽  
Real Data ◽  
Chaotic Time Series ◽  
Polynomial Coefficient ◽  
State Dependent ◽  
Forecasting Performance ◽  
One Step ◽  
Chaotic Time Series Prediction

We apply the polynomial function to approximate the functional coefficients of the state-dependent autoregressive model for chaotic time series prediction. We present a novel local nonlinear model called local polynomial coefficient autoregressive prediction (LPP) model based on the phase space reconstruction. The LPP model can effectively fit nonlinear characteristics of chaotic time series with simple structure and have excellent one-step forecasting performance. We have also proposed a kernel LPP (KLPP) model which applies the kernel technique for the LPP model to obtain better multistep forecasting performance. The proposed models are flexible to analyze complex and multivariate nonlinear structures. Both simulated and real data examples are used for illustration.

scholarly journals Reconstruction of the Lorenz system using the method of perspective coefficients

2018 ◽  
Author(s):  
V. G. Gorodetskiy ◽  
N. P. Osadchuk
Keyword(s):  
Time Series ◽  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Lorenz System ◽  
Ode System ◽  
Right Hand ◽  
Observable Variable ◽  
The Lorenz System

Reconstruction of the Lorenz ordinary differential equations system is performed by using perspective coefficients method. Four systems that have structures different from Lorenz system and can reproduce time series of one variable of Lorenz system were found. In many areas of science, the problem of identifying a system of ordinary differential equations (ODE) from a time series of one observable variable is relevant. If the right-hand sides of an ODE system are polynomials, then solving such a problem only by numerical methods allows to obtain a model containing, in most cases, redundant terms and not reflecting the physics of the process. The preliminary choice of the structure of the system allows to improve the precision of the reconstruction. Since this study considers only the single time series of the observable variable, and there are no additional requirements for candidate systems, we will look only for systems of ODE's that have the least number of terms in the equations. We will look for candidate systems among particular cases of the system with quadratic polynomial right-hand sides. To solve this problem, we will use a combination of analytical and numerical methods proposed in [12, 11]. We call the original system (OS) the ODE system, which precisely describes the dynamics of the process under study. We also use another type of ODE system-standard system (SS), which has the polynomial or rational function only in one equation. The number of OS variables is equal to the number of SS variables. The observable variable of the SS coincides with the observable variable of the OS. The SS must correspond to the OS. Namely, all the SS coefficients can be analytically expressed in terms of the OS coefficients. In addition, there is a numerical method [12], which allows to determine the SS coefficients from a time series. To find only the simplest OS, one can use the perspective coefficients method [10], which means the following. Initially, the SS is reconstructed from a time series using a numerical method. Then, using analytical relations and the structure of the SS, we determine which OS coefficients are strictly zero and strictly non-zero and form the initial system (IS), which includes only strictly non-zero coefficients. After that, the IS is supplemented with OS coefficients until the corresponding SS coincides with the SS obtained by a numerical method. The result will be one or more OS’s. Using this approach, we have found 4 OS structures with 7 coefficients that differ from the Lorenz system [17], but are able to reproduce exactly the time series of X variable of the Lorenz system. Numerical values of the part of the coefficients and relations connecting the rest of the coefficients were found for each OS

Improved Estimation of Embedding Parameters of Nonlinear Time Series by Structural Learning of Neural Network with Fuzzy Regularizer

2007 ◽  
Vol 11 (6) ◽  
pp. 600-609
Author(s):  
Yusuke Manabe ◽  
◽  
Basabi Chakraborty
Keyword(s):  
Neural Network ◽  
Time Series ◽  
Regularization Parameter ◽  
Nonlinear Time Series ◽  
Computational Time ◽  
Structural Learning ◽  
Feed Forward Neural Network ◽  
One Step ◽  
Short Term Prediction ◽  
Selection Of

This work proposes an improved refinement scheme of estimation of optimal embedding parameters of a nonlinear time series by a feed-forward neural network trained by structural learning with a fuzzy regularizer (FR). The newly proposed fuzzy rules for tuning regularization parameter enables automatic selection of optimal model with lesser computational load than the basic refinement scheme with RNS proposed by authors earlier. From the simulation results, it has been found that the proposed scheme is very efficient in estimation of optimal embedding parameters in lesser computational time. The short term prediction results also show that the estimated embedding parameters produce better and stable one step prediction.

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