scholarly journals Optimizing the Kaplan–Yorke Dimension of Chaotic Oscillators Applying DE and PSO

Technologies ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 38 ◽  
Author(s):  
Alejandro Silva-Juarez ◽  
Gustavo Rodriguez-Gomez ◽  
Luis Gerardo de la Fraga ◽  
Omar Guillen-Fernandez ◽  
Esteban Tlelo-Cuautle
Keyword(s):  
Mathematical Model ◽  
Time Series ◽  
Color Image ◽  
Equilibrium Points ◽  
Computing Time ◽  
Chaotic Oscillator ◽  
State Variables ◽  
Huge Number ◽  
Chaotic Oscillators ◽  
Swarm Optimization

When a new chaotic oscillator is introduced, it must accomplish characteristics like guaranteeing the existence of a positive Lyapunov exponent and a high Kaplan–Yorke dimension. In some cases, the coefficients of a mathematical model can be varied to increase the values of those characteristics but it is not a trivial task because a very huge number of combinations arise and the required computing time can be unreachable. In this manner, we introduced the optimization of the Kaplan–Yorke dimension of chaotic oscillators by applying metaheuristics, e.g., differential evolution (DE) and particle swarm optimization (PSO) algorithms. We showed the equilibrium points and eigenvalues of three chaotic oscillators that are simulated applying ODE45, and the Kaplan–Yorke dimension was evaluated by Wolf’s method. The chaotic time series of the state variables associated to the highest Kaplan–Yorke dimension provided by DE and PSO are used to encrypt a color image to demonstrate that they are useful in implementing a secure chaotic communication system. Finally, the very low correlation between the chaotic channel and the original color image confirmed the usefulness of optimizing Kaplan–Yorke dimension for cryptographic applications.

Chaotic Oscillator Based on Fractional Order Memcapacitor

2019 ◽  
Vol 28 (14) ◽  
pp. 1950239 ◽  
Author(s):  
Akif Akgul
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponents ◽  
Fractional Order ◽  
Nonlinear Oscillator ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chaotic Oscillator ◽  
State Equations ◽  
Chaotic Oscillators ◽  
Eigen Values

Many literatures have discussed fractional order memristor and memcapacitor-based chaotic oscillators but the entire oscillator model is considered to be of fractional order. My interest is to propose a nonlinear oscillator with considering only the memcapacitor element of fractional order. Hence, I propose a fractional order memcapacitor (FMC)-based novel chaotic oscillator. The complete mathematical model for the proposed oscillator is derived and presented in this paper. The dimensionless state equations are then analyzed by using the equilibrium points and their stability, Eigen values, Kaplan–Yorke dimensions and Lyapunov exponents. To understand the complete dynamical behavior, bifurcation graphs are obtained and presented. Finally, the proposed fractional memcapacitor oscillator is implemented by using the shelf components.

scholarly journals About Extracting Dynamic Information of Unknown Complex Systems by Neural Networks

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Eloy Irigoyen ◽  
Antonio Javier Barragán ◽  
Mikel Larrea ◽  
José Manuel Andújar
Keyword(s):  
Neural Networks ◽  
Mathematical Model ◽  
Nonlinear Systems ◽  
Dynamic Behaviour ◽  
Equilibrium Points ◽  
Multilayer Perceptrons ◽  
State Variables ◽  
Dynamic Information ◽  
Universal Approximators ◽  
Operating Points

This work presents a straightforward methodology based on neural networks (NN) which allows to obtain relevant dynamic information of unknown nonlinear systems. It provides an approach for cases in which the complex task of analyzing the dynamic behaviour of nonlinear systems makes it excessively challenging to obtain an accurate mathematical model. After reviewing the suitability of multilayer perceptrons (MLPs) as universal approximators to replace a mathematical model, the first part of this work presents a system representation using a model formulated with state variables which can be exported to a NN structure. Considering the linearization of the NN model in a mesh of operating points, the second part of this work presents the study of equilibrium states in such points by calculating the Jacobian matrix of the system through the NN model. The results analyzed in three case studies provide representative examples of the strengths of the proposed method. Conclusively, it is feasible to study the system behaviour based on MLPs, which enables the analysis of the local stability of the equilibrium points, as well as the system dynamics in its environment, therefore obtaining valuable information of the system dynamic behaviour.

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 Complex dynamical behaviors of a novel exponential hyper–chaotic system and its application in fast synchronization and color image encryption

2021 ◽  
Vol 104 (1) ◽  
pp. 003685042110033
Author(s):  
Javad Mostafaee ◽  
Saleh Mobayen ◽  
Behrouz Vaseghi ◽  
Mohammad Vahedi ◽  
Afef Fekih
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Sliding Mode ◽  
Color Image ◽  
Equilibrium Points ◽  
Lyapunov Stability Theory ◽  
System Stability ◽  
Color Image Encryption ◽  
Terminal Sliding Mode ◽  
Finite Time Stability

This paper proposes a novel exponential hyper–chaotic system with complex dynamic behaviors. It also analyzes the chaotic attractor, bifurcation diagram, equilibrium points, Poincare map, Kaplan–Yorke dimension, and Lyapunov exponent behaviors. A fast terminal sliding mode control scheme is then designed to ensure the fast synchronization and stability of the new exponential hyper–chaotic system. Stability analysis was performed using the Lyapunov stability theory. One of the main features of the proposed controller is the finite time stability of the terminal sliding surface designed with high–order power function of error and derivative of error. The approach was implemented for image cryptosystem. Color image encryption was carried out to confirm the performance of the new hyper–chaotic system. For image encryption, the DNA encryption-based RGB algorithm was used. Performance assessment of the proposed approach confirmed the ability of the proposed hyper–chaotic system to increase the security of image encryption.

scholarly journals Stochastic and deterministic mathematical model of cholera disease dynamics with direct transmission

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Getachew Teshome Tilahun ◽  
Woldegebriel Assefa Woldegerima ◽  
Aychew Wondifraw
Keyword(s):  
Mathematical Model ◽  
Stochastic Model ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Invariant Solution ◽  
Stochastic Approach ◽  
Disease Dynamics ◽  
Treatment Rate ◽  
Invariant Region ◽  
Contact Transmission

AbstractIn this paper we develop a stochastic mathematical model of cholera disease dynamics by considering direct contact transmission pathway. The model considers four compartments, namely susceptible humans, infectious humans, treated humans, and recovered humans. Firstly, we develop a deterministic mathematical model of cholera. Since the deterministic model does not consider the randomness process or environmental factors, we converted it to a stochastic model. Then, for both types of models, the qualitative behaviors, such as the invariant region, the existence of a positive invariant solution, the two equilibrium points (disease-free and endemic equilibrium), and their stabilities (local as well as global stability) of the model are studied. Moreover, the basic reproduction numbers are obtained for both models and compared. From the comparison, we obtained that the basic reproduction number of the stochastic model is much smaller than that of the deterministic one, which means that the stochastic approach is more realistic. Finally, we performed sensitivity analysis and numerical simulations. The numerical simulation results show that reducing contact rate, improving treatment rate, and environmental sanitation are the most crucial activities to eradicate cholera disease from the community.

Cellular Neural Network Based on Hybrid Linear Matrix Inequality and Particle Swarm Optimization for Noise Removal of Color Image

2011 ◽  
Vol 422 ◽  
pp. 771-774
Author(s):  
Te Jen Su ◽  
Jui Chuan Cheng ◽  
Yu Jen Lin
Keyword(s):  
Neural Network ◽  
Particle Swarm Optimization ◽  
Linear Matrix Inequality ◽  
Color Image ◽  
Particle Swarm ◽  
Cellular Neural Network ◽  
Noise Removal ◽  
Linear Matrix ◽  
Matrix Inequality ◽  
Swarm Optimization

This paper presents a color image noise removal technique that employs a cellular neural network (CNN) based on hybrid linear matrix inequality (LMI) and particle swarm optimization (PSO). For designing templates of CNN, the Lyapunov stability theorem is applied to derive the criterion for the uniqueness and global asymptotic stability of the CNN’s equilibrium point. The template design is characterized as a standard LMI problem, and the parameters of templates are optimized by PSO. The input templates are obtained by employing the CNN’s property of saturation nonlinearity, which can be used to eliminate noise from arbitrary corrupted images. The demonstrated examples are compared favorably with other available methods, which illustrate the better performance of the proposed LMI-PSO-CNN methodology.

scholarly journals Identification of monthly municipal water demand system based on autoregressive integrated moving average model tuned by particle swarm optimization

2017 ◽  
Vol 19 (2) ◽  
pp. 261-281 ◽  
Author(s):  
Sahbi Boubaker
Keyword(s):  
Time Series ◽  
Particle Swarm Optimization ◽  
Water Demand ◽  
Moving Average ◽  
Arima Model ◽  
Particle Swarm ◽  
Demand System ◽  
Swarm Optimization ◽  
Municipal Water ◽  
Average Absolute Relative Error

In this paper, a modeling-identification approach for the monthly municipal water demand system in Hail region, Saudi Arabia, is developed. This approach is based on an auto-regressive integrated moving average (ARIMA) model tuned by the particle swarm optimization (PSO). The ARIMA (p, d, q) modeling requires estimation of the integer orders p and q of the AR and MA parts; and the real coefficients of the model. More than being simple, easy to implement and effective, the PSO-ARIMA model does not require data pre-processing (original time-series normalization for artificial neural network (ANN) or data stationarization for traditional stochastic time-series (STS)). Moreover, its performance indicators such as the mean absolute percentage error (MAPE), coefficient of determination (R2), root mean squared error (RMSE) and average absolute relative error (AARE) are compared with those of ANN and STS. The obtained results show that the PSO-ARIMA outperforms the ANN and STS approaches since it can optimize simultaneously integer and real parameters and provides better accuracy in terms of MAPE (5.2832%), R2 (0.9375), RMSE (2.2111 × 105m3) and AARE (5.2911%). The PSO-ARIMA model has been implemented using 69 records (for both training and testing). The results can help local water decision makers to better manage the current water resources and to plan extensions in response to the increasing need.

scholarly journals MATHEMATICAL MODEL OF THREE SPECIES FOOD CHAIN INTERACTION WITH MIXED FUNCTIONAL RESPONSE

2012 ◽  
Vol 09 ◽  
pp. 334-340 ◽  
Author(s):  
MADA SANJAYA WS ◽  
ISMAIL BIN MOHD ◽  
MUSTAFA MAMAT ◽  
ZABIDIN SALLEH
Keyword(s):  
Mathematical Model ◽  
Functional Response ◽  
Food Chain ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Bifurcation Points ◽  
Dynamical Behaviors ◽  
Practical Life ◽  
Parameter Values ◽  
Chain Interaction

In this paper, we study mathematical model of ecology with a tritrophic food chain composed of a classical Lotka-Volterra functional response for prey and predator, and a Holling type-III functional response for predator and super predator. There are two equilibrium points of the system. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. Furthermore the dynamical behaviors of this model are investigated. Models for biologically reasonable parameter values, exhibits stable, unstable periodic and limit cycles. The dynamical behavior is found to be very sensitive to parameter values as well as the parameters of the practical life. Computer simulations are carried out to explain the analytical findings.

scholarly journals A Control Based Mathematical Model for the Evaluation of Intervention Lines in COVID-19 Epidemic Spread: The Italian Case Study

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 890
Author(s):  
Paolo Di Giamberardino ◽  
Rita Caldarella ◽  
Daniela Iacoviello
Keyword(s):  
Mathematical Model ◽  
Equilibrium Points ◽  
Italian Case ◽  
Control Actions ◽  
Containment Measures ◽  
Asymptomatic Subjects ◽  
The Impact

This paper addresses the problem of describing the spread of COVID-19 by a mathematical model introducing all the possible control actions as prevention (informative campaign, use of masks, social distancing, vaccination) and medication. The model adopted is similar to SEIQR, with the infected patients split into groups of asymptomatic subjects and isolated ones. This distinction is particularly important in the current pandemic, due to the fundamental the role of asymptomatic subjects in the virus diffusion. The influence of the control actions is considered in analysing the model, from the calculus of the equilibrium points to the determination of the reproduction number. This choice is motivated by the fact that the available organised data have been collected since from the end of February 2020, and almost simultaneously containment measures, increasing in typology and effectiveness, have been applied. The characteristics of COVID-19, not fully understood yet, suggest an asymmetric diffusion among countries and among categories of subjects. Referring to the Italian situation, the containment measures, as applied by the population, have been identified, showing their relation with the government's decisions; this allows the study of possible scenarios, comparing the impact of different possible choices.

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