scholarly journals Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1194
Author(s):  
Jose-Cruz Nuñez-Perez ◽  
Vincent-Ademola Adeyemi ◽  
Yuma Sandoval-Ibarra ◽  
Francisco-Javier Perez-Pinal ◽  
Esteban Tlelo-Cuautle
Keyword(s):  
Evolutionary Algorithms ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Optimization Algorithms ◽  
Dynamical Behavior ◽  
Optimization Techniques ◽  
Maximum Lyapunov Exponent ◽  
Invasive Weed Optimization ◽  
Chen System ◽  
Invasive Weed

This paper presents the application of three optimization algorithms to increase the chaotic behavior of the fractional order chaotic Chen system. This is achieved by optimizing the maximum Lyapunov exponent (MLE). The applied optimization techniques are evolutionary algorithms (EAs), namely: differential evolution (DE), particle swarm optimization (PSO), and invasive weed optimization (IWO). In each algorithm, the optimization process is performed using 100 individuals and generations from 50 to 500, with a step of 50, which makes a total of ten independent runs. The results show that the optimized fractional order chaotic Chen systems have higher maximum Lyapunov exponents than the non-optimized system, with the DE giving the highest MLE. Additionally, the results indicate that the chaotic behavior of the fractional order Chen system is multifaceted with respect to the parameter and fractional order values. The dynamical behavior and complexity of the optimized systems are verified using properties, such as bifurcation, LE spectrum, equilibrium point, eigenvalue, and sample entropy. Moreover, the optimized systems are compared with a hyper-chaotic Chen system on the basis of their prediction times. The results show that the optimized systems have a shorter prediction time than the hyper-chaotic system. The optimized results are suitable for developing a secure communication system and a random number generator. Finally, the Halstead parameters measure the complexity of the three optimization algorithms that were implemented in MATLAB. The results reveal that the invasive weed optimization has the simplest implementation.

scholarly journals Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 165
Author(s):  
Zai-Yin He ◽  
Abderrahmane Abbes ◽  
Hadi Jahanshahi ◽  
Naif D. Alotaibi ◽  
Ye Wang
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Epidemic Model ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Maximum Lyapunov Exponent ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Reasonable Range ◽  
Fractional Orders

This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

Application of chaos-based chaotic invasive weed optimization techniques for environmental OPF problems in the power system

2014 ◽  
Vol 69 ◽  
pp. 271-284 ◽  
Author(s):  
Mojtaba Ghasemi ◽  
Sahand Ghavidel ◽  
Jamshid Aghaei ◽  
Mohsen Gitizadeh ◽  
Hasan Falah
Keyword(s):  
Power System ◽  
Optimization Techniques ◽  
Invasive Weed Optimization ◽  
Invasive Weed

scholarly journals On an Optimal Control Applied in MEMS Oscillator with Chaotic Behavior including Fractional Order

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Angelo Marcelo Tusset ◽  
Frederic Conrad Janzen ◽  
Rodrigo Tumolin Rocha ◽  
Jose Manoel Balthazar
Keyword(s):  
Fractional Order ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Dynamical Analysis ◽  
Linear Feedback ◽  
Nonlinear Stiffness ◽  
Mems Resonator ◽  
Optimal Linear ◽  
Mems Oscillator ◽  
Resonator System

The dynamical analysis and control of a nonlinear MEMS resonator system is considered. Phase diagram, power spectral density (FFT), bifurcation diagram, and the 0-1 test were applied to analyze the influence of the nonlinear stiffness term related to the dynamics of the system. In addition, the dynamical behavior of the system is considered in fractional order. Numerical results showed that the nonlinear stiffness parameter and the order of the fractional order were significant, indicating that the response can be either a chaotic or periodic behavior. In order to bring the system from a chaotic state to a periodic orbit, the optimal linear feedback control (OLFC) is considered. The robustness of the proposed control is tested by a sensitivity analysis to parametric uncertainties.

scholarly journals A Discrete Fractional-Order Prion Model Motivated by Parkinson’s Disease

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
M. F. Elettreby ◽  
E. Ahmed ◽  
A. S. Alqahtani
Keyword(s):  
Parkinson’S Disease ◽  
Parkinson's Disease ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Disease Model ◽  
Equation Model ◽  
Wide Range ◽  
The Stability ◽  
Uniqueness Of A Solution

A prion differential equation model motivated by Parkinson’s disease (PD) is studied. A fractional-order form of this model is proposed. After that, we discretized fractional-order Parkinson’s disease model. A sufficient condition for the existence and the uniqueness of a solution to the system is obtained. The stability of the fixed points of the system is achieved by using the Jury test. The impacts of varying the parameters of the system are examined. Under certain conditions, the system undergoes some kinds of bifurcations. We observe that the model loses its stability through double-period bifurcation to chaotic behavior as the growth rate increases. Also, the system stabilizes by increasing the memory parameter, and the contact rate between the two types of prions increases. The system shows rich dynamical behavior for a wide range of the values of the parameters.

scholarly journals Dynamical analysis and adaptive fuzzy control for the fractional-order financial risk chaotic system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sukono ◽  
Aceng Sambas ◽  
Shaobo He ◽  
Heng Liu ◽  
Sundarapandian Vaidyanathan ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Fuzzy Control ◽  
Fractional Order ◽  
Chaotic System ◽  
Financial Risk ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Adaptive Fuzzy Control ◽  
Dynamical Analysis ◽  
Adaptive Fuzzy

AbstractIn this paper, a fractional-order model of a financial risk dynamical system is proposed and the complex behavior of such a system is presented. The basic dynamical behavior of this financial risk dynamic system, such as chaotic attractor, Lyapunov exponents, and bifurcation analysis, is investigated. We find that numerical results display periodic behavior and chaotic behavior of the system. The results of theoretical models and numerical simulation are helpful for better understanding of other similar nonlinear financial risk dynamic systems. Furthermore, the adaptive fuzzy control for the fractional-order financial risk chaotic system is investigated on the fractional Lyapunov stability criterion. Finally, numerical simulation is given to confirm the effectiveness of the proposed method.

scholarly journals Fractional Dynamics in Calcium Oscillation Model

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yoothana Suansook ◽  
Kitti Paithoonwattanakij
Keyword(s):  
Fractional Order ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Cell Types ◽  
Cytosolic Calcium ◽  
Cardiac Cells ◽  
Calcium Oscillations ◽  
Calcium Oscillation ◽  
Fractional Dynamics ◽  
Oscillation Model

The calcium oscillations have many important roles to perform many specific functions ranging from fertilization to cell death. The oscillation mechanisms have been observed in many cell types including cardiac cells, oocytes, and hepatocytes. There are many mathematical models proposed to describe the oscillatory changes of cytosolic calcium concentration in cytosol. Many experiments were observed in various kinds of living cells. Most of the experimental data show simple periodic oscillations. In certain type of cell, there exists the complex periodic bursting behavior. In this paper, we have studied further the fractional chaotic behavior in calcium oscillations model based on experimental study of hepatocytes proposed by Kummer et al. Our aim is to explore fractional-order chaotic pattern in this oscillation model. Numerical calculation of bifurcation parameters is carried out using modified trapezoidal rule for fractional integral. Fractional-order phase space and time series at fractional order are present. Numerical results are characterizing the dynamical behavior at different fractional order. Chaotic behavior of the model can be analyzed from the bifurcation pattern.

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